Aerospace Science and Technology 29 (2013) 175–184

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Aerospace Science and Technology

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Analysis of a structural-aerodynamic fully-coupled formulationfor aeroelastic response of rotorcraft

G. Bernardini, J. Serafini, M. Molica Colella, M. Gennaretti ∗

Mechanical & Industrial Engineering Department, University Roma Tre, Via della Vasca Navale 79, 00146 Rome, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 May 2012Received in revised form 20 February 2013Accepted 8 March 2013Available online 21 March 2013

Keywords:Computational aeroelasticityRotary wing aeroelastic responseBoundary-element-method aerodynamics

This paper deals with a computational aeroelastic tool aimed at the analysis of the response of rotarywings in arbitrary steady motion. It has been developed by coupling a nonlinear beam model forblades structural dynamics with a potential-flow boundary integral equation solver for the predictionof unsteady aerodynamic loads around three-dimensional, lifting bodies. The Galerkin method is usedfor the spatial integration of the resulting differential aeroelastic system, whereas the periodic bladeresponse is determined by a harmonic balance approach. This aeroelastic model yields a unified approachfor aeroelastic response and blade pressure prediction, that may conveniently be used for aeroacousticpurposes. It is able to examine configurations where blade–vortex interactions occur. Numerical resultsshow the capability of the aeroelastic tool to evaluate blade response and vibratory hub loads for ahelicopter main rotor in level and descent flight conditions, and examine the efficiency and robustnessof the different numerical solution algorithms that may be applied in the developed aeroelastic solver.Comparisons among aeroelastic predictions based on different aerodynamic models are also presented.

© 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

The aim of this work is to present some features and recentadvances of a computational tool for the analysis of the aeroelasticresponse of rotary wings. Rotorcrafts are affected by strong cou-plings between elastic deformations and aerodynamic loads, whichsignificantly contribute to the vibrations transmitted to the fuse-lage, as well as, to the emitted noise. A great variety of flightconditions may be experienced by rotary-wing aircraft, each yield-ing specific aerodynamic environment and corresponding elasticresponse. For instance, during descent flights of helicopters, strongBVI (Blade–Vortex Interaction) characterizes both vibrations andnoise levels and accurate simulation tools are required for reliablepredictions of the corresponding aeroelastic responses.

Commonly, in computational tools aimed at the aeroelasticanalysis of rotors the aerodynamic loads are determined throughsectional theories that take into account the effects of the in-flow induced by the wake [13]. Wake inflow may be evaluated inseveral ways, ranging from the application of semi-empirical static-inflow analytical models [5], up to free-wake computational solu-tions [14], including the use of dynamic inflow models [20]. At thesame time, the sectional load models may take into considerationthe complex dynamic stall effects through semi-empirical models[16,15]. In the last decade, the inclusion of free-wake inflow within

* Corresponding author. Tel.: +39 (06) 57333260; fax: +39 (06) 5593732.E-mail address: m.gennaretti@uniroma3.it (M. Gennaretti).

1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.ast.2013.03.002

tools for the analysis of rotor aeroelastic response has become alargely applied solution procedure (see, for instance, Refs. [17,4],and Ref. [19] where a dual vortex free-wake code is applied forrotor aeroelastic and aeroacoustics purposes).

The computational aeroelastic tool for rotary wings analyzedhere is the result of the research work presented and validatedby the authors during the last years [6,7,2]. It uses a three-dimensional, unsteady, Boundary Element Method (BEM) aerody-namic solver based on an integral formulation for potential flowssuited for the prediction of strong BVI effects [7]. This solver canbe applied both for direct evaluation of the aerodynamic loads andfor calculation of the wake inflow to be used within a sectionalaerodynamic load model. For this work, the blade structural modelis based on a nonlinear, bending-torsional, theory for nonuniform,homogeneous, isotropic, rotating, straight beams undergoing mod-erate deflections [10]. In the past decades, several authors havefaced the problem of developing more advanced structural dynam-ics models, that are valid for applications concerning composite-material blades, curved-axis blades and blades undergoing largedeflections (see, for instance, Refs. [1,3,25,12,22,21]). However, herethe emphasis is on the structure/aerodynamics coupling and nu-merical strategies to solve the resulting differential formulation,and hence the introduction of a more complex blade structuralmodel is beyond the scope of the paper and would appear asan unnecessary complication of the problem. The Galerkin ap-proach is applied for the space integration of the resulting set ofintegro-differential equations, with the natural modes of vibrationof a cantilever, nonrotating beam used as test and trial functions

176 G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184

(rigid-body modes are included when hinged blades are exam-ined). The aerodynamic and the structural formulations are fullycoupled by forcing the structural equations with the aerodynamicgeneralized loads computed by the BEM solver, and taking intoaccount the elastic displacements in evaluating the aerodynamicboundary conditions (non-fully-coupled solutions that will be con-sidered in the following are those obtained using the wake inflowfrom the BEM solver for the evaluation of sectional loads from 2Dtheory). Considering arbitrary steady flight conditions, the aeroe-lastic responses given in terms of blade deflections and vibratoryloads (in rotating and nonrotating frames) are obtained through aharmonic balance procedure that iteratively updates periodic de-formations and aerodynamic loads until convergence. When theaerodynamic solver is applied for the direct evaluation of loads(fully-coupled mode solution), among the outputs of the solutionprocess there is the blade pressure distribution which, in turn, maybe applied in aeroacoustic formulations for the evaluation of theemitted noise field.

In the following, first, aerodynamic and structural models willbe briefly outlined, then the techniques applied for the numericalintegration of the fully-coupled aeroelastic system will be illus-trated. Finally, the results of a detailed numerical investigation onthe efficiency and robustness of the different computational algo-rithms that may be applied in the fully-coupled mode aeroelasticsolution will be presented, along with those concerning the anal-ysis of the effects of aerodynamic modeling on aeroelastic predic-tions.

2. Rotor aeroelastic modeling

A beam-like model is applied to describe the structural dynam-ics of rotor blades. This is based on the nonlinear bending-torsionformulation [10], that is valid for straight, slender, homogeneous,isotropic, nonuniform, twisted blades, undergoing moderate dis-placements.

The radial displacement is eliminated from the set of equationsby solving it in terms of local tension, and thus the resulting struc-tural operator consists of a set of coupled nonlinear differentialequations governing the bending of the elastic axis and the bladetorsion [11]. If present, the effects of blade pre-cone angle, hingeoffset, torque offset and mass offset are included in the model.

Combining this structural dynamics model with a model de-scribing the distributed aerodynamic loads yields the aeroelasticformulation that may be used for rotor response and stabilityanalyses. In this work, the computational tool considered is an en-hanced version of that presented in Ref. [6]. Indeed, it is based ona potential-flow, three-dimensional, free-wake, BEM aerodynamicsolver suited for the prediction of BVI effects [7], and therefore isapplicable to a wider range of rotor steady flight configurations,including descent patterns. The aerodynamic BEM tool is appliedeither for the direct evaluation of blade pressure distributions andcorresponding generalized forces or for the identification of thewake inflow on rotor blades, to be coupled with aerodynamic sec-tional theories to determine the generalized forces.

2.1. 3D BEM rotor aerodynamics model

In several common flight conditions the aerodynamics of heli-copter rotors is significantly affected by the interactions (includingimpacts) between blades and wake vortices. This phenomenon iswidely known as blade–vortex interaction (BVI). It produces im-pulsive pressures on the blade surface, thus strongly contributingto the vibratory loads transmitted to the airframe and generatingextremely annoying acoustic effects. Therefore, a simulation toolapplied for the optimal design of new generation helicopters has

Fig. 1. Far-wake and near-wake decomposition.

to be able to predict BVI occurrence and corresponding aeroelas-tic and aeroacoustic effects. In view of this, a boundary integralformulation for potential flows suited for the prediction of strongaerodynamic body–wake interaction effects has been presented inRef. [7].

This formulation, for the flow velocity field v described asv = ∇ϕ , assumes the potential field, ϕ , to be given by the su-perposition of an incident field, ϕI , with a scattered field, ϕS(i.e., ϕ = ϕI + ϕS ). The scattered potential is generated by sourcesand doublets over the surfaces of the blades, S B , and by doubletsover the wake portion that is very close to the trailing edge fromwhich emanated (near wake, S N

W ). The incident potential is dueto doublets distributed over the complementary wake region thatcompose the far wake, S F

W (see Fig. 1). The division of the wakesurface is such that the far wake is the only wake portion that maycome in contact with blades. The scattered potential is discontin-uous across S N

W , whereas the incident potential is discontinuousacross S F

W .As demonstrated in Ref. [7], the scattered potential is repre-

sented by

ϕS(x, t) =∫S B

[G(χ − χI ) − ϕS

∂G

∂n

]dS(y)

−∫

S NW

�ϕS∂G

∂ndS(y) (1)

where G = −1/4πr is the unit-source solution of the three-dimensional Laplace equation, with r = ‖y − x‖, while �ϕS is thepotential jump across the wake surface, that is known from thepast history of the potential discontinuity at the trailing edge ofthe blades (through the Kutta–Joukowski condition), observing thatit remains constant following a material point of the wake [8]. Inaddition, χ = vB · n, with vB representing the blade velocity and nits outward unit normal, whereas χI = uI · n, with uI denoting thevelocity induced by the far wake.

In turn, considering the far wake discretized into M panels,assuming the potential jump constant over each panel, and recall-ing the equivalence between surface distribution of doublets andvortices, the incident velocity field is evaluated through the Biot–Savart law applied to the vortices having the shape of the far-wakepanel contours as [7]

uI (x, t) ≈ −M∑

k=1

�ϕS(yTE

Wk, t − ϑk

) ∫Ck

∇xG × dy (2)

where yTEWk

is the trailing edge position where the wake materialpoint currently in yW emanated at time t − ϑk , Ck denotes the

k

G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184 177

contour line of the k-th far wake panel, and ∇x denotes gradientoperator with respect to the variable x.

The equations above show that the incident potential affects thescattered potential through the induced-velocity term, χI , whilethe scattered potential affects the incident potential by its trailing-edge discontinuity that is convected along the wake and yields theintensity of the vortices of the far wake.

Eq. (2) is applied to evaluate both the term χI in Eq. (1) andthe velocity field from which the wake shape evolution is deter-mined in a free-wake analysis. Note that, for an accurate predictionof BVI phenomena, the accurate evaluation of the wake distortedshape is essential in that a crucial role is played by the relativepositions between body and wake. In order to assure a regular dis-tribution of the induced velocity within the vortex core, and thus astable and regular solution even in blade–vortex impact conditions,a Rankine finite-thickness vortex model is introduced in Eq. (2) [7].

Akin to the far-wake contribution, Eq. (1) is solved numericallyby boundary elements, i.e., by dividing S B and S N

W into quadrilat-eral panels, assuming ϕS , χ , χI and �ϕS to be piecewise constant(zeroth-order boundary element method), and imposing that theequation be satisfied at the center of each body element (colloca-tion method) [7].

Once the potential field is known, the Bernoulli theorem yieldsthe pressure distribution and hence blade loads may be evaluatedthrough integration (see Appendix A for a detailed description ofthe algorithm applied for the evaluation of the blade pressure,which is a crucial issue of the presented formulation, not ad-dressed in Ref. [7]).

2.2. Aerodynamic loads from airfoil theories

In the aeroelastic solver examined in this work, solution proce-dures based on aerodynamic generalized forces evaluated throughspanwise integration of loads from airfoil unsteady aerodynam-ics theories are available. Three-dimensional effects due to wakevortices are taken into account in terms of either analytical wakeinflow models, or wake inflow determined by the BEM solver (thusyielding the non-fully-coupled solution).

For a thin, straight airfoil moving in an incompressible flow,following the Greenberg theory [9] (which is the extension of theTheodorsen theory [23] to pulsating airstream) it is possible to de-termine the aerodynamic force acting on it by combining the non-circulatory lift, Lnc , orthogonal to the chord, with the circulatorylift, Lc , orthogonal to the relative wind (see, for instance, Ref. [11]).Specifically, for ρ denoting the air density and b denoting the air-foil semi-chord length, the non-circulatory lift is expressed as

Lnc(t) = πρb2 vn1/2 (3)

where vn1/2 denotes the time derivative of the normal component

of the relative wind evaluated at the airfoil mid point (positiveupwards). The circulatory lift, that in practical applications is themost relevant load contribution, is given by

Lc(t) = 2πρbVF−1[C(k)vn3/4

](4)

where V denotes the relative wind velocity, F denotes Fouriertransformation and vn

3/4 = F(vn3/4), with vn

3/4 representing thenormal component of the relative wind evaluated at the airfoil3/4-chord point (it is worth noting that, wake inflow is takeninto account as a perturbation to vn

3/4 and to V , as well). In ad-dition, C(k) is the lift deficiency (complex) function defined byTheodorsen [23] in terms of the reduced frequency, k = ωb/V ,where ω is the variable in the Fourier domain and V is the mean(or reference) value of V which is responsible for the shed vortic-ity convection (see Ref. [24] for a detailed analysis of the effects

of nonuniform convection of the shed vorticity along the wake).For a harmonic vn

3/4 input, C(k) defines gain and phase shift of theLc/V harmonic response. The set of the aerodynamic loads actingon the airfoil includes also the pitching moment about the 1/4-chord point (positive clockwise for the relative wind directed fromleft to right), which is given by [9]

M1/4(t) = −b

2Lnc − πρ

b3

2

(V ωa + b

4ωa

)(5)

where ωa is the angular velocity of the airfoil. Once vn1/2, vn

3/4,ωa

and V are expressed in terms of the blade degrees of freedom,Eqs. (3), (4) and (5), combined with a model taking into accountdrag effects, may be applied to develop rotary wing aeroelastic for-mulations.

3. Harmonic balance for aeroelastic response evaluation

The equations governing the blade structural dynamics cou-pled with the unsteady aerodynamic load model yield the aeroe-lastic integro-partial differential equations to be integrated. Here,the space integration is performed through the Galerkin approach,starting from the description of the elastic axis deformation as alinear combination of shape functions that satisfy homogeneousboundary conditions. The resulting aeroelastic system consists of aset of nonlinear ordinary differential equations of the type

M(t)q + C(t)q + K(t)q = fnlstr(t,q) + faer(t,q) (6)

where q denotes the vector of the Lagrangian coordinates, whereasM,C, and K are time-periodic, mass, damping, and stiffness struc-tural matrices representing the linear structural terms (note thatthese matrices are time-variant because of the cyclic pitch con-tribution). Nonlinear structural contributions are collected in theforcing vector fnl

str(t,q), whereas vector faer(t,q) collects the gener-alized aerodynamic forces. As already mentioned, the aerodynamicloads may be obtained either by integration along blades surfacesof the pressure distribution given by the BEM solver outlined above(fully-coupled solution), or by spanwise integration of the expres-sions in Eqs. (3)–(5) derived from airfoil theories, combined withthe wake inflow predicted by the BEM solver (non-fully-coupledsolution).

Since, here, the aim is the prediction of the aeroelastic peri-odic response during steady forward flight, the aeroelastic systemin Eq. (6) is solved by using the harmonic balance approach. It isa methodology suitable for the analysis of the asymptotic solution(as time goes to infinity) of differential equations forced by peri-odic terms, as in the present case. The harmonic balance solutionconsists of: (i) express LHS and RHS of the aeroelastic system inEq. (6) in terms of their Fourier series; (ii) equate the resulting har-monic coefficients; (iii) solve the corresponding algebraic system interms of the unknown Fourier coefficients of the Lagrangian coor-dinates of the problem. Specifically, expressing q and f = fnl

str + faer

in terms of the following Fourier series

q(t) = q0 +N∑

n=1

[qc

n cos(Ωnt) + qsn sin(Ωnt)

]

f(t) = f0 +N∑

n=1

[f cn cos(Ωnt) + f s

n sin(Ωnt)]

where qcn,qs

n, f cn and f s

n denote cosine and sine components of then-th harmonic of q and f, whereas Ωn = nΩ , with Ω representingthe fundamental frequency (i.e., the rotor angular velocity), andthen combining with Eq. (6) yields the following harmonic repre-sentation of the aeroelastic system

178 G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184

[M + C + K]q = f (7)

where

qT = {qT

0 qc1

T qs1

T qc2

T qs2

T · · ·}and

fT = {fT0 f c

1T f s

1T f c

2T f s

2T · · ·}

Matrices M, C and K in Eq. (7) come out from Eq. (6) by combiningthe harmonics of the q, q and q terms with the harmonics of thematrices M,C, and K (see Ref. [6] for details). In particular, if M,C,and K were constant matrices, the matrices in Eq. (7) would beof block-diagonal type and each harmonic of q would depend onlyon the same harmonic of f (the q-harmonics equations would beuncoupled). Instead, in the problem under examination the struc-tural matrices are periodic and hence, once expressed in terms ofthe Fourier series and combined with the harmonics of q, q and q,yield fully-populated M, C and K matrices, thus coupling the alge-braic equations for the unknown harmonics in q.

Because of the presence of the nonlinear structural terms and ofthe aerodynamic contributions in f, Eq. (7) has to be solved usingan iterative procedure. To this aim, the Newton–Raphson method isapplied. Specifically, the harmonic aeroelastic solution is obtainedfrom convergence of the following iterative procedure (with n in-dicating the iteration step number)

qn = [M + C + K − Jaer

n−1

]−1[fn−1 − Jaer

n−1qn−1]

(8)

where Jaern−1 is the aerodynamic Jacobian evaluated at q = qn−1,

while fn−1 denotes the sum of the nonlinear structural terms andof the aerodynamic loads evaluated at q = qn−1 (i.e., correspondingto the Lagrangian coordinates given by the previous iteration step).Under this assumption, the aerodynamic portion of the forcingterm is equivalent to f aer

0 + fnlaer , with f aer

0 denoting the aerody-namic load portion that is not affected by the structural deforma-tions. For the sake of computational time saving, the approxima-tion of maintaining constant the aerodynamic Jacobian might beapplied (indeed, this would avoid the evaluation of the inversionof the global Jacobian matrix at each step of the iterative process).

3.1. Evaluation of aerodynamic Jacobian

The aerodynamic Jacobian is evaluated numerically. Observingthat, at a given iteration step, it relates the perturbation harmonicsof each structural Lagrangian variable to the harmonics of the cor-responding perturbations of the generalized aerodynamic forces, itis identified through a sequence of time-marching harmonic re-sponses of the BEM solver. Specifically, considering the solutioncomputed at the previous iteration step as the reference set of La-grangian variables Fourier components, the harmonic responses to(small amplitude) perturbations to each of the 2N +1 Fourier com-ponents of each of the Lagrangian variables are evaluated. Then,the Fourier components of the resulting multi-harmonic, perturba-tion forces divided by the corresponding input perturbation yieldone column of the aerodynamic Jacobian matrix to be used inEq. (8), as given by

Jaern, j = f aer(qn + �q j) − f aer(qn)

�q j(9)

where Jaern, j denotes the j-th column of the aerodynamic Jacobian at

the n-iteration step, whereas qn + �q j denotes the solution at then-th step with the j-th element perturbed.

Table 1Rotor control settings.

μ = 0.15 μ = 0.25 μ = 0.30

Collective pitch, θ0 (deg) 6.8 4.2 8.4Longitudinal pitch, θs (deg) −2.4 −2.5 −5.4Lateral pitch, θc (deg) 1.8 1.5 1.8Shaft angle, αs (deg) −1.2 1.5 −4.0

4. Numerical results

Here, results obtained by the harmonic-balance fully-coupledaeroelastic formulation outlined above are presented and dis-cussed, with the aim of assessing efficiency and robustness of dif-ferent algorithms that might be used in the solution process. Inaddition, the effects of the applied aerodynamic model on aeroe-lastic responses (blade tip deflections and vibratory hub loads) isexamined.

For these analyses, the four-bladed rotor introduced in Ref. [26]is considered. In the past, it has been examined by the authors tovalidate the aeroelastic formulation applied here [6]. It is a simplerotor model equivalent to the Bo-105 main rotor, with blades hav-ing radius R = 4.93 m, constant chord c = 0.395 m, NACA 0012section profile, linear pre-twist of −8◦ and mass and stiffnessproperties uniformly distributed such that the resulting dynamiccharacteristics are of soft-in-plane blades with first lag frequencyωl = 0.74/rev, first flap frequency ω f = 1.15/rev, and first torsionfrequency ωt = 3.55/rev. The flight conditions investigated are twolevel flights at advance ratios μ = 0.15 and μ = 0.3, and one de-scent flight at μ = 0.25. The rotor angular speed is Ω = 40 rad/s,while shaft angle and pitch control settings are those given inTable 1. Note that, for μ = 0.15 and μ = 0.3, these have been ob-tained from Ref. [26], while for μ = 0.25 they have been identifiedthrough a momentum trim based on quasi-steady aerodynamics.Indeed, in order to highlight the differences among the aeroelasticmodels examined, here the aeroelastic predictions are carried outkeeping constant the control settings, without concern about theflight trim state.

Both for level flight and descent flight analyses, BEM predic-tions have been obtained through the free-wake solution mode.Note that, in the following, ‘quasi-steady’ aeroelastic solutions referto predictions given by quasi-steady aerodynamics combined withDrees linear wake inflow [5]. Further, all the results presented inthe following have been obtained using six shape functions for de-scribing each structural unknown (i.e., blade lag, flap and torsiondeformations), and five harmonics to represent each correspondingLagrangian variable.

4.1. Level flight, advance ratio μ = 0.30

First, in Fig. 2 convergence analyses of the iterative aeroelas-tic solution algorithm are presented in terms of the error decay,for different values of the relaxation factor used in the Newton–Raphson method (here, the relaxation factor is defined as a coef-ficient that multiplies the vector fn−1 in Eq. (8)). Denoting withNB the number of panels on each rotor blade, NR the number ofpanels along the radial direction, NW the number of a blade wakepanels along the azimuth direction, N S the number of wake spiralsconsidered and NT the number of time steps per blade revolu-tion, these results have been obtained with NB = 384, NR = 12,NW = 180, N S = 3, NT = 136. Note that, the relative quadraticerror is evaluated in terms of the difference between the vibra-tory hub loads computed at two consecutive iteration steps. Thischoice is motivated by the fact that, from the practical point ofview, the interesting outcomes of the aeroelastic response anal-ysis are vibratory hub loads and blade deflections, and that theconvergence of the former implies the convergence of the latter.

G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184 179

Fig. 2. Effect on convergence history of relaxation factor. μ = 0.3.

Fig. 3. Effect on convergence history of aerodynamic Jacobian modeling. μ = 0.3.

For computational time saving, the aerodynamic Jacobian is keptconstant during the iterative process (thus yielding the so-calledfixed-slope iteration procedure), equal to that evaluated numeri-cally about the undeformed configuration (i.e., at q = 0), adoptinga prescribed-wake solution mode (i.e., considering the wake sur-face as the surface swept by the trailing edge of the blades duringtheir motion). The solution histories show that this approach iseffective and leads to convergence of the solution for any valueof the relaxation factor, although a relaxation factor close to 1seems to give a slightly faster convergence rate. It is worth notingthat, a similar convergence history has been achieved by evalu-ating the aerodynamic Jacobian about the deformed rotor config-uration given by the simpler quasi-steady aeroelastic solver. Thismeans that the aerodynamic Jacobian is not subject to signifi-cant variations within the domain of variation of the states q, anddemonstrates that the fixed-slope iteration procedure based on theundeformed-configuration Jacobian is appropriate for the problem.

Next, Fig. 3 shows the effect on the convergence rate of theapplication of aerodynamic Jacobians from different aerodynamicmodels (the same mesh applied for Fig. 2 has been considered).Specifically, the results obtained by the Jacobian evaluated throughthe prescribed-wake, BEM solver are compared with those ob-tained by the quasi-steady aerodynamics Jacobian, as well as withthose determined with null Jacobian. As expected, no convergenceis achieved when the aerodynamic Jacobian is not included in thesolution algorithm, but it is interesting to observe that a very sim-ilar negative result is obtained when the Jacobian from the quasi-

Fig. 4. Effect on convergence history of BEM mesh used for evaluation of Jacobian.μ = 0.3.

Table 2Computational costs for Jacobian evaluation.

NB NR NW N S NT Run time (min)

Coarse mesh 48 6 30 1 50 4Finer mesh 96 8 60 2 90 15Solution mesh 384 12 180 3 136 200

steady aerodynamic model is applied. This shows that the use ofthe same aerodynamic solver for solution and Jacobian evaluationis mandatory.

Then, in order to examine the way for reducing computationalcosts, the impact on the convergence rate of the BEM mesh usedfor the evaluation of the Jacobian matrix has been investigated.Three different meshes have been considered, as presented in Ta-ble 2 (where ‘Solution mesh’ indicates the mesh applied for theevaluation of the aerodynamic solution that yields the loads ap-pearing in the aeroelastic equations). The results of this analysisare illustrated in Fig. 4 and demonstrate that the application of avery high fidelity (and computationally expensive) Jacobian doesnot yield a convergence rate significantly faster than those cor-responding to the use of less accurate (and less computationallyexpensive) Jacobians. Indeed, even the convergence rate of the so-lution based on the Jacobian from a coarse mesh evaluation isacceptable (Table 2 gives the computational costs correspondingto the specific meshes examined).

Finally, considering the ‘Solution mesh’, the aeroelastic responseobtained by using the proposed fully-coupled aeroelastic model iscompared with those obtained through the application of the non-fully-coupled model and the quasi-steady formulation. The resultsare given in terms of blade tip deflections (Fig. 5) and vibratoryhub loads (Fig. 6). While the tip deflections time histories fromsolvers based on the sectional aerodynamic theory show a simi-lar behavior, different blade tip displacements are predicted by thefully-coupled model, especially in terms of torsion. This is likely tooccur because of the different accuracy in capturing the wake ef-fects that play a significant role on this class of problems (similardiscrepancies are observed between experimental data and numer-ical predictions given by solvers based on non-fully-coupled-typemodels in Ref. [18]). Quite large discrepancies appear among the4/rev vibratory hub loads evaluated through the three approaches.These are high-frequency effects that are strongly sensitive tothe accuracy in capturing the wake influence. In-plane forces aresimilarly predicted by non-fully-coupled and quasi-steady models,while similar predictions of the out-of-plane force are given bynon-fully- and fully-coupled solvers, with the quasi-steady model

180 G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184

Fig. 5. Blade tip deflections. μ = 0.3.

Fig. 6. 4/rev vibratory hub loads. μ = 0.3.

yielding a significant underestimation. Moment components tendto be overestimated by the non-fully-coupled solver, with respectto the results from the other two approaches.

4.2. Level flight, advance ratio μ = 0.15

Next, the above analyses have been repeated considering a ro-tor operating at lower advance ratio, for which the wake remainscloser to the rotor disk and the influence of the wake-inducedvelocity is stronger. The same meshes applied for the results pre-sented in Section 4.1 have been adopted.

Akin to the investigation presented for μ = 0.3, first resultsconcerning the effects of the relaxation parameter on the solu-tion convergence history are presented in Fig. 7. It shows that,also in this case, the error decay is sped up when higher val-ues of the relaxation factor (close to 1) are used. In the overall,the convergence error threshold tends to be reached more slowlywith respect to the case at μ = 0.3. Although not shown here, theanalysis on the effect of aerodynamic Jacobian determined fromdifferent aerodynamics confirms that solution convergence is notachieved neither when the Jacobian is not included in the iterativeprocess, nor when it is obtained by a quasi-steady aerodynamicssolver.

Fig. 8 presents the sensitivity of the aeroelastic convergencerate on the accuracy of the prescribed-wake, BEM solver used todetermine the Jacobian. For the present advance ratio the fastestsolution is clearly obtained using the finest ‘Solution mesh’. How-

Fig. 7. Effect on convergence history of relaxation factor. μ = 0.15.

Fig. 8. Effect on convergence history of BEM mesh used for evaluation of Jacobian.μ = 0.15.

Fig. 9. Blade tip deflections. μ = 0.15.

ever, the trade-off between convergence speed and computationalcosts would suggest the use of the coarse mesh.

The aeroelastic response results, given in terms of blade tipdeflections and vibratory hub loads, are presented in Figs. 9and 10. The observation of the tip deflections reveals that lag pre-dicted by the three aeroelastic approaches considered are in quitegood agreement, flap predictions from fully-coupled and non-fully-

G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184 181

Fig. 10. 4/rev vibratory hub loads. μ = 0.15.

coupled solutions are in fair agreement whereas, akin to whatobserved at μ = 0.3, greater amplitude of the higher harmoniccontent of the torsion deformation is present in the fully-coupledsimulation. Concerning the 4/rev vibratory hub loads, Fig. 10presents a very strong underestimation of quasi-steady predictionswith respect to those from fully-coupled and non-fully-coupledsolutions. This is in agreement with a similar comparison shownin Ref. [26]. Fully-coupled and non-fully-coupled solutions are inquite good agreement, except for the prediction of the out-of-planeforce, for which the non-fully-coupled approach yields a value thatis about twice that given by the fully-coupled approach.

4.3. Descent flight, advance ratio μ = 0.25

Finally, a fast descent flight condition has been analyzed, wherethe use of a free-wake aerodynamic solver is mandatory becauseof the presence of blade–vortex interaction (BVI) phenomena.

In this flight condition, the onset of impulsive loads due toBVI and the wake deformation might deeply affect the conver-gence rate of the numerical scheme. However, due to the highcomputational costs that would be required by the evaluation ofthe free-wake aerodynamic Jacobian, the possibility of applying theJacobian given by the BEM prescribed-wake analysis has been ex-plored also in this case. In particular, fully-coupled aeroelastic so-lutions have been obtained by the prescribed-wake Jacobian (eval-uated through the coarser mesh previously considered) applied incombination with free-wake aerodynamic predictions determinedthrough four different discretization meshes. Fig. 11 presents theconvergence analysis of the relative quadratic error for the casesexamined, with the four curves identified by NB , NR , NW , N S , NT .Although derived from a prescribed-wake model, the selected Jaco-bian guarantees convergence of the solution, with faster rates cor-responding to the finer discretization meshes of the aerodynamicproblem. These results show that, regardless the aerodynamic so-lution mesh and even in BVI conditions, coarse-mesh, prescribed-wake Jacobians are suited for free-wake aeroelastic analyses, thusallowing a good trade-off between accuracy and computationalcosts. The use of more accurate Jacobians given by free-wake, morerefined mesh analyses is not convenient in terms of computationaltime.

Considering predictions obtained by the discretization meshthat is the finest among those of Fig. 11, Figs. 12 and 13 presentthe aeroelastic response in terms of blade tip deflections and hubvibratory loads, respectively, obtained by the fully-coupled, thenon-fully-coupled and the quasi-steady solvers. Similarly to thepredictions concerning the flight at μ = 0.15, lag and flap re-sponses are in fair agreement, whereas the torsion is significantly

Fig. 11. Effect on convergence history of the BEM mesh used for aeroelastic solutionat μ = 0.25. Curves identified by NB , NR , NW , N S , NT .

Fig. 12. Blade tip deflections. μ = 0.25.

Fig. 13. 4/rev vibratory hub loads. μ = 0.25.

overestimated by the fully-coupled method with respect to whatobtained by the other two approaches. Fig. 13 presents significantdiscrepancies among the vibratory loads predicted by the threesolutions considered here. In the overall, those given by the quasi-steady solver are much lower that those from fully-coupled andnon-fully-coupled solvers (except for Fx) which, in turn, show afair agreement only for the in-plane forces. Such differences among

182 G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184

the predictions examined are expected in this case, in particular,because of the higher frequency interactions occurring betweenbody and wake that make the accurate simulation of wake effectsof crucial importance.

5. Conclusions

A fully-coupled aeroelastic tool for the analysis of the responseof rotors in steady flight conditions has been examined. Specifi-cally, performance and robustness of the presented iterative nu-merical solution algorithm for nonlinear systems that combinesharmonic blade dynamics with BEM aerodynamics harmonic loadshave been analyzed for a four-bladed rotor in level and descentflight conditions. Particular attention has been paid on the effectsof relaxation factor and fidelity of aerodynamic Jacobian evalua-tion on the convergence rate of the fixed-slope solution methodapplied. The use of Jacobians numerically evaluated through theprescribed-wake BEM solver guarantees convergence of the iter-ative aeroelastic solution process, with speed depending on therelaxation factor introduced and on the accuracy of the BEMsolver for Jacobian identification. No convergence is achieved atall, if the aerodynamic Jacobian is evaluated by a simpler aero-dynamic solver (a quasi-steady model has been considered) oris even neglected. It has been shown that, for Jacobian identifi-cation the use of a mesh that is significantly coarser than thatconsidered for the aerodynamic solution yields a quite fast con-vergence rate of the iterative procedure at acceptable computa-tional costs. The algorithm has been demonstrated to be suc-cessful also in descent flight analysis, when BVI occurs with theonset of blade impulsive loads. Also in this case, in which it ismandatory to apply a free-wake solution mode in order to cap-ture the aerodynamic phenomena, the investigation has shownthat the convergence of the process is guaranteed also by Jaco-bians identified by prescribed-wake analysis and that, therefore,the computational efficiency of the aeroelastic solver is main-tained in that the extremely costly evaluation of free-wake Ja-cobians may be avoided. Finally, in order to examine the sensi-tivity of the aeroelastic response to the unsteady aerodynamicsmodel applied, blade tip displacements and vibratory hub loadsobtained by the proposed fully-coupled formulation have beencompared with those from non-fully-coupled and quasi-steady so-lutions. Among displacements, the blade torsion is that more af-fected by the aerodynamic model considered, with predictionsfrom the fully-coupled formulation having much greater ampli-tudes of the higher harmonics. Significant discrepancies have beenobserved also among the vibratory loads evaluated through thethree approaches considered. Loads from the quasi-steady formu-lation are generally much smaller than those from the other twosolvers which, in turn, present an overall better agreement with,anyway, strong discrepancies at specific loads for each of the flightsexamined. The predictions of the vibratory loads, which are sig-nificantly affected by the wake inflow, has been shown to bestrongly dependent on the accuracy of the aerodynamic model ap-plied.

Acknowledgements

This work has been partially supported by the European UnionProject ARISTOTEL (Grant agreement No. 266073), financed underthe Seventh Framework Programme (FP7/2007-2013).

Appendix A. Pressure evaluation from Bernoulli theorem

From the decomposition of the potential field into a scatteredcomponent, ϕS , and an incident component, ϕI (see Section 2.1),

Fig. 14. Sketch of wake coordinates.

in a frame of reference fixed with the undisturbed air (air frame)the Bernoulli theorem for incompressible flows reads

∂ϕS

∂t

∣∣∣∣A

+ ∂ϕI

∂t

∣∣∣∣A

+ ‖∇ϕS + uI‖2

2+ p

ρ= p0

ρ(A.1)

where ∂/∂t|A denotes time derivative observed in the air frame,whereas p0 is the pressure of the undisturbed medium. Observingthat, at each time step, the integral formulation yields the scat-tered potential over the body surface, and recalling that, for ∂/∂t|Bdenoting time derivative observed in a frame of reference fixedwith the body,

∂(·)∂t

∣∣∣∣A

= ∂(·)∂t

∣∣∣∣B

− vB · ∇(·)

Eq. (A.1) is conveniently recast in the form

∂ϕS

∂t

∣∣∣∣B

+ ∂ϕI

∂t

∣∣∣∣B

− vB · (∇ϕS + uI ) + ‖∇ϕS + uI‖2

2+ p

ρ

= p0

ρ(A.2)

Therefore, in the determination of the pressure over the body sur-face, the critical issue remains the evaluation of the time derivativeof the incident potential, which is the only term in Eq. (A.2) notdirectly derivable from the aerodynamic formulation presented inSection 2.1.

In principle, the incident potential (and the corresponding timederivative) could be obtained from the doublet distribution overthe far wake, S F

W , as [7]

ϕI (x, t) = −∫

S FW

�ϕS∂G

∂ndS(y) (A.3)

However, the above expression cannot be applied when the wakecomes in contact with the body or passes very close to it (BVIoccurrence), in that numerically unstable wake contributions ariseyielding unrealistic distributions of the incident potential and cor-responding time derivative [7].

In order to avoid the outcome of this problem in Eq. (A.2), ananalytic expression of the incident-potential time derivative thatallows a regular and stable numerical solution is derived in the fol-lowing. To this aim, a system of curvilinear material coordinates,(ξ1, ξ2), is introduced on the wake surface, such that ξ1 and ξ2

coincide, respectively, with time instant and position on a curvilin-ear abscissa defined along the trailing edge at which the materialpoint emanated from the blade (see Fig. 14). Then, for t − τ de-noting the age of the wake points that lie on the boundary linebetween near and far wakes, and xP = xP (t) denoting the positionvector of an arbitrary point P ∈ S B , Eq. (A.3) yields

∂ϕI

∂t

∣∣∣∣B(xP , t) = − ∂

∂t

t−τ∫ R∫�ϕS

∂G

∂nJ dξ1 dξ2

0 rc

G. Bernardini et al. / Aerospace Science and Technology 29 (2013) 175–184 183

= −t−τ∫0

R∫rc

∂

∂t

(�ϕS

∂G

∂nJ

)dξ1 dξ2

−R∫

rc

�ϕS∂G

∂nJ

∣∣∣∣ξ1=t−τ

dξ2 (A.4)

where rc denotes the blade root cut-off, R denotes the bladeradius, �ϕS = �ϕS(ξ

1, ξ2), G = G[xP (t),y(ξ1, ξ2, t)] and J =J (ξ1, ξ2, t) is the Jacobian of the mapping from (ξ1, ξ2) to y ∈ SW .Next, note that from the principles of conservation of mass andlinear momentum across a surface of discontinuity (like the poten-tial wake) it is possible to deduce that the potential discontinuityremains constant following a material point of the wake [8], andhence

∂�ϕS

∂t= 0 (A.5)

Further, the time derivative applied to the doublet distributionyields

∂

∂t

(∂G

∂n

)= ∇x

(∂G

∂n

)· ∂xP

∂t+ ∇y

(∂G

∂n

)· ∂y

∂t(A.6)

which, observing that ∂xP /∂t = vB at P ∈ S B , that ∂y/∂t = vW isthe velocity of the wake point at (ξ1, ξ2), and that

∇x

(∂G

∂n

)= −∇y

(∂G

∂n

)

becomes

∂

∂t

(∂G

∂n

)= (vB − vW ) · ∇x

(∂G

∂n

)(A.7)

Finally, combining Eq. (A.4) with Eqs. (A.5) and (A.7), the followingconvenient expression for the evaluation of the time derivative ofthe incident potential appearing in the Bernoulli theorem

∂ϕI

∂t

∣∣∣∣B(xP , t) = −

∫

S FW

�ϕS(vB − vW ) · ∇x

(∂G

∂n

)dS(y)

−∫

S FW

�ϕS∂G

∂n

∂(ln J )

∂tdS(y)

−R∫

rc

�ϕS∂G

∂nJ

∣∣∣∣ξ1=t−τ

dξ2 (A.8)

where vB and ∂G/∂n are intended as evaluated at the bodypoint xP .

It is possible to observe that the first term in Eq. (A.8) is re-lated to the velocity induced by the vorticity distributed along thefar wake. Indeed, introducing the zeroth-order BEM approach withM panels to discretize the far wake surface and recalling the equiv-alence between the velocity field induced by surface distributionsof doublets and vortices, it may be approximated through the fol-lowing form [7] (see also Section 2.1)

M∑k=1

�ϕS(yT E

Wk, t − ϑk

)(vB − vWk ) ·

∫

S FWk

∇x

(∂G

∂n

)dS(y)

=M∑

k=1

�ϕS(yT E

Wk, t − ϑk

)(vB − vWk ) ·

∫Ck

∇xG × dy

= −M∑

(vB − vWk ) · uIk (A.9)

k=1

where Ck denotes the contour line of the k-th far wake panel,S F

Wk, that induces the velocity field uIk at the body point xP . As

for Eq. (2) of Section 2.1, the introduction of a Rankine finite-thickness vortex model in Eq. (A.9) assures a regular distributionof the induced velocity within the vortex core, and hence a stableand regular contribution even in blade–vortex impact conditions.

The second contribution to the incident potential time deriva-tive arises if the wake is subject to deformation, as it occurs whenfree-wake numerical aerodynamic solutions are carried out. Onceapplied the BEM approach for the numerical solution, it would giveraise to numerical instabilities when wake panels come too closeto or impinge the body at xP (as it may occur in BVI conditions).However, in this case such contribution is not taken into accountin that becomes negligible with respect to the one from Eq. (A.9)(typically the rate of change of the wake shape is quite slow).

Finally, the third contribution in Eq. (A.8) is a line integral eval-uated along the boundary that separates the near wake from thefar wake. It takes into account the rate of change of the incidentpotential due to the flux of wake points through this boundary(note that it would vanish if the line was a wake material line, i.e.,τ increased as t; nevertheless, especially for BVI applications, it isconvenient to consider a small, constant portion of near wake andhence a small, constant value of τ ).

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