TOWARD A COMPUTATIONAL FRAMEWORK FOR ROTORCRAFTMULTI-PHYSICS ANALYSIS: ADDING COMPUTATIONAL

AERODYNAMICS TO MULTIBODY ROTOR MODELS

Giuseppe Quaranta†, Giampiero Bindolino, Pierangelo Masarati, and Paolo Mantegazza

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano, Italy

Abstract

This paper presents a computational framework for the integrated simulation of complex multi-physicsproblems. It results from interfacing existing structural dynamics, multidisciplinary software and differentcomputational fluid dynamics codes by means of a newly developed, broadly general communicationscheme that accounts both for solution synchronization and field interpolation. A first application to thefluid structure interaction problem based on a free wake code is presented. More complex applicationswill be presented in the near future.

Introduction

Rotorcraft integrated dynamics analysis is a verydifficult task, which showed only partially successfulresults in current practice. However, the need for reli-able tools that couple the structural dynamics and theaerodynamics of rotorcraft is strong, and satisfactoryresults cannot be obtained with existing software. Itis worth recalling that a general reduction of vibratoryloads transmitted to the cabin and of the generatednoise are perceived as major goals to be achieved bythe rotary wing industry. The Blade Vortex Inter-action (BVI) problem is dominated by the “missingdistance” parameter, which is strongly influenced bytip vortex trajectory, blade elastic deformation and in-duced velocity distribution (see Yu [1]). As stated inthe papers of Bousman [2], in the last fifteen yearsrotor loads prediction capability improved, but, espe-cially for vibratory loads, the comparison of numericalmethods with flight tests is still not satisfactory. Theload prediction problem has been correctly character-ized by Dr. W. Johnson [2] who said, almost twentyyears ago, in 1985: “for a good prediction of loads it isnecessary to do everything right, all of the time. Withcurrent technology it is possible to do some of thethings right, some of the time”. This statement callsfor more and more integrated analysis tools. Both

Submitted to the 30th European Rotorcraft Forum Mar-seilles, France, September 14–16 2004

†Corresponding author. Address: via La Masa 34, 20156Milano, Italy. Tel: +39 02 2399 8387. Fax: +39 02 2399 8334.email: quaranta@aero.polimi.it

theory and computational tools are improving dramat-ically, not to mention hardware capabilities. It appearsthat time has come to try putting everything togetherand see how far we are from complete deformable ro-torcraft aero-servoelastic analysis. This is required toachieve the major goals of rotorcraft dynamics predic-tion, e.g. those related to control systems analysis anddesign, flight control system synthesis, vibration con-trol, load reduction and flutter suppression. Amongthe works that are going in this direction, those byPomin and Wagner [3], van der Ven and Boelens [4],Yang et al. [5] deserve a mention.

The structural dynamics problem is posed on soundfoundations, and it has been developed to a high de-gree of accuracy. Today, the multibody approach tothe simulation of the dynamics of rotor blades, rotorsand entire rotorcraft is gaining broad acceptance bothin the academic and in the industrial field [6, 7]. Atthe same time, CFD for rotors is gaining momentumand it is becoming a practical tool for helicopter ro-tor aerodynamics analysis and design. A new idealtool, required to tackle this challenge, needs to com-bine all the expertise developed in each field to reacha satisfactory global solution. As a consequence, itappears reasonable to prospect the possibility and thefeasibility of effective and efficient coupling of existing,state-of-the-art software. This paper presents a viablesolution for coupling existing software in an integratedsolution scheme, without particular requirements onthe functionality of the codes that are being coupled.The presented approach allows the user to choose in a

18.1

Free wake

Local CFD domain

Multibody deformable blade

Figure 1: Domain decomposition on a physical basis.

wide range of possible coupling schemes, going fromwhat is known as “tight coupling” of the entire mul-tidisciplinary problem, up to “loose coupling” wherethere is no feedback interaction between the differentfields [8]. The tight procedure is nonetheless imple-mented using separate pieces of software. This al-lows to achieve the best trade-off between accuracyand efficiency, depending on the requirements of theproblem under investigation, while dealing always withthe same set of one’s preferred tools. Currently, onlythe time marching solution of initial value problems isaddressed.

Approach

Until recently, the problem of modeling the aerody-namic forces in aeromechanics comprehensive codesfor rotors has been mainly tackled by means of theclassical blade section element theory, coupled to aninflow model, especially within the multibody ap-proach [9, 7]. However, it is well known that thereare flight conditions where these simplified models failat giving correct prediction, especially when vibratoryloads and noise problems are addressed [2]. Typicalcomprehensive rotorcraft codes, which heavily rely onsimplified theories for the aerodynamics, enabled thedevelopment of a rational engineering process, yet al-lowing significant uncertainties, costly testing and un-pleasant surprises, as reported by Caradonna [10].

The main object of this work is the extension ofthe modeling capability for both, the computation ofthe aerodynamic loads, and the description of the flowfield. In fact, if the synthesis of automatic control sys-tems for the noise reduction needs to be addressed, itis necessary to model also the flow field and more pre-cisely the position of vortical elements. The idea pur-

sued here is to create a toolbox of modeling paradigmsfor aerodynamic loads prediction, where each one iscapable to address by itself a specific aspect of thefluid flow, but at the same time to interact with otherparadigms to achieve a higher degree of precision forthe whole model. The goal is to develop a sort of ”do-main decomposition on a physical basis” of the flowfield (Figure 1), so that, depending on the specificweight of each phenomena inside the flow field, it ispossible to choose the model which represents an op-timal trade-off between accuracy and computationalcosts.

To realize this concurrent simulation environment,an approach based on the creation of a new process,called “broker”, is proposed. The broker is used tosynchronize the execution of all the other participatingprocesses. The communications are based on the pop-ular Message Passing Interface (MPI) software com-munication package, available in open source form onvirtually any platform.

As soon as all the other processes join the pool,the broker schedules their correct execution and takescare of exchanging the required data, according to“blocks” defined for each communication pattern.This allows highly heterogeneous software to partic-ipate to the execution pool, ensuring that only thedata specific to each participant is visible to them andthus minimizing the communication effort. Moreover,whenever required, the broker can be instructed onhow to interpolate the data in field form that it is re-quired to pass. This frees the participating processesfrom the need to know whom they are communicatingwith, and thus to implement a common intercommu-nication format.

The structural dynamics and multi-field partici-pants to the execution pool need not be unique: there

18.2

may be multiple concurring structural models, e.g.multiple rotorcraft in a helicopter interference simu-lation, or the same structural model can present mul-tiple communication patterns, e.g. one for each rotorin a multi-rotor model.

At the same time, the CFD participants need not beunique: there may be multiple concurring aerodynam-ics domains, e.g. one Euler model for each blade, andone free-wake model that provides far-field boundaryconditions for all (Figure 1).

The envisaged communication patterns aresketched in Figure 2 and described in Table I; onlythe “driver” structural pattern is currently imple-mented. The “driven” pattern is intended to couplethe current multibody analysis simulation, basedon an implicit integration scheme, to an explicitintegration process that models fast dynamics suchas crash landing or impact problems.

Figure 2: Communication Patterns

The aim of this work is to allow the integrated solu-tion of multi-field initial value problems, with the finalobjective of simulating transients in rotorcraft analy-sis. In fact, integrated accurate rotorcraft analysis areinherently transient, because of the concurring effectof different sources of time-dependent excitation op-erating at different time periods, e.g. main and tailrotor in conventional helicopters, rotor transmissionsand more. As a consequence, nearly steady solutionscan only be achieved as asymptotic conditions of sta-ble transient solutions, if no simplifications are to beaccepted.

Structural Dynamics Model

The structural dynamics software that is used inthe current simulations is the general purpose multi-body/multidisciplinary code MBDyn, which is cur-rently available as “freedom software” from the website http://www.aero.polimi.it/~mbdyn/. It has

been successfully applied to the simulation of rotor-craft dynamics; particular focus has been dedicated totiltrotor aircraft systems, including the 1/5 scale V-22wind tunnel model known as WRATS, and its recentadaptation to soft-inplane investigations [7, 11], theERICA tiltrotor concept developed by Agusta S.p.A.,now AgustaWestland, and the ADYN wind tunnelmodel of the European Tiltrotor Concept, which isscheduled for experimental whirl-flutter investigationin late 2005/early 2006. Figure 3 shows a multibodymodel of the WRATS, where the structural dynamics,the hydraulic control system and basic aerodynamicforces were modeled.

The MBDyn code is a complete multibody analysistool, particularly suited for rotorcraft dynamics model-ing because it features geometrically exact beam mod-els, essential for rotor blade modeling, including com-posite beam analysis. In fact, the multibody formal-ism provides arbitrary topology modeling capabilities,included multi-path kinematic chains, and a correctrepresentation of the nonlinear behavior of mecha-nisms. Within this framework, one can progress fromrigid body models, for preliminary performance def-inition, up to fully detailed aeroservoelastic analysismodels, through intermediate steps encompassing de-tailed mechanism definition, the introduction of de-formable elements, servohydraulic actuators, and con-trol systems components. As a consequence, this ap-proach naturally leads to the generation of “modularmodels” that concur in creating a complete systemwith the required details for each subpart.

Figure 3: MBDyn model of a tiltrotor system.

Free-Wake Model

The free wake method, which is widely used in therotorcraft community, derives its appeal from the pos-sibility to solve the whole flow field by representingonly the regions where vortical singularities arise.

18.3

Table I: Communication patterns

Connector Input Output NotesStructural Dynamics loads configuration simulation driverStructural Dynamics configuration loads driven simulation

Fluid Dynamics configuration loads driven simulation/dev/null loads none “rigid” simulation driver; visualization/dev/null configuration none no loads; visualization

To compute an unsteady, time dependent, incom-pressible flow, Laplace’s equation, ∇2φ = 0, must besolved in the multiply connected region outside thevortical layers Ω. This equation by itself does not de-pend on time, but it must be solved together with theappropriate boundary conditions, which prescribe theno-penetration condition on material surfaces

(∇φ(x, t)− vB(x, t)) · n(x, t) = 0, (1)

where vB is the local body velocity and n is localsurface normal. Equation ∇2φ = 0 can be solved inthe potential variable using Green’s classical theoremfor harmonic fields (see Morino [12])

φ(x) =

1E(x)

∮δΩ

(φ

∂

∂n

(1

4πr

)− 1

4πr

∂φ

∂n

)dA(x′)

(2)

where r = |x − x′|, and E(x) is an operator whichis equal to 1,1/2 or 0, respectively if x is positionedinside, on the boundary or outside Ω.

Vortical layers, such as the wake, are surfaces ofdiscontinuity for the kinetic potential. However, theyare also stream surfaces, meaning that they are alwaystangential to the local velocity vector, so

(v−vV L)·nV L = ∆(∇φ)·nV L = ∆(

∂φ

∂n

)V L

= 0.

(3)

where the velocity of the vortical layer is vV L = (vl +vu)/2, with the subscripts l and u to indicate the twosides of the wake. This last condition means that forvortical layers the second term of Eq. (2) is equal tozero.

Lifting bodies can be usually represented by theirmean surface, ignoring the effects of the thicknessin the approximate model. This choice is reasonablefor thin surfaces, such as typical wings. In this case,both the wake and the body are vortical layers, soany term associated to the discontinuity of the normalderivative of the potential may be dropped.

The velocity field can be reconstructed by consid-

ering the gradient of the potential

v(x, t) = ∇xφ =

14π

∫SB

∆φ∇x

(∇x′

(1r

)· n

)dA(x′)+

14π

∫SW

∆φ∇x

(∇x′

(1r

)· n

)dA(x′)

(4)

In the virtual singularities theory, the terms associ-ated to a discontinuity in the potential field, calleddoublets, are mainly responsible for the rise of liftingforces, while the terms associated with the normalderivative of the potential, called sources, are respon-sible of the effect of the thickness. Because of theequivalence between doublets and vorticity distribu-tions, the same result of Eq. (4) can be obtained usingthe Biot-Savart law

v(x) = − 14π

∫(x− x′)×ω(x′)

|x− x′|3dx′. (5)

A constant distribution of doublets over a flat surfaceis completely equivalent to a closed loop of straightvortices lying on its perimeter, with constant circula-tion ΓL = ∆φ (see Konstadinopoulos et al. [13]). Us-ing closed loop vortices guarantees the satisfaction ofthe divergence-free condition for the vorticity. The un-known circulations of the bound vortex lattice whichrepresents the lifting surface may be obtained by ap-plying the boundary condition Eq. (1). As a conse-quence, the loop circulation on the lifting surface pan-els is related to the velocity of the fluid at the controlpoints on the boundary. The vorticity generated onthe blade is simply convected in the flow field start-ing from the originating points, which are representedby the trailing edge of the lifting surface. What isneeded is a condition to obtain the initial value of thevorticity. The Kutta condition states that the pres-sure jump between the top and the bottom surface atthe trailing edge must be equal to zero. This implieswhat is known as the Joukowsky hypothesis of smoothflow at the trailing edge, which simply states that thevorticity convected into the flow field is equal to thejump of potential at the trailing edge (see Morino andBernardini [14]).

18.4

The capability of these types of codes to correctlyrepresent the vorticity in the flow field without requir-ing extremely fine grids, makes them good candidatesfor the representation of the wake dynamics even whencompressibility effects are present in the flow field. Infact, even when large Mach numbers (close to 1) areconsidered, the asymptotic behavior of vortical wakesis essentially incompressible, as confirmed by experi-mental results, see Gai et al. [15]. This happens be-cause the wake dynamics is governed by the Machnumber related to the relative velocity, which is of-ten very low since the wake elements are stationary ormove slowly compared to the sound celerity.

The free-wake analysis software that is used in thecurrent simulations is called NUVOLA [16]. It fea-tures a nonlinear unsteady incompressible vortex lat-tice method, capable to predict the instantaneous con-figuration of the wake and the distribution of the aero-dynamic loads on flexible rotor blades during arbitraryunsteady flight conditions.

The aerodynamic loads computation is based onthe use of Bernoulli’s theorem

∂φ(x)∂t

+v(x)2

2+

p(x)ρ

=v2∞2

+p∞ρ

. (6)

By means of simple mathematical elaborations thepressure jump across each surface panel appears to bemade of two contributions: one related to the velocitypotential jump in the local reference frame, and thusequal to the time derivative of the loop circulation; theother, which can be referred to as the stationary part,related to the vorticity value can be computed usingthe classical Kutta-Joukowsky formula which statesthat a vortical flow generates a load F = ρv×Γ. Theseparate computation of these contributions gives theopportunity to improve the aerodynamic loads evalu-ation since the Kutta-Joukowsky formula can be ap-plied to a more refined set of points. A better rep-resentation of the leading edge suction can be ob-tained using the Kutta-Joucowsky theorem togetherwith the unsteady contribution. This effect, typicalof relatively thick airfoils, is important since it mayconsiderably affect the induced drag.

Fluid-Structure Interface

To solve a coupled fluid structure problem, the ca-pability to compute the solution of the structural andof the aerodynamic model is not sufficient. It is alsonecessary to exchange information between the twomodels: the modification of the boundary conditionsmust be transferred from the deformable structure tothe aerodynamic boundary, and conversely, the loadsresulting from the aerodynamic field must be applied

to the discretized structural model. The adoptionof two different codes for the simulation of the twophysical domains gives the possibility to (or raises theproblem of) dealing with “non-compatible” discretiza-tions, e.g. beam elements for the structure and flatlifting surfaces for the aerodynamics (see Figure 4);the exchange layer must connect entities which maybe extremely different from a topological point ofview.

The field interpolation is a key task, which is ac-complished by the broker. An accurate method, ca-pable of interfacing any possible structural or aerody-namic discretization scheme, is required. The basictechnique is inferred from surface reconstruction the-ory, which deals with reconstructing regular surfacesfrom scattered data. As opposed to what has beenpresented by Farhat et al. [17], a complete indepen-dence from the formulation of the structural system isachieved with the proposed technique, while retainingthe required conservation properties.

The fluid boundary movement is indicated by yf ;ys denotes the structural boundary movement; p thepressure acting on it, σs and σf respectively the struc-ture stress tensor and the fluid viscous stress tensor,and n the normal vector to the interface boundary Γ.The compatibility conditions are

σs · n = −p n + σf · nys = yf

ys = yf

on Γ. (7)

The first of Eqs. (7) expresses the dynamic equilibriumbetween the stresses on the structure and those on thefluid side. The other two are kinematic compatibilityconditions. The last one, for inviscid flows, is replacedby a condition only on the normal component of thevelocity

∂ys

∂n=

∂yf

∂n.

These conditions are valid regardless of the formula-tion used for the description of each field; the aerody-namics can be approximated either by the Euler equa-tions or any other CFD model, or even by a simplepanel method.

Eqs. (7) are valid for a continuous system. How-ever, the two fields are solved resorting to a discreteapproximation based on finite approximations of thefunctional space under which the solutions fall. Asa consequence, the correct expression of Eqs. (7) forthe discretized problem must be derived, under theconstraint of retaining the conservation properties ofthe original problem, see van Brummelen et al. [18].The mass conservation is trivially satisfied since there

18.5

Figure 4: Possible fluid-structure interfaces.

is no mass exchange. The change of energy in thefluid-structure system equals the energy supplied (orabsorbed) by an external force. Consequently, atany time t, the energy released/absorbed from thestructure, except for the contribution dissipated bythe structural damping, must balance the energy ab-sorbed/released by the fluid, or, in other words, thework equivalence must hold. Finally, the change ofmomentum of the fluid-structure system may be in-duced only by external forces.

The conservation properties are retained when thetwo computational domains of the fluid and structurehave matching discrete interfaces and compatible ap-proximation spaces, such as identical spaces for thestresses at either side of the interface. However, inrealistic applications, the fluid and structural meshesare not compatible along the interface, either becausediscretized with major geometric discrepancies, or be-cause each problem has different resolution require-ments; typically, the fluid grid, at the interface, is finerthan the structural mesh. In the following, Γf and Γs

will respectively indicate the discrete fluid/structurenon-matching representation of Γ.

In order to satisfy the conservation between the twomodels, the correct strategy would be to enforce thecoupling conditions only in a weak sense, through theuse of simple variational principles such as the VirtualWorks one. Let δyf and δys be two admissible virtualdisplacements for each field. Admissible means thatthe trace of these two fields on Γ must be equal.

Regardless of the interpolation method chosen toenforce the compatibility of the displacements, therelationship between the admissible virtual displace-ments can be written as

(δyf )i =js∑

j=1

hij (δys)j , (8)

where (δyf )i are the discrete values of δyf at thefluid nodes of the grid on the wet surface, and hij

are the coefficients of the displacements interpolationmatrix H. The resulting virtual displacement of theboundary surface will be

δyf =if∑

i=1

Ni

js∑j=1

hij (δys)j . (9)

Shape functions Ni belong to the approximation spaceof the aerodynamic field discretization, and if is thenumber of nodes which belong to the interface surfaceΓf . The virtual work of the aerodynamic load is equalto

δWf =∫

Γf

(−p n + σf · n) · δyf dA,

=∫

Γf

(−p n + σf · n) ·if∑

i=1

Ni

js∑j=1

hij (δys)j dA.

(10)

18.6

On the other side, the virtual work of forces and mo-ments acting on Γs is equal to

δWs =js∑

j=1

fj · (δys)j . (11)

Imposing the equality of the virtual works and callingFi the quantity given by

Fi =∫

Γf

(−p n + σf · n)Ni dA, (12)

the nodal loads applied to the structure are

fj =if∑

i=1

Fi hij . (13)

Eq. (13) states that, after computing the nodal loadsfor the aerodynamic boundary grid points using thecorrect approximation space, Fi, the loads on thestructural nodes, fj , can be obtained by simply multi-plying Fi for the transpose of the interpolation matrixH computed to connect the two grid displacements.The conservation problem is now shifted on the defi-nition of the appropriate interpolation matrix H.

To build a conservative interpolation matrix whichenforces the compatibility, Eq. (7), a weak/variationalformulation can be used as well. The problem can beexpressed in weighted least square form

Minimize

∫Γ

φ (Tr (δyf )|Γ − Tr (δys)|Γ)2 dA (14)

where Tr (·) is the trace operator. Additional proper-ties like smoothness of the resulting field after interpo-lation, computational efficiency, and some control onthe interpolation error, can be sought for. A solutionwhich possesses all these qualities can be obtainedusing the Moving Least Square (MLS) technique; re-cently, this type of approximation attracted the re-searchers’ attention also for meshless methods for thenumerical solution of PDEs, see Belytschko et al. [19].After calling f(x) the function which represents theTr (() δys), the first step is to build a local approx-imation of f as a sum of monomial basis functionspi(x)

f =m∑

i=1

pi(x)ai(x) ≡ pT (x) a(x), (15)

where m in the number of basis functions, andai(x) are their coefficients, which are obtained by aweighted least square fit for the approximation

minx

J(x) =

minx

∫Ω

φ(x− x)(f − f(x)

)2

dΩ(x)(16)

under the linear constraint

f(x) =m∑

i=1

pi(x)ai(x). (17)

This equation is completely equivalent to Eq. (14)which expresses the interface problem (all the de-tails may be found in [20]). The problem is localizedby choosing weight functions that are smooth non-negative Radial Basis Function (RBF) with compactsupport [21]. The adoption of different RBF func-tions, together with the definition of the local supportdimension, allows to achieve the required regularity ofthe interpolated function.

The resulting interface matrix satisfies all the re-quirements. It is effective, because the major compu-tational complexity, which lies in the geometric part ofthe computation, can be accomplished in an efficientmanner; it can produce surfaces with any prescribedsmoothness; it allows a high control on the behaviorfor each local region. The interface method is com-pletely independent from the implementation detailsof the codes that are interfaced. There is no need toknow what shape functions are used for each field; asa consequence, the implementation of the structuralelements or the code used to solve the aerodynamicfield can be changed without affecting the interfacecode.

The last problem that requires special attentionis related to the treatment of rotations, since weare dealing with structures that can experience largechanges in orientation. Rotations need to be ac-counted for, e.g., when a beam structural model isinterfaced to an aerodynamic surface boundary. Ro-tations are complex tensorial entities that can be sub-jected to singularity problems when parameterized. Toavoid the need to build a specialized class of interpo-lation matrices for these entities, since the updatedposition of an aerodynamic surface node is obtainedby combining the effect of linear displacements and ro-tations, a simple procedure has been developed. Theidea is to add three rigid arms to each structural beamnode, directed along the node local reference axes,creating a sort of “fish-bone” structure. The effect ofthe rotation is accounted for by the displacements ofthe added dummy nodes at the end of each arm, andthe matrix H is computed using displacements only.

As an example (more can be found in [20]) the be-havior of the interface is analyzed for a simple straight,unswept helicopter blade with a NACA 0012 airfoiland an aspect ratio of 10. This choice is intendedto show the interfacing capabilities of the proposedscheme when the underlying structure is representedby means of beams. The structural model is madeof five beam elements, while the aerodynamic sur-

18.7

Table II: Helicopter blade linear twist: twist angle atstation 9 m.

Scheme Angle % ErrorAnalytic 18 0Linear C0 10 pts. 17.9886 0.06Linear C0 20 pts. 17.2094 4.40Linear C4 20 pts. 17.7743 1.25Quadratic C4 20 pts. 17.9880 0.06Quadratic C4 30 pts. 17.9905 0.05

face is represented by a structured grid of 200 × 21nodes. The fish-bone structure is obtained by addingfour rigid arms at each node, a couple for each direc-tion perpendicular to the beam axis. The two armsin each direction have opposite sign to obtain a finallayout which respects the symmetry. As a result, therotation of each node is correctly transmitted to theaerodynamic wet surface. For the first test case, alinear twist has been applied along the blade, from 0deg at the root to 20 deg at the tip. Table II comparesthe analytical twist angle of the airfoil nodes on thedeformed blade at 90% of the blade span, with thecorresponding results obtainable with different inter-facing schemes.

Coupled Integration

Up to this point we have analyzed all the detailsconcerning the modeling of each field which repre-sents the physics under investigation, or how to ex-change information between the different fields. How-ever, when tight interactions between the componentfields take place, the response of the overall systemmust be calculated concurrently. Various strategiescan be adopted to solve this problem, with specificpros and cons, depending on the respective imple-mentation. Usually each domain has its peculiarities,which require special numerical strategies to success-fully accomplish the integration. At the same time,different space and time scales may arise in each field,so they must be correctly either represented or filteredout in the coupled simulation.

Often, a staggered procedure is used for the cou-pled simulations, in which separate fluid and struc-tural analysis codes are run in a strictly sequentialfashion, and they exchange interface data such as sur-face stresses and velocities at each time step (Pipernoet al. [22] and Zwaan and Prananta [23]). A classicalstaggered algorithm working cycle is composed by:

a) advance the structural system under the pres-sure loads computed at the previous step(xn+1

s , xn+1s );

b) update the fluid domain boundary conditions(and eventually the mesh) (xn+1

f , xn+1f );

c) compute the new pressure loads pn+1.

In this case at each time step there is no balancebetween the energy developed by the fluid force onthe structure and the energy gained by the fluid onthe interface, since∫

∂ΩB

(pnn · xn+1s − pn+1n · xn+1

f ) dA =∫∂ΩB

(pn − pn+1)n · xn+1s dA 6= 0

(18)

As a result, using unconditionally stable time integra-tion algorithms in each field analyzer does not guar-antee the unconditional stability of the overall stag-gered solution algorithm. These schemes may becomeenergy increasing and, hence, numerically unstable.To enlarge the stability boundaries, preserving at thesame time the modularity that characterizes the par-titioned methods, a new class of methods which canbe called “fully implicit partitioned methods”. For thesolution of the fluid-structure interaction problem un-der investigation, an implicit partitioned method hasbeen chosen, based on a relaxation technique similarto the one presented by Le Tallec and Mouro [24],which has two advantages:

A) it can be easily integrated with the solution al-gorithm currently implemented for the multibodycode;

B) by using the relaxation factor, the method canbe adapted to different operative conditions.

The algorithm at each time step performs the follow-ing operations

1) guess the position and velocity of the structuralelements by using the explicit second-order pre-dictor of MBDyn [25];

2) obtain the position and velocity of thefluid boundary elements using the conservativescheme here presented;

3) solve the fluid problem;

4) compute the interface loads;

5) transform the loads computed at the previouspoint into nodal forces using the correct approx-imation space, which is the constant functionspace for pressures on panels and forces per unitlength on lines;

6) evaluate the residual of the nonlinear structuralproblem applying the aerodynamic loads com-puted by the interface process and check for con-vergence: if the algorithm has converged stop thecycle;

18.8

7) solve the structural problem;

8) update the structural interface position accordingto the relaxation equation

xn+1s = (1− ω)xn

s + ωxn+1s ; (19)

9) go back to step (2).

Le Tallec and Mouro [24] have shown that this typeof scheme is equivalent to a preconditioned descentalgorithm for which the stability of the method withω = 1 is threatened when the parameter Kref∆t2

is small, where Kref is a characteristic stiffness ofthe problem under investigation. This means that theiterative scheme may encounter convergence problemswhen dealing with soft structures or small time steps.For these cases, under-relaxation may be used.

Often, the displacements of the structural nodesduring the correction phase are small. In these casesit is convenient, in terms of computational time sav-ings, to avoid updating the velocity that is inducedat each panel by the wake vortices. By simply pro-jecting the induced velocity computed during the pre-diction phase along the new normal direction of eachpanel, large time savings can be achieved with a verysmall error. This error can be estimated, by lookingat the Biot-Savart formula, Eq. (5), as proportionalto O(δr/r3), where δr is the position variation of thecontrol node of the panel.

Applications

Validation: Caradonna-Tung Rotor

The experimental results of Caradonna and Tung[26] are typically chosen as validation test for he-licopter aerodynamic codes. In this case the rotorblades are rigid, so no deformability is involved; how-ever, the test is useful to asses how the coupled pro-cedure, which involves the multibody code, the brokerand the free wake code, behaves. The test here ad-dressed for validation purposes was run with 8 deg ofcollective pitch at 1250 rpm; the tip Mach numberwas 0.439, so the flow field can be considered incom-pressible. Higher Mach numbers will be tested afterthe integration with CFD codes is completed. Therotor is in hover, and no initial wake is assigned, sothe system must reach the regime condition after de-veloping is own wake from the wind-up. For the hoversimulation with time marching algorithms, the initialwake state is critical because the the wake can becomeunstable owing to strong starting vortices generatedby the impulsively starting blades. The problem canbecome unstable especially for the vortex released by

Figure 5: Wake developed by Caradonna-Tung rotor:instability of the inner vortex ring.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 1 2 3 4 5 6

CT

rev

Figure 6: Traction coefficient for Caradonna-Tung un-stable solution; the dashed line represents the experi-mental value.

the inner extremity of the blade at the root cut-out.In fact, the start-up vortex at the inner root creates avortex ring with a small radius, which rapidly becomesunstable and causes the rise of an upward flow in theinner disk around the hub; an example of this wakephenomenon is shown in Figure 5. After the start-up phase, the inflow velocity induced by the develop-ing wake is capable of pulling the new wake elementsdownward in the external region, but not in the innerregion, where a fountain effect appears. This effectmay cause a global instability of the wake, as shown inFigure 6 for the time history of the traction coefficientCT which corresponds to the wake of Figure 5. In thepresent work, different methods, which do no requirean initial solution for the wake and are at the sametime computationally efficient, have been tested. Astable and correct solution may be obtained by simplyimposing a very fast growing law for the core radius of

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Figure 7: Wake developed by Caradonna-Tung rotor:stable algorithm.

0

0.002

0.004

0.006

0.008

0.01

0 2 4 6 8 10 12 14

CT

rev

Figure 8: Traction coefficient for Caradonna-Tung ro-tor stable solution; the dashed line represents the ex-perimental value.

the start-up and the inner vortex, somehow simulat-ing the vorticity diffusion which acts in the real exper-iment. The solution obtained in this case is shown inFigure 7, with the corresponding traction coefficientin Figure 8. Figure 9 shows the good correlation interms of spanwise section lift coefficient, while Fig-ure 10 shows the good correlation in terms of vortexvertical position as function of the azimuth.

Puma Rotor in Forward Flight

A full scale articulated helicopter rotor has beenconsidered as the final test. The Aerospatiale AS 330Puma helicopter aerodynamic behavior was investi-gated during the end of the ’80s by different worldwideresearch agencies to assess the accuracy of compre-hensive analysis codes and CFD Full Potential meth-ods for the prediction of airloads on a helicopter blade

0

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CL

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Figure 9: Caradonna-Tung rotor: section lift coeffi-cient.

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Figure 10: Caradonna-Tung rotor: position of the tipvortex.

in high-speed flight. All the experimental and numer-ical results obtained, together with a detailed reportof the aircraft configuration can be found in Bousmanet al. [27].

This test case has been chosen here in order to ob-tain detailed numerical results that can be comparedwith experimental data. An articulated rotor repre-sents a very interesting case for Fluid Structure In-teraction (FSI) coupled analysis, since large displace-ments due to rotations about the hub hinges may takeplace, essentialy governed by the aerodynamic hingemoments. Furthermore, the high-speed tests repre-sent an interesting base for comparison with the CFDtechniques that will be developed in the following.

As a first assessment, rigid blades have been con-sidered. The results obtained by the FSI procedureare compared with what can be obtained by a sim-ple blade section method with uniform inflow. At thisstage, the tests have been mainly considered a wayto assess the correctness and the ffectiveness of theFSI procedure. Future tests will address the model-ing of flexible blades, comparisons with the experi-

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Table III: Puma flight 123 parameters.

Advance ratio, µ 0.321Shaft angle of Attack, αs -6.0Collective pitch, θc 13.2Fore-aft cyclic pitch, θfa -7.15Lateral cyclic pitch, θl 2.1

mental results available so far, and the coupled inte-gration with CFD techniques. The baseline forwardflight case made with the standard unswept blade,Flight Test 123 (see [27]) has been selected for theinitial computations. The rotor configuration data hasbeen extracted from the report [27]. More informa-tion about the helicopter global characteristics maybe found in [28]. The rotor model is made of fourarticulated blades. The multibody model of the hubincludes:

a complete, kinematically exact model of theswashplate;

all the blade hinges;

a rigid pitch link is used, because the report doesnot contain enough information on the controlsystem stiffness;

a viscous damper acting about the lead-lag hinge;

the rigid blades.

The flight condition parameters are presented in Ta-ble III. The rotor control angles were computed bytrimming the simple (blade section aerodynamics +uniform inflow) model. The coupled model has beenrun imposing the controls computed from the simplermodel without looking for a specific trim condition.

A coarse grid on each blade, made by 5 elementsalong the chord and 10 along the span, has been usedtogether with a coarse time discretization, since only45 time steps for each rotor turn were used. Thismeans that the results presented in the following areonly a first assessment of the capabilities of the al-gorithm. More tests with a finer grid, together witha flexible blade model need to be conducted beforemaking a comparison with flight test measurements.However, note that the rotor is fully articulated andthe motion of the hinges is not imposed; on the con-trary, it results from the equilibrium, so the test canbe considered significant for the assessment of thestability and robustness of the interaction procedure.For this case, the rotor wake has been “forgotten”after three rotor turns, without significantly affectingthe velocity induced by the wake on the rotor disk.Figure 11.

The overall Z force generated with the vortex lat-tice model is slightly lower than that resulting from

the inflow models. Looking at the load distributionsobtained by the two models, Figure 12, the main dif-ferences for the vortex lattice model are slightly higherloads in the inner part of the blade, and lower loadsnear the tip due to the tip loss effects, which are notcorrectly accounted for in the simpler model.

Toward CFD Fluid-Structure Interaction

The element that characterizes the rotorcraft fluidflows and makes their modeling so difficult is thewake. The interaction between blades and vortices,expecially at low rotorcraft speed, gives rise to majorimportant phenomena, such as vibratory loads andBVI noise. At the same time, at high forward speed,compressibility phenomena need to be accounted for.The previous sections showed how the use of a vortexmethod allows to simulate the dynamics of the rotorwake. However, this type of approach suffers fromtwo limitations:

A) the aerodynamic field is considered incompress-ible;

B) the local interaction between vortices and solidboundaries cannot be easily represented.

On the CFD side, the integration of the free wakemodel for generic deformable blades represents a firststep toward the complete aeroelastic analysis of he-licopter rotors and tiltrotors. In fact, accrding tothe driving principle of applying the most appropriatephysical model for each fields or sub-field, the free-wake model can be used to compute the “far-blade”sub-field (or “far-wing” for tiltrotors), where the cor-rect approximation of the vorticity is the main goal,while CFD based on Euler or Navier-Stokes equationsshould be used in the “near-blade” sub-field to cor-rectly account for compressibility effects and interac-tions with solid boundaries. It will be easy to add aCFD code in the framework here presented using thebroker as the interface and interaction coordinator be-tween the different pieces of codes that will concur inthe simulation.

Conclusions and Future Work

This paper illustrates the application of a com-putational framework for the simultaneous solutionof multi-field initial value problems to the cou-pling of a free-wake modeling software to a multi-body/multidisciplinary analysis software. A novel ap-proach to solution coupling is presented, based ona communication synchronization and field interpo-lation process called “broker”. This is expected toshow a minimal impact on the participating analysis

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Figure 11: Puma rotor in forward flight: vorticity over the wake.

45 90 135 180 225 270 315Azimuth, deg

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315Azimuth, deg

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a) Vortex lattice (b) Blade section + inflow

Figure 12: Puma forward flight: rotor Z sectional force distribution.

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software, and in most cases it should allow their usewithout even minor changes, which may be a manda-tory requirement in case the source code is not avail-able. Different communication patterns are availableto allow the broadest applicability of the scheme. Theframework has been applied to the integrated solutionof aeroelastic problems, and is being applied to rotor-craft dynamics analysis.

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