DOC375008_s1

Conservatoire National des Arts et Metiers

These

Specialite: Mecanique

presente par

Biel Ortun

pour obtenir le grade de Docteur

CSM/CFD Coupling for the DynamicAnalysis of Helicopter Rotors

—Numerical Simulations of Fluid-Structure Interactionby coupling a Navier-Stokes Finite Volume code and

a Non Linear Structure Finite Element code

Soutenue le 18 decembre 2008,

devant le jury compose de:

Marc Berthillier Universite Franche-Comte RapporteurChristophe Pierre Universite McGill RapporteurPhilippe Devinant Universite Orleans President du juryRogelio Ferrer Eurocopter ExaminateurJean-Pierre Grisval ONERA ExaminateurDidier Petot ONERA Co-encadrant de theseRoger Ohayon CNAM Paris Directeur de these

Remerciements

La periode decembre 2005 - decembre 2008 restera pour moi un excellent souvenir. Cestrois ans de these passes a l’ONERA et au Laboratoire de Mecanique des Structures etdes Systemes Couples du Conservatoire National des Arts et Metiers ont ete une sourcede grande satisfaction, en grande partie grace aux equipes avec lesquelles j’ai travaille.

Je tiens a remercier M. Jean-Pierre Grisval, directeur du Departement d’Aeroelasticiteet Dynamique des Structures et M. Nicolas Piet, chef de l’unite Modelisation et Simula-tion, de m’avoir accueilli au sein de leur equipe.

J’ai eu l’appui incontestable de M. Roger Ohayon, mon directeur de these et professeurtitulaire de chaire de Mecanique du CNAM ainsi que de M. Didier Petot, mon encadranta l’ONERA. J’ai apprecie leur accessibilite et rapidite a repondre soigneusement a toutesmes questions. J’ai beaucoup appris grace a eux.

J’exprime ma gratitude a M. Christophe Pierre, doyen de la faculte d’ingenierie del’Universite de McGill et au professeur Marc Berthillier, de l’Universite de Franche-Comte,d’avoir accepte d’etre les rapporteurs de ce travail.

Je remercie egalement M. Rogelio Ferrer, representant d’Eurocopter, et M. PhilippeDevinant, professeur a l’Universite d’Orleans, d’avoir consacre leur temps et interet a laparticipation de mon jury de soutenance le 18 decembre 2008.

Le bon deroulement de ma these n’aurait pas ete possible sans la collaboration deKhiem-Van Truong. Je souhaite aussi remercier Alain Dugeai, Christophe Blondeau etYves Gorge pour leur appui informatique. a Marc Rapin pour la participation au congresde Reno en 2007. Et plus generalement, a l’ensemble de mes collegues a DADS, qui ontrendu l’ambiance de travail aussi accueillante.

Mes remerciements vont aussi a d’autres departements de l’ONERA : le DSNA, ou j’aieu la chance de travailler avec Marc Poinot, Christophe Benoit, Stephanie Peron, JacquesSides et d’autres. Et le DAAP, avec Michel Costes, Philippe Beaumier, Benoit Rodriguezet Thomas Renaud. Je ne cite pas tous les noms par souci de brievete.

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Au dela de l’ONERA je souhaite remercier l’equipe du Laboratoire de Mecanique desStructures et des Systemes Couples du CNAM pour les echanges toujours enrichissantssous forme de seminaires. Et les ingenieurs d’Eurocopter France, Eurocopter Deutschlandet le DLR que j’ai eu le plaisir de rencontrer regulierement au cours des reunions du projetSHANEL.

Merci a Marion, qui a ecoute mes innombrables repetitions, pour son soutien et sessuggestions.

Enfin, merci au Centre Aeronautique de Beynes, ou j’ai decouvert le vol a voile et laserenite de caresser les nuages.

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Contents

Introduction 1

1 Introduction to the Physics of a Helicopter Rotor 61.1 Rotor aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Introduction to Rotorcraft Nonlinear and Unsteady Aerodynamics . 61.2 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Introduction to rotor dynamics. . . . . . . . . . . . . . . . . . . . . 11

2 The Numerical Analyses 182.1 Advanced finite element analysis for rotor dynamics . . . . . . . . . . . . . 182.2 The finite element solver MSC.Marc . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Adapting finite element analysis to rotor modeling . . . . . . . . . . 212.2.2 About the MSC.Marc simulations in an inertial frame . . . . . . . . 262.2.3 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 The CFD code elsA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 The rotorcraft comprehensive analysis code HOST . . . . . . . . . . . . . . 32

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 The basic HOST algorithm . . . . . . . . . . . . . . . . . . . . . . . 332.4.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.4 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.5 Review of the start point: the HOST/CFD coupling . . . . . . . . . 362.4.6 Concurrent comprehensive/CFD couplings . . . . . . . . . . . . . . 40

3 Development of a Framework for Code Coupling 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Coupling specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Adopted solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Programming model . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 Data model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.3 Programming language for the framework . . . . . . . . . . . . . . 47

3.4 Architecture of the framework . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.1 Network distributed computing . . . . . . . . . . . . . . . . . . . . 483.4.2 Properties of the new coupling framework . . . . . . . . . . . . . . 48

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CONTENTS

3.5 An interface for MSC.Marc . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5.1 Coupling regions and the user subroutines . . . . . . . . . . . . . . 493.5.2 The python interface . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6 An interface for HOST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Fluid-Structure Interaction in a Time-Accurate Coupling 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Staggered algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 The conventional serial collocated scheme . . . . . . . . . . . . . . 574.2.2 The parallel collocated scheme . . . . . . . . . . . . . . . . . . . . . 584.2.3 The serial non-collocated scheme . . . . . . . . . . . . . . . . . . . 594.2.4 Staggered algorithms in rotorcraft aeroelasticity . . . . . . . . . . . 60

4.3 The fluid/structure interface . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 CFD grid deformation technique . . . . . . . . . . . . . . . . . . . . 644.3.2 CFD airloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.3 HOST airloads for an external structure . . . . . . . . . . . . . . . 664.3.4 Definition of a fluid-structure interface for the beam models . . . . 674.3.5 Transfer of the structure motion of the beam FE models . . . . . . 674.3.6 Definition of a fluid-structure interface for the 3D models . . . . . . 684.3.7 Transfer of the structure motion of the 3D FE models . . . . . . . . 694.3.8 Transfer of airloads from the CFD to the 3D FE model . . . . . . . 73

4.4 Rotor control and trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.1 Amplification of the rotor controls . . . . . . . . . . . . . . . . . . . 79

5 Applications 835.1 7A rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1.2 Fluid grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 ERATO rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2.2 Fluid grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.3 Conservation of the energy in the loads and motion transfers . . . . 975.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.5 Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . 1055.2.6 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3 Comparison of coupling staggered algorithms . . . . . . . . . . . . . . . . . 1105.3.1 Remarks on the comparison of staggered schemes . . . . . . . . . . 115

Conclusions and Perspectives 117

References 124

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CONTENTS

A Frames of reference 125

vi

Introduction

Description of the Problem

The analysis of the aeroelastic behaviour of helicopter rotors describes the interactionbetween the structural dynamics and the aerodynamics. Aeroelastic simulation predictsboth the structural loads acting on the rotor and its aerodynamic performance. A thor-ough understanding of rotor aeroelastics is of utmost importance because the design ofa rotor is the largest single factor determining the flight performance, handling qualities,ride comfort and exploitation costs of the complete helicopter. Indeed, the rotor is thecore component of the helicopter because it accomplishes three basic roles: (1) generatevertical lifting force; (2) provide propulsive force; and (3) provide steering control.

This functional compactness comes at a price, though. The flowfield around the rotoris very complex because it is highly three-dimensional, asymmetric and dominated byunsteady aerodynamics. Additionally, depending on the flight condition, one or more ofthe following phenomena may be encountered: blades flying into their own wake, transonicspeeds with shocks at the blade tip, flow separation, flow reversal and dynamic stall. Theunsteady airloads generated by the rotor are an important source of vibration.

Rotor blades undergo large amplitude motions in flapping (out-of-rotor-plane), lead-lag (in-plane) and feathering (pitch). The rotor blades are stiffened by the centrifugalloading. Yet the deflections of the blades are still significant and, contrarily to manyturbomachinery problems, Coriolis forces cannot be neglected. In addition, the pitch ofa rotor blade is changing constantly to control the rotor. The forces proper to a rotatingbody, combined with the unsteady airloads and the pitch control input, trigger a sharpstructural response. The sudden changes of shape and position of the rotor blades imply amodification of the flow around them, setting the scenario for a challenging fluid-structureinteraction problem.

Current rotorcraft aeroelastic simulations rely mostly on the so-called ‘rotorcraft com-prehensive analysis codes’. Comprehensive codes owe their name to their capacity toencompass all of the technical disciplines that are necessary to perform aeroelastic sim-ulations: structure, aerodynamics and flight mechanics models. Comprehensive codesare popular among the helicopter manufacturers. The industry needs low computationalcosts and a design-oriented, engineering usability. As a result, the fluid and structuremodels that are implemented in comprehensive codes are reduced to the minimum com-

1

Introduction

plexity that still yields reliable results. Typically, structure models are based on beamtheory. Aerodynamic models are based on blade element theory, which basically assigns2D experimental polars to a number of blade span-stations, plus reduced models for theinduced velocity and rotor wake.

However, the physics-fidelity of the models in comprehensive codes is not enough topredict accurately the aeroelastic phenomena taking place in important -and everyday-flight conditions, such as high-speed or steep-descent flight (read landing). For these twoexamples the shortcomings arise especially from failing to capture accurately the unsteadyairloads that follow the blades’ dynamics. Steep-descents when landing in urban areas area problem because they are noisy due to blade-vortex interaction phenomena. Expandingmaximum cruise speed is also very important for the helicopter to gain a larger share ofthe transportation market by competing with fixed-wing aircraft. In short, the accurateanalysis of the aeroelastic phenomena of a rotor is a key step towards improving thedesigns of future rotors and hence, produce better helicopters.

But the accuracy of comprehensive codes is not only undermined by the type of flightcondition. The beam assumptions of the structure models do not fit certain structuralelements that are not slender, like the rotor hubs. What is more, beam models may fallshort as well to represent advanced concepts such as aeroelastic tailoring of the rotorblade via piezoelectric controls.

As a consequence of the limitations of comprehensive codes, the capacity of the heli-copter industry to produce innovative designs and major breakthroughs -fuel consumption,noise, ride comfort, etc- with respect to todays helicopters is also diminished.

Past Efforts

In order to tackle the shortcomings of the aerodynamic models of comprehensive codes,computational fluid dynamics (CFD) analyses are being increasingly used for rotorcraftapplications by coupling them to the comprehensive codes. CFD analyses can solve the 3Dmass, momentum and energy conservation equations of a fluid, thus yielding high-fidelityflowfield representations.

The CFD coupling strategy has brought significant improvements in aeroelastic pre-diction accuracy. Recent efforts in coupling CFD with comprehensive analyses have beenpublished by Beaumier et al. [7], Pahlke and van der Wall [37], Potsdam et al. [46] andBhagwat et al. [10], among others.

Motivations of the Research

So far, research efforts in improving the prediction capabilities of comprehensive codeshave focused on the fluid side by coupling with CFD methods. Yet however accurateCFD methods might be, aeroelasticity is a coupled problem, where the structure responsematters just as much as the aerodynamics. Investing on computationally expensive CFD

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makes sense as long as the structural dynamics keeps pace with the CFD’s accuracy.In this respect, the classical comprehensive structural dynamics models based on beamtheory have proved reliable enough.

Yet future structural models should embrace arbitrary geometries. In a rotor systemcomposed by blades, hub and control mechanisms, there are many elements unsuitable fora beam representation. The newest bearingless hubs would benefit from three-dimensionalstructural models. Elastic swashplates would open the way to more accurate studieson pitch control mechanisms. Even the latest blade designs, seeking a lower acousticsignature, feature increasingly complex geometries, which puts a growing strain on beammodels for blades. Advanced structural models of the fuselage could also provide newinsight into rotorcraft vibration. The advent of morphing profiles in the blades will requirethe rotor blades to be modeled at least as a bi-dimensional structure.

These advanced structural models will be referred to as Computational StructuralMechanics (CSM) in the remainder of this document.

Objectives and Context of the Research

The objective of this work is two-fold. First, to introduce 3D finite element (FE) basedstructural dynamics models in the simulation of rotorcraft aeroelasticity. Second, todevelop the basis for a new, highly versatile tool for the analysis of rotorcraft aeroelasticitywith the capability to perform high-accuracy analyses.

These two objectives will be achieved by developing a computing environment orframework to couple HOST, a rotorcraft comprehensive analysis developed by Eurocopter,with a 3D finite element based structural analysis (CSM) and a CFD analysis.

Effective coupling will be accomplished through the use of a partitioned procedure.In a partitioned procedure for rotorcraft aeroelasticity, each of the subsystems -fluid,structure, flight mechanics- is time-integrated by a different scheme that is tailored to itsmathematical model. Then the global problem is solved by a staggered algorithm thatexchanges information between the subsystems at each time-step.

For the coupling to have a practical utility and be user-friendly, the following threeconstraints shall be observed:

1. The coupling must be modular, this is, it must be possible to replace, add or re-move any of the participating codes. This is achieved by developing a modularsoftware architecture, in which every technical discipline is solved by a dedicatedcode. Communication between the codes solves a multi-disciplinary problem.

2. The coupling must be able to analyze any random rotorcraft configuration: isolatedrotors, complete helicopters, tiltrotors and alike. This condition requires not onlyan open software architecture, but also physical models that can represent anygeometry. This second condition is met by the CFD and 3D CSM methods.

3

Introduction

3. The fidelity of the aeroelastic analyses must be scaleable. This means that simpleor autonomous HOST computations must remain available. Yet, when greater ac-curacy is needed in the fluid and/or structure model, a coupling with an externalcode CFD and/or CSM code is activated.

This research belongs to the SHANEL (Simulation of Helicopter Aerodynamics, Noiseand ELasticity) programme, a five-year Franco-German cooperation between ONERA,DLR1, Eurocopter France2 and Eurocopter Deutschland2 launched in 2006.

The developments engaged during this research will be pursued during the remainderof the SHANEL programme, which ends in 2011. This work has been led in collaborationwith the Eurocopter Group. Therefore, the tools and methods here developed must bein part regarded as a contribution to Eurocopter’s technological edge. ONERA and DLRare also to gain better insight into rotorcraft aeroelasticity problems.

The multidisciplinary nature of aeroelasticity makes it a multi-team work. This studyhas also been done in close teamwork with two other ONERA departments: the Ap-plied Aerodynamics Department (DAAP) and the Computational Fluid Dynamics andAeroacoustics Department (DSNA).

This study takes advantage of the developments of a previous cooperation completedin 2005, the CHANCE (Complete Helicopter AdvaNced Computational Environment)project [17], in which a HOST/CFD coupling was developed.

What is new

This research introduces for the first time:

• 3D finite element based structure models in rotorcraft aeroelasticity analysis.

• A new method to modify automatically the rotor controls in order to yield a trimmedsolution while using a time-accurate coupling procedure. Rotor trim designates theequilibrium of the rotor in steady flight (weight-lift, drag-thrust, pitch and rollmoments).

• An innovative approach for the software architecture in partitioned procedures. Thisincludes software modularity, distributed computing and the use of public and in-ternationally regulated data models to describe the data exchanged between codes.

Organisation of the Research

Naturally, there were many unknowns at the start of this project: from the most basicquestions on how to get several codes to communicate and exchange data to the details

1Deutsches Zentrum fur Luft- und Raumfahrt, Germany’s national research centre for aeronautics andspace.

2Eurocopter France and Eurocopter Deutschland GmbH belong to the Eurocopter Group and are ahelicopter manufacturer. The Eurocopter Group is 100% owned by the European Aeronautic Defenceand Space Company (EADS).

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related to fluid/structure interaction; which information was to be exchanged, when andhow. Would the simulations be stable? How to maximize accuracy?

As in every scientific work, the method consisted in discretising the large problem intosimpler units and addressing them in a scalating level of complexity.

The goal of the first actions was to answer the most basic questions on the practi-cal implementation of the coupling. The first priorities were to set up a basic softwarecoupling environment, to choose a set of code-independent variables to be exchanged andto learn how to get the CSM software (MSC.Marc) and HOST to exchange data. Code-independent variables matter because the coupling must not be limited to a specific CSMor CFD solver.

In order to ease the early developments, only beam elements were used for the firstfinite element model. This simple beam model represented nevertheless the 7A rotor, arotor that had already been analyzed by the HOST/CFD coupling during the CHANCEprogramme. The results from that first coupling were to be used as a comparison bench-mark to validate the new developments.

Thus, the beam model became a development support that was used to implementthe minimum software infrastructure to get the codes running and exchanging data. Thisimplementation was carried out by reviewing the literature and consulting ONERA’sspecialists in code coupling while observing the requirements of the SHANEL programme.

The beam-based rotor model in MSC.Marc was first coupled with HOST. HOST pro-vided the airloads and MSC.Marc the structural response. The next step consisted incoupling the finite element model with CFD airloads.

Once the most basic software problems were solved, work resumed by extending thefluid/structure coupling to a third discipline, namely, the flight mechanics. A methodcalled ‘active trim’ was developed. The active trim allows to modify the rotor controlsduring the simulation until the forces generated by the rotor respect the in-flight equilib-rium in terms of lift, thrust and pitch and roll moments, for example.

The developments for the beam finite element model were validated by comparing withthe equivalent HOST/CFD coupling of the CHANCE project. Then work resumed byintroducing a 3D finite element model of a rotor, yet a fictitious rotor -i.e., non tested- ofsimple geometry to ease developments: rectangular blade planform and no twist. The goalhere was to upgrade the coupling capabilities in order to be able to handle 3D FE models.Next, the work moved on to a more geometrically complex rotor for which experimentalmeasurements in the wind tunnel were available: the ERATO rotor.

The 3D finite element model of the ERATO rotor was coupled to CFD aerodynamicsand to HOST’s flight mechanics. Numerical solutions were compared to experimentalmeasurements. Investigations were carried out on the different staggered schemes bycomparing the solutions and also by looking at the conservation of the energy on thefluid/structure interface.

The organisation of the chapters does not follow strictly the chronology of the research;instead, it was deemed more convenient to present first all the developments and secondlythe results.

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Chapter 1

Introduction to the Physics of aHelicopter Rotor

This chapter introduces the dynamic and aerodynamic physical phenomena affecting ro-torcraft aeroelasticity that are relevant to this work.

1.1 Rotor aerodynamics

The purpose of this section is to introduce the main phenomena in rotor aerodynamics thatwill be later observed in the results chapter, with a special focus on nonlinear unsteadyaerodynamics, which is a dominant feature in rotor aerodynamics.

1.1.1 Introduction to Rotorcraft Nonlinear and Unsteady Aero-dynamics

The value of a helicopter lies in its capacity to perform vertical flight: take-off and landvertically and sustain hover flight. In these conditions the rotor works much as a conven-tional propeller, with an inflow that is perpendicular to the rotor disk plane. The thrustgenerated by the rotor is fully vertical and therefore, no lateral forces push the helicopterto move horizontally.

However, as soon as the pilot tilts the rotor forward to get moving, the rotor inflowis skewed by the helicopter airspeed and the flowfield becomes asymmetric. The comingparagraphs describe the source of this asymmetry and its consequences as a generator ofnonlinear and unsteady aerodynamics.

In forward flight, the rotor blade sees a component of the helicopter forward velocityas well as the velocity due to its own rotation. On the advancing side of the rotor disk theairspeed of the blade is increased above the rotation speed by the forward speed, whileon the retreating side the opposite is true, as shown in Figure 1.1. The most inboardregion of the retreating blade sees reversed flow. The different airspeeds of the advancing

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and retreating blades create an asymmetric flowfield. The asymmetric flowfield triggersin turn oscillatory airloads on the blades.

Figure 1.1: Velocity distribution on a rotor in forward flight.

As the helicopter forward speed increases, the advancing blade sees greater airspeeds,and eventually the blade tip reaches transonic speeds. Similarly, the fraction of theretreating blade under reversed flow grows larger with helicopter speed. The fraction ofthe retreating blade on the outboard side that still sees normal flow generates lots of dragbecause, as it will be seen later, the pitch of the retreating blade is very high to produceenough lift despite the low airspeeds. In addition, above a certain helicopter speed, theonset of shock waves in the advancing blade makes the drag to leap there. Since both thewave drag of the advancing blade and the induced drag of the retreating blade take placenear the blade tip, the resulting drag-torque at the rotor head is significantly magnified.At some point the engine fails to provide enough torque, and it is this limit in enginepower that dictates the maximal cruise velocity.

In order to keep a straight forward flight, the rotor disk must not tilt sideways. Ifthe blade pitch were constant, the advancing blade would generate more lift than theretreating blade and hence, the lift asymmetry would create a roll moment that wouldbreak the level flight. Lift asymmetry is prevented by setting a minimum pitch to theforward blade (pitch control is explained later in Section 1.2.1.1) and a maximum pitchto the retreating blade. The difference in pitch offsets the difference in airspeed. Bladetwist is also used to counter the radial speed distribution on the blades.

The asymmetrical flowfield of the rotor in forward flight is only one of the multiplecomplexities of rotor aerodynamics. There are at least three more key aerodynamic phe-nomena that drive the critical rotor loads. These key aerodynamic phenomena are: wakeinteraction, dynamic stall and tip transonic effects. Depending on the flight regime, oneout of the three phenomena dominates over the other two and determines the boundariesof the flight envelope. The following paragraphs describe each of these phenomena andtheir consequences on flight performance. An in-depth review of the subject was publishedby Datta, Nixon and Chopra in [22].

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1. INTRODUCTION TO THE PHYSICS OF A HELICOPTER ROTOR

Before proceeding into more detail, let us define the azimuthal angle of a rotor bladewith the support of Figure 1.1. The azimuthal angle is an useful notation to indicatethe location of the blade on the rotor disk. It is denoted by Ψ and the origin lies on themost downstream part of the rotor. Ψ = 90deg corresponds to full advancing blade andΨ = 270deg to full retreating blade.

Wake effects. Rotor blades fly in the vicinity of the wake of their preceding blades inthe majority of flight conditions. This phenomenon has to be taken into account becauseit affects significantly the flowfield around the blades.

Transition from hover to forward flight is greatly influenced by wake effects. The rotorwake impinges on the whole rear half of the fuselage and on the tail boom, which can leadto aeroelastic phenomena such as tail shake and other cross-coupling phenomena betweenthe various degrees of freedom of the helicopter. Vibrations in a certain range of lowspeeds are so strong that the helicopter has to be flown through quickly. The flowfield ofthe tail rotor may be also perturbed by the wake depending on the flight condition.

In descending or low speed level flight the blades may encounter the vortices shedby the preceding blades. This constitutes the so-called blade-vortex interaction (BVI).Blade-vortex interaction is characterized by rotor vibratory loads and a loud ‘popping’noise. BVI constitutes one of the largest sources of noise of the helicopter and is especiallyrelevant because it can be heard during the landing final descent.

During high-speed cruise flight the wake is quickly convected downstream and thereforeless interaction occurs. Nevertheless, when the advancing blade flies close to the vorticesshed by the previous blades, it receives a high-frequency excitation that strengthens theunsteady transonic phenomena described in the next paragraph.

Tip transonic effects. During high speed flight the advancing blade tip reaches tran-sonic speeds, and the onset of shock waves limits the helicopter maximal cruising speed.The three dimensional unsteady transonic flow field creates moving shock waves on theadvancing blade, which causes the aerodynamic center to shift towards the trailing edge,giving rise to large nose-down pitching moments. The shock-induced pitching momentforces the blade to twist nose-down, thus reducing the lift at the blade tip.

Transonic effects are easily noticeable in most wind tunnel measurements reproduc-ing high-speed flight. For instance, for the ERATO rotor, the evolution of the pitchingmoment coefficient Cm at r/R=0.975 section over one rotor revolution is shown in Fig-ure 1.2(a). It can be seen how, shortly after crossing the Ψ = 90deg azimuth (advancingside of the rotor), the pitching moment at the blade tip becomes negative. In the samefigure it can be observed a second, larger, negative pulse around Ψ = 300deg. That is dueto dynamic stall, which is presented later.

Note that Figure 1.2(b) contains the aerodynamic pitching moment coefficient, Cm,times the squared local Mach number, M2. Giving the product CmM

2, rather than thebare coefficient Cm of Figure 1.2(a), is more telling about the actual magnitude of the

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(a) Pitching coefficient Cm. (b) Pitching moment CmM2.

Figure 1.2: Measured sectional airloads on the ERATO blade in high-speed flight.

airloads because the inclusion of the squared Mach number accounts for the dynamicpressure.

Since the speed of the blade tip -and thereby the dynamic pressure- is much greaterin the advancing blade (Ψ = 90deg) than in the retreating blade (Ψ = 270deg), thesmaller of the two negative peaks in pitching coefficient Cm in Figure 1.2(a), located atΨ ≈ 90deg, becomes actually a larger pitching moment CmM

2 in Figure 1.2(b) than thatdue to dynamic stall.

Dynamic Stall. Dynamic stall is an unsteady flow separation phenomenon that occurson heavily loaded rotors -i.e., operating at high thrust or high altitude-. Dynamic stallappears mostly on the retreating blade due to high angles of attack, and on the advancingblade because of shock-boundary layer interaction.

As the airfoil pitches up, the trailing edge flow separates progressively due to negativepressure gradient in the boundary layer, and a leading edge vortex appears. This vortex isconvected downstream along the upper side of the airfoil, yielding a negative nose-downstall pitching moment. However, the lift does not stall while the low-pressure vortextraverses over the airfoil. As soon as the vortex leaves the trailing edge the lift stalls,but simultaneously the negative pitching moment reaches its maximum and the flow re-attaches as the airfoil pitches down. This cycle forms hysteresis loops on the airloadsversus the angle of attack, see Figure 1.31. High-frequency torsion-oscillating profiles canproduce greater lift coefficients at greater angles of attack without succumbing to stall.

1Source: Imperial College London, Department of Aeronautics.

9


Insects such as the dragonfly take advantage of this phenomenon [51], since their wingswould not be able to lift them on a quasi-steady aerodynamics basis.

Dynamic stall triggers transient high-frequency torsional response and it is a source ofimportant blade vibratory loads, which can eventually limit the speed and loading of therotor. Because of its impact on rotorcraft performance (e.g. vibration, pushrod loads),dynamic stall has benefited from intensive research. Popular dynamic stall models havebeen proposed by Friedmann [27], Leishman [31], Petot [38] and Truong [53].

Figure 1.3: Lift coefficient in dynamic stall.

Aerodynamic interactions The fuselage and the tail rotor also affect the flow pastthe main rotor. In forward flight the fuselage induces an upwash in the innerboard partof the blade, around the 180 azimuth, which increases the local angle of attack. Anincrease of angle of attack leads to an increase in lift force. Then, by gyroscopic effect,the blade flaps a quarter of revolution later, generating a roll moment around the rotorhub.

1.2 Structural dynamics

This section presents the main phenomena in rotor dynamics that are relevant to the testapplications of this work.

10


1.2.1 Introduction to rotor dynamics.

1.2.1.1 Mechanical description of a rotor.

The standard helicopter main rotor is composed of the blades, the hub, the pitch controlsystem and the lead-lag adaptors.

It has been seen in the aerodynamics section that, in forward flight, the airloads actingon the blade are oscillatory and periodic over one revolution. These varying airloadsinduce considerable bending motion on the rotor blades because the blades are slenderand hence, highly flexible. The largest bending motion takes place out of the plane of therotor disk and is called flapping motion. If the blades were simply clamped to the rotorhub, root bending moments and blade stresses would quickly lead to structural failure ofthe blade root, something that the early pioneers in rotary-wing flight quickly realized.This problem was fixed by introducing a flap hinge between the blade root and the rotorhub. However, the out-of-plane flapping motion of the blades becomes even greater withthe hinge and as a consequence, Coriolis forces induce important in-plane motion, calledlag. The amplitude of the lag motion is large enough as to require a second hinge toalleviate the lag bending stresses.

Rotors equipped with flap and lag hinges are known as articulated rotors. Placinga flap hinge and a lag hinge between the blade root and the rotor hub, as shown inFigure 1.42, alleviates the root bending moments by letting the blade have rigid-bodymotion. In addition of the flap and lag hinges, there is a third articulation: the pitchbearing. The pitch bearing allows to control the angle of attack of the blades (see belowSection 1.2.1.1) and hence, to control rotor aerodynamics.

The most recent rotor designs have replaced the articulations between blade rootand rotor hub by highly flexible materials that still allow near rigid body motion of theblades. The advantage of this new technology, baptised ‘bearingless’ rotors, is that itreduces the mechanical complexity of the rotor hub (fewer elements, no maintenance-demanding bearings). Pictures of an articulated rotor and a bearingless rotor are shownin Figure 1.5(a) and Figure 1.5(b), respectively.

Rotor control and gyroscopic effects. In a helicopter rotor each blade is actuatedin pitch. Blade pitch is used to regulate the angle of attack of the blade and hence, toregulate how much lift it generates. Blade pitch is expressed as a Fourier series truncatedat the first harmonic,

θ = θ0 + θ1CcosΨ + θ1SsinΨ, (1.1)

where Ψ is the blade azimuth, θ0 is called the collective control and the first harmonics θ1C

and θ1S are called the lateral and longitudinal cyclic controls, respectively. The reasonbehind these names is explained shortly. The cyclic pitch is used by the pilot to tiltthe rotor plane towards the desired heading. The collective pitch is used to regulate the

2Source: Helicopter Theory, by Wayne Johnson. Dover Publications, 1994.

11


Figure 1.4: Schematic of an articulated rotor hub showing one blade only.

(a) Mil Mi-4’s articulated rotor. (b) EC135’s bearingless rotor.

Figure 1.5: The evolution of technology in rotor hubs.

12


rotor thrust. The vertical component of the thrust is the lift force, while the horizontalcomponent is the propulsive force.

Figure 1.4 showed how a pushrod sets the blade pitch. This section has shown thatblade pitch control has three components: one collective and two cyclical angles. Theswashplate is the mechanical device that actuates the pushrods according to the threepitch control angles. Figure 1.6 contains a schematic illustration of a swashplate. Aswashplate consists in two parallel disks separated by bearings. The upper disk is con-nected to the blades via the pushrods and it rotates with them. The lower disk is connectedto the control input rods. In addition, the two disks can be tilted and displaced along therotor shaft. Collective pitch is adjusted by displacing the swashplate up and down alongthe shaft thanks to a simultaneous and homogeneous displacement of the control inputrods. Cyclic pitch is achieved by tilting the swashplate thanks to the individual action ofcertain control rods; θ1C actuates only the control inputs rods that lie in the longitudi-nal direction (Ψ = 0deg to Ψ = 180deg), thus tilting the swashplate longitudinally. θ1S

actuates the lateral rods only.

Figure 1.6: A swashplate.

Gyroscopic effects. The angular momentum of a spinning rotor is very large and soare the gyroscopic effects. The most practical consequence of gyroscopics affects how therotor controls are mounted. It has been explained that the helicopter is steered by tiltingthe rotor in the desired heading direction. Rotor tilt is defined by the blade tip flappingpath. For example, engaging a left-hand turn while in forward flight requires the bladeflapping angle to be greater on the right side of the disk than on the left side. Sinceflapping is proportional to lifting force, it follows that flapping is controlled by the pitchangle of the blade.

A blade with positive angle of attack will generate lift and consequently, flap upwards.Yet, as a consequence of gyroscopic precession, the flapping will only happen a quarter of

13


a revolution after the lift force. For an increased flapping at Ψ = 90deg, blade pitch mustbe increased at Ψ = 0deg. That is why the cyclic control angle θ1C , although actuated inthe longitudinal direction of the disk -i.e., Ψ = 0deg to Ψ = 180deg-, is called the lateralcontrol angle. Similarly, the cyclic pitch angle θ1S is called the longitudinal control angledespite acting on the lateral sides of the disk.

1.2.1.2 Blade dynamics

The purpose of this section is to introduce concepts and terminology that will be referredto later in this document. For a comprehensive treatise in rotary-wing dynamics, thebooks of Johnson [30] and Bielawa [12] are classics.

Notation of blade motion. The flapping motion is denoted with the angle β. The lagmotion is denoted by δ. Blade motion is typically described by Fourier series, like for theblade pitch. For example, for the blade motion, the flap angle of the blade is expressedas a function of the azimuth angle Ψ by the expression

β = β0 + β1CcosΨ + β1SsinΨ, (1.2)

the same formulation applies for the lag angle δ. Note that the harmonics higher thanone are neglected for being small.

Forces acting on a rotor blade. Consider a spanwise segment of length dr and massm centered at a fraction r/R of a blade with radius R. The blade is articulated in lagand flap. The lag and flap hinges are collocated. The angular speed of the rotor is Ω.The loads acting on this segment are:

• An inertial force with three components for the flap, lag and pitch accelerations.

• A centrifugal force mΩ2r.

• An aerodynamic force.

• A Coriolis force. The Coriolis acceleration is twice the cross product of the angularvelocity vector and the velocity vector relative to the rotating frame.

Typical eigenfrequencies of a rotating blade. The centrifugal loads stiffen the bladeand hence, the eigenfrequencies of the blade tend to increase with the angular speed ofthe rotor Ω. The inertial loads due to rotation of the blade approximately increase thesquare of the natural frequency ω2

n in proportion to the square of the rotation speed Ω2.This effect can be expressed as ω2

n = ω20n + κnΩ2, where ω0n is the non-rotating natural

frequency and κn depends on the mode.The constant κn differs for the flap and lag motions. While flapping, the centrifugal

loads always have the same direction, as shown in Figure 1.7. However, in lag motion, the

14


centrifugal force is a follower force; the blade hinges around a point offset by a distancee to the rotation center. Since this hinge offset e is relatively small with respect to theblade radius, it follows that the centrifugal force is nearly aligned with the blade axis. Asa result, the restoring moment generated by the centrifugal force tends to zero as e → 0for the lag motion, whereas in flap motion the restoring moment depends on the angle β.

Figure 1.7: Rotor blade out-of-plane flapping.

Figure 1.8: Rotor blade in-plane lag.

The smaller restoring centrifugal moment that lag motion receives means that lagmotion is less stiffened than flap motion. As a result, the lag eigenfrequencies of a rotatingblade rise with a smaller slope -i.e., smaller κn- than flap eigenfrequencies as the rotorangular speed Ω is increased.

The κn constant is equal for lag motion and torsion motion. Imagine a flat blade fullycontained in the rotor disk spin plane and whose pitch axis is also contained in that plane.If the blade rotates around the pitch axis, its leading and trailing edges will leave the rotordisk spin plane and the restoring moment of the centrifugal force will act on them in thesame manner than for the flap motion. Nevertheless, the non-rotating natural frequencyof torsion ω0 is much higher than those of flap and lag because there is no pitch hinge, sothe blade is not free to pitch.

Figure 1.9 illustrates the increase of an articulated blade’s eigenfrequencies -in red-with rotation speed (assuming the blade in the vacuum). The straight yellow lines areconstant frequency curves, noted by 1/rev, 2/rev and so on. The notation n/rev meansthat frequency is n times the rotor angular frequency. At Ω = 0, there is a mode withzero eigenfrequency; it corresponds to the rigid flap motion of the blade around the flap

15


hinge. This same mode follows the 1/rev curve as Ω increases. Therefore, the κ of therigid flap mode is equal to one. The second lowest frequency at Ω = 0 corresponds to thefirst lag mode. The lag motion, although also articulated, has a non-null non-rotatingeigenfrequency (ω0 ≈ 4Hz in the figure) due to the presence of the lead-lag damper. Thelag mode is observed to be nearly independent of the rotation speed. The first torsionmode appears above 100Hz for Ω = 0 and presents a very mild slope as Ω increases.Its κ value is equal to one and it is equal to that of the first flap mode. Yet since itsnon-rotating frequency is higher, ω0 ≈ 6/rev, the proportional effect of Ω is reduced,ωΩ=1 =

√(6Ω)2 + 1Ω2 = 6.08/rev.

It is also interesting to observe how certain modes exchange their behaviour. Forexample, at around 60Hz mode 6 (for 6th largest non-rotating eigenfrequency) starts aslag but switches to flap beyond Ω = 30rad/s. At the same rotation speed, flap mode 5switches to lag.

Figure 1.9: Campbell diagram.

Sources of couplings. One of the greatest complexities of rotating blade dynamics isthe variety of couplings between the flap, lag and pitch motions. Coupling effects arise inblade dynamics principally from five main sources:

16


1. General and usual misalignments of the blade section principal axes (both elasticand inertial) from the spin plane due to all components of blade pitch angles andbuilt-in geometric twist;

2. Noncoincidence of the various elastomechanical centers within the blade sections(mass center, elastic axis, neutral axis);

3. Skewness of the blade’s radial axis from the spin plane due to built-in coning orflapping;

4. Nonlinear effects of combined flatwise and edgewise bending;

5. Nonlinear coupling between flap and lag motion due to the blade Coriolis forces.

In addition, the dynamic behaviour of articulated rotors depends strongly on theorder of the articulations -when not collocated- and on the offset of the articulations tothe rotation centre.

Another source of coupling is the kinematic k-coupling. The k-coupling is best ex-plained by looking at Figure 1.4, in which the pushrod actuating the blade pitch bearingis outboard of the flap hinge. If the pushrod is fixed and the blade flaps, the pitch ofthe blade will be modified. This can be useful; pitch may be automatically reduced whenthe blade flaps upwards and inversely. Yet generally helicopter manufacturers counter thek-coupling with a simple solution: extending the pitch actuator to the flap axis and lettingthe pushrod be coplanar with the flap axis. A close look on the rotor of Figure 1.5(a)reveals this construction.

17

Chapter 2

The Numerical Analyses

This work used three large software analyses: the rotorcraft comprehensive analysis codeHOST, the structural finite element solver MSC.Marc and the CFD method elsA.

HOST is a work tool at the aeromechanical engineering departments of Eurocopter.ONERA has made and continues to make contributions to the development of HOST.

The choice of MSC.Marc, a structural finite element solver, was done prior to thisthesis and it was based mostly on its non-linear analysis capability. This is a must inorder to handle the large displacements, rotations and elastic deflections of a rotor blade.

In the case of the CFD software, elsA is the most sensible choice for two reasons.First, it is perfectly suited to rotorcraft applications because it integrates state-of-the-art methods in the domain of rotorcraft CFD. Second, elsA is developed at ONERA forresearch purposes but is also used by many of ONERA’s industrial partners.

This chapter presents the numerical analyses HOST, elsA and MSC.Marc, includingfor the latter the approach that was used to apply a general structural finite elementanalysis to rotor dynamics simulation.

An overview of the rotorcraft comprehensive analysis code HOST is offered. TheHOST/CFD coupling developed in 2005 is described and compared to concurrent com-prehensive/CFD analyses.

2.1 Advanced finite element analysis for rotor dy-

namics

It has been seen that the dynamics of a rotating blade feature large amplitude displace-ments and rotations, nonlinear geometrical deformations and nonlinear couplings. Despitethe complexity of rotor dynamics, for most of the rotor designs flying today, beam modelsgive satisfying predictions of their dynamic behaviour. However, future technologies willcall for refined capabilities in structural modeling for reasons that are not strictly relatedto dynamics. Two examples are offered next.

Modern helicopter blades are made of composite materials. Composite materials not

18

2.2 The finite element solver MSC.Marc

only are lightweight, but also offer a high degree of customisation. By appropriate plyorientation, the designer can exploit the anisotropic properties of the fibers and renderthe blade more aeroelastically stable. This potential for aeroelastic tailoring has not beenadopted by helicopter manufacturers yet, who rely on composite blades mostly becauseof their excellent fatigue characteristics, damage tolerance and reduced weight. But asaeroelastic tailoring technologies become more mature, there will be an increasing needin being able to model complex material behaviours.

Another technology likely to reach soon a satisfying degree of maturity is smart ma-terials. The morphing of the blades by piezoelectric devices or alike will bring enormousprogress in terms of flight performance, vibration reduction and mechanical design. Mod-eling a blade with morphing cross-sections requires to use at least 2D models such asshells or plates.


MSC.Marc is a commercial off-the-shelf software for structural finite element analysis.There are three good reasons to use it. The first is its non-linear analysis capability, anecessary requirement due to the large rotations and displacements that characterize rotordynamics. A second reason is the availability of user subroutines in Fortran that ease thetask of coupling MSC.Marc with external software (e.g. aerodynamics solvers). Thirdly,ONERA already had positive experience with the use of MSC.Marc for the analysis ofrotor blade dynamics [55][56]. Another asset of MSC.Marc is that, since it belongs toMSC software, the same company that develops Nastran, another popular structuralfinite element package, finite element models for Nastran can be easily imported intoMSC.Marc.

On the minus side, the disadvantages related to the use of a commercial softwarestem from the lack of complete mastery on the numerical procedures and algorithmsimplemented in it. The source code is not available and the documentation is not alwaysas comprehensive or even accurate as one may desire. The consequences are that someanalysis options, short of exhaustive documentation, demand a time-consuming approachbased on trial-and-error to work out their application.

In this work MSC.Marc was used to solve transient dynamics problems via directintegration of the nonlinear equation of motion,

Mu+Cu+K(u)u = F (u), (2.1)

where M is the mass matrix, C the damping matrix, K the stiffness matrix (of whichmore below), F the structural loading and u the nodal displacement. The overdot denotestime derivative.

(2.1) is said to be nonlinear because the stiffness matrix depends on the deformationof the structure system. Indeed, K(u) is called the tangent nonlinear stiffness matrix andincludes three terms: (1) the linear or small displacement stiffness matrix; (2) the geo-metric or initial stress stiffening matrix (centrifugal stiffening); and (3) the large or initial

19

2. THE NUMERICAL ANALYSES

displacement matrix. In addition, the structural loading (airloads, centrifugal loads) alsodepends on the deformation.

A typical operator for integrating the equation of motion in time is the Newmarkscheme. The Newmark scheme is available in MSC.Marc and was used in this work. Itsgeneralized form is

un+1 = un + (1− γ) ∆tun + γ∆tun+1

un+1 = un + ∆tun +

(1

2− β

)∆t2un + β∆t2un+1 (2.2)

where u is the displacement vector, a dot denotes time derivative and ∆t is the time step,so that time tn+1 = tn + ∆t. The particular form of the the Newmark scheme for γ = 1/2and β = 1/4, applied to the equation of motion (2.1) results in(

4

∆t2M +

2

∆tC +K

)∆u = F n+1 −Rn +M

(un +

4

∆tun)

+Cun (2.3)

where the internal load-vector R is obtained from the internal stresses as

R =∑elem

∫V

βTσdv, (2.4)

where β is the strain-displacement relation and σ the stress tensor. (2.4) adds, for everyelement, the integration of the internal stresses over the volume of the element.

The solution of (2.3) for ∆u allows the implicit solution of the system,

un+1i = un + ∆ui (2.5)

Notice that the operator (2.3) includes K(u), the tangent nonlinear stiffness matrix.Hence, (2.3) is solved iteratively by a full Newton-Raphson approach, in which the tangentstiffness matrix is reassembled at each iteration i. It is important to reassemble thestiffness matrix at each iteration in order to take accurately into account the geometricalnonlinear phenomena of the rotor dynamics.

In this work the kinematics of deformation were solved by MSC.Marc using a totalLagrangian formulation. The term Lagrangian implies that the finite element mesh isattached to the material. And in the total Lagrangian approach the equilibrium is ex-pressed with the original undeformed state as the reference. This method is appropriatefor rotor dynamics because of its moderately large rotations yet small and elastic strains.

MSC.Marc includes an extensive element library. This work used beam, shell and solidelements. Although rotor blades undergo moderately large bending deflections, they arealso very elastic. Therefore, the deformations are considered as a large displacement, smallstrain problem. The advantage of having small strains is that changes in the stress-strainlaw of elasticity of the materials can be neglected.

20


Centrifugal and Coriolis forces. In simulating the dynamic behaviour of a rotorblade, it is essential to take into account the effect of the rotor rotational speed. Whensolving the finite element problem in a non-inertial frame, the centrifugal and Coriolisforces appear in the equations of motion as three additional terms,

Mu+ (C − ΩCc) u+(K − Ω2Kc

)u = F + Ω2Kcu0, (2.6)

where u0 is the position vector of the stationary system. The term ΩCcu is commonlyreferred to as the Coriolis force and the term Ω2Kc (u0 +Bu) is referred to as the cen-trifugal force. Note that the centrifugal force term in the right hand side of the equationbehaves as a body force during the simulations.

This thesis detected that MSC.Marc fails to account for the Coriolis forces in itstime-domain simulations (whereas it should). This problem emerged after a simple inves-tigation summarized in Section 2.2.1.6.

2.2.1 Adapting finite element analysis to rotor modeling

2.2.1.1 Rotor articulations.

The rotors that have been modeled in this study are research rotors in which the bladesare linked to the rotor hub by three articulations: the flap hinge, the lag hinge and thepitch bearing. These articulations were schematically described by Figure 1.4, in whichthe flap and lag hinges are confounded as a cardan joint.

In the finite element models, the three articulations were all concentrated into aspherical joint because this option is easier than modeling the separate articulations.A schematic of this system is shown in Figure 2.1. It has been checked that the sphericaljoint approximation does not alter significantly the modeshapes of the blade by comparingthe predicted non-rotating eigenfrequencies with measured data. No pitch-flap couplingis introduced either because the pitch articulation is collocated with the flap hinge. Thespherical joint is modeled by two matching nodes (but belonging to contiguous beamelements) free to rotate but rigidly attached in displacement.

2.2.1.2 The blade pitch control system

Like in a real rotor, each of the blades in the finite element model is equipped with apitch control system composed of a pitch horn articulated with a pushrod, as shown inFigure 2.2. Since the mechanical properties of the wind tunnel rods were not available, itwas preferred to make them rigid and nearly weightless, so that they would not deform norundergo important inertial loads. The system works by prescribing a vertical displacementupon the lower node of the pushrod (right at the point where it would be attached to theswashplate). Horizontal displacements for this node are blocked, so it works as a slideelement. The pushrod is linked to the pitch horn by a spherical joint. The pitch horn issolidly linked to the blade and acts as a lever.

21


Figure 2.1: Schematic of the spherical joint used in the finite element models.

Figure 2.2: The blade pitch control system in the finite element model.

22


For a null pitch angle (θ = 0), the pitch horn is assumed to be horizontal and perpen-dicular to the pushrod. Thus, for a desired pitch input angle θ, the vertical displacementh to be prescribed on the pushrod is a simple trigonometric relation

α = arcsin

(r2(1− cosθ)

r1

)(2.7)

h = d+ e = r2sinθ + r1(1− cosα) (2.8)

2.2.1.3 The lead-lag damper

Like in a real rotor, each of the blades in the finite element model is equipped with alead-lag spring-damper system.

The spring/damper system is declared in MSC.Marc as a spring equipped with adamping dashpot. One end of the spring/damper system is attached to the blade root,a few centimeters outboard of the articulations, and the other end is attached to therotating frame of the blade. The force generated by this system is proportional to thespring stiffness k times the lag displacement ∆δ plus the damping coefficient c timesthe lag velocity δ of the point of the blade where the spring is attached, as given byF = k∆δ + cδ.

It is sometimes convenient to raise artificially the damping coefficient of the lead-lagdamper at the beginning of a simulation to help stabilize the lag motion, which has a lowfrequency (around 1/3/rev).

2.2.1.4 Initializing the simulation

In the geometry definition of the FE model the blades are oriented with an initial pitch,i.e., the root chord of the blade is rotated with respect to the rotor plane by a certain angle.Ideally, this ‘built-in’ pitch should be close to the typical operating values to minimisethe amount of pitch input into the FE model, because pitch input adds computation costto the FE simulation.

In any case, the pitch setting θ of each blade must be individually adjusted to beready for the coupling with the external airloads. The initial value of the pitch has beenpreviously estimated with an autonomous-HOST computation.

Initializing the simulation consists in setting the initial blade pitch by means of aramp function. Pitch input drives a rigid motion of the entire blade and therefore, pitchvariations must be smooth for the FE solver not to diverge.

The ramp period also serves to damp out the high-frequency oscillations in bladeelongation that arise as a consequence of the onset of the centrifugal loading. Thishigh-frequency damping is provided by the time-integration scheme of the FE solver,as explained later in the time integration section, Section 2.2.3.

If a particular flight condition is to be simulated several times, it would be interestingto save the restart file containing the already initialized solution because this initializationis computationally expensive when using 3D FE models. Restart options are possible in

23


MSC.Marc and this work tried to use them, yet it failed due to obscure problems withlibraries in the user subroutines.

2.2.1.5 Non-inertial frames

In the finite element solver MSC.Marc the standard method to modeling rotating struc-tures is to make use of a non-inertial frame of reference, this is, a frame fixed to therotating body. A non-inertial frame of reference in MSC.Marc is defined by declaring arotation axis that is fixed in space.

An advantage of using rotation-fixed frames of reference is that the specification of theFE model (the input file or card) is simpler compared to that of a rotating structure inan inertial frame. The use of an inertial frame requires more intervention from the user,an aspect that is demonstrated later in Section 2.2.2, where the inertial frame approachwas tested.

The disadvantage of using rotation-fixed frames of reference is that phenomena suchas rotor shaft vibration -and thereby change of the spatial orientation of the rotation axis-are neglected.

For the isolated rotor applications of this study a fixed axis of rotation could beassumed. However, shaft dynamics cannot always be neglected. Indeed, future work willmodel complete helicopter configurations (fuselage plus rotor), and for that the vibrationof the rotor shaft will have to be taken into account.

Worries about the capacity of MSC.Marc to simulate structures rotating around amoving axis were raised. It was then decided to conduct a small investigation, based onthe use of inertial frames, see Section 2.2.2, on how rotor shaft vibration could one daybe taken into account in MSC.Marc’s simulations.

2.2.1.6 Testing Coriolis forces in the non-inertial frame

A small investigation was carried to check that, when MSC.Marc rotor models are solvedin a non-inertial frame, the Coriolis forces are correctly calculated.

The test was conducted with a very simple model representing an isotropic blade withconstant and symmetric cross-section. The blade was straight but presented a certainangle with respect to the horizontal plane, which would be the rotor spin plane. Thelength of the blade was equal to 1m and the FE model was discretised into four beamelements of 0.25m each with constant mechanical properties. The bending stiffness wasequal to 10000Nm2. The total weight of the beam was 20kg. The rotation speed Ω wasequal to 1000rev/min. Figure 2.3 shows a side view of the undeformed geometry of theblade (line coloured in lilac).

According to the MSC.Marc manual, a distributed load of type 103 (which includescentrifugal and Coriolis loading after entering the angular speed ω2 in rad/s and definingan axis of rotation) was applied to all four beam elements of the blade. One end of theblade was clamped to the rotation axis, which was vertical. The other end was free.

24


A simulation in the time-domain was performed. The centrifugal loading was notramped but fully applied from time zero. Since the undeformed blade lay out of therotation plane, the onset of the centrifugal loading triggered large amplitude oscillationsin flapping. The amplitude of the flapping oscillations was almost equal to 0.36m (36%of the blade span) and the frequency slightly superior to 1/rev.

Figure 2.3: No Coriolis forces showing up.

As a consequence of the blade velocity relative to the rotating-frame associated to theflapping motion, it was expected to observe in-plane motion due to the Coriolis force.The Coriolis acceleration is twice the cross product of the angular velocity vector and thevelocity vector relative to the rotating frame.

Disappointingly, no Coriolis coupling appeared. The blade flapped significantly, butthis motion remained confined in the xz plane, with a velocity v = (vx, 0, vz). The crossproduct of this velocity vector with the angular velocity vector on the z-axis (Ω = (0, 0, ω))should have generated a Coriolis force acting on the y-direction. As a result of these y-direction Coriolis forces, the blade should have underwent displacements, however small,in the y-direction. Yet y-displacements were strictly zero throughout the simulation. Ashot of the deformed blade is shown in Figure 2.3. The zeros above the deformed blade-in black- are the y-displacements.

The fact that no coupling is observed between flap and lag motion due to Coriolisforces suggests a malfunctioning of the Coriolis loading in MSC.Marc’s simulations in anon-inertial frame in the time-domain. This problem has been duly reported to the MSCSoftware company, who is now fixing it.

This result will need to be taken into account when assessing the numerical solutions.

25


(a) Simple rotor model (b) Gyroscopic precession

Figure 2.4: Model for testing MSC.Marc in a fixed frame.

2.2.2 About the MSC.Marc simulations in an inertial frame

The solution to simulate rotors rotating around a moving axis consists in using an inertialframe of reference. Instead of declaring a fixed axis of rotation, it is possible to prescribean angular velocity to a node and let this node drive the rotation of the rest of thestructure. Then, by prescribing the angular velocity on a node of the rotor shaft, itis possible to drive the rotation of the entire rotor while allowing for shaft vibration.This approach has been tested with a simple exercise; the model shown in Figure 2.4(a)represents an horizontal-axis rotor. A counter-clockwise angular velocity is prescribedthrough the right end of the rotor shaft (see red arrows). This same node is fixed inall three displacements but free to rotate. Gravity is prescribed along the vertical axis(Z-axis). As the simulation starts, the rotor head starts to drop, but then the gyroscopiceffect induces the precession ω towards the right, shown in Figure 2.4(b), in agreementwith the vectorial product ω = M g ∧Hs, where M g is the gravity moment and Hs therotor spin angular momentum.

Unfortunately, the use of inertial frames in which the structure rotates brings addi-tional issues.

One of them is the angular acceleration. The large rotational inertia of the rotor hasto be accelerated from rest to the nominal rotation speed, which at best would be com-putationally expensive, particularly for 3D FE models. Indeed, it would take too long tostabilize the response, especially for the low-frequency lead-lag motion. Angular acceler-ation can be bypassed by prescribing an initial velocity -corresponding to the stabilizedrotation regime- to each of the nodes of the mesh at the simulation start. But these initialvelocities would have to be calculated by the user, which for large FE models is a tediousand error-prone operation.

26


A second issue is related to the frames of reference for the input and output quanti-ties during the coupling. Even with the use of strategically placed local frames, transferoperations would not be as direct as when using a rotating rotor frame.

In conclusion, the non-inertial frame used in MSC.Marc for the simulation of isolatedrotors was preferred to the inertial frame because: (1) from a dynamics point of view it iseasier to initialize because accelerations due to rotor torque are avoided; (2) it simplifiesthe specification or input card of the finite element model; and (3) it provides user-friendlier frames of reference for inputs/outputs.

2.2.3 Time integration

For dynamic transient analysis, MSC.Marc offers three time-integration operators:

1. Newmark-beta operator

2. Houbolt operator

3. Central difference operator

The central difference operator is an explicit method limited to rapid dynamics analy-ses with very small time-steps, like crash-shock phenomena. The Houbolt operator isinteresting in that it provides high-frequency damping, but so does the Newmark schemewith the right choice of parameters and at a better accuracy. The Newmark scheme isa widely accepted time-integrator for structural dynamics problems because its implicitcharacter guarantees the stability of the response irrespectively of the time-step, whereasthe stability of explicit methods is conditioned to the use of small time-steps relative tothe frequencies of the studied problem. These frequencies are not only physical, but alsoinclude the very high frequencies that follow the spatial discretisation used in the finiteelement method.

The Newmark algorithm is based on a set of two relations expressing the forwarddisplacement un+1 and velocity un+1 in terms of their current values and the forward andcurrent values of the acceleration,

un+1 = un + (1− γ)hun + γhun+1

un+1 = un + hun +

(1

2− β

)h2un + βh2un+1 (2.9)

where h is the time step, so that tn+1 = tn+h. A dot denotes time derivative. The choiceof the parameters β and γ determines the accuracy and stability properties of the scheme.

This work used γ = 5/6 and β = 4/9. This choice of parameters provides good accu-racy while introducing numerical damping for the higher-frequencies. How much exactly

27


is argued in the coming paragraphs, following the analysis by Geradin and Rixen in [28].

Writing the equations of motion at times tn and tn+1

Mun = −Cun −Kun − F n

Mun+1 = −Cun+1 −Kun+1 − F n+1 (2.10)

and introducing the Newmark relations, previously multiplied by M , in them yields

Mun+1 = Mun + h (1− γ) (−Cun −Kun + F n) + hγ(−Cun+1 −Kun+1 + F n+1

)Mun+1 = Mun + hMun + h2

(1

2− β

)(−Cun −Kun + F n)

+ h2β(−Cun+1 −Kun+1 + F n+1

)(2.11)

By noting qT = [uTuT ], Eqs. 2.11 can be recast in matrix form

qn+1 = Aqn + gn+1 (2.12)

where

A = H1−1H0 (2.13)

gn+1 = H1−1bn+1 (2.14)

H1 =

[M + γhC γhKβh2C M + βh2k

](2.15)

H0 =

[(1− γ)hC −M (1− γ)hK(12− β

)h2C − hM

(12− β

)h2K −M

](2.16)

bn+1 =

[(1− γ)hF n + γhF n+1(12− β

)h2F n + βh2F n+1

](2.17)

Matrix A is called the amplification matrix of the integration scheme. Stability analysescan be performed from its eigenvalues.

For zero structural damping (C = 0), and decomposing the equations of motion intoa system of normalized decoupled equations, the amplification matrix can be written forevery mode of angular frequency ωj as

A =

[1− γ ω2h2

1+βω2h2 −ω2h(

1− γ2

ω2h2

1+βω2h2

)h

1+βω2h2 1− 12

ω2h2

1+βω2h2

](2.18)

its eigenvalues λi being the solution of its characteristic equation

λ2 − λ

2−(γ +

1

2

)(ω2h2

1 + βω2h2

)︸︷︷︸

ε2

+ 1−(γ − 1

2

)(ω2h2

1 + βω2h2

)︸︷︷︸

ε2

= 0 (2.19)

28


The eigenvalues can be expressedλ1,2 = ρe±iϕ (2.20)

where

ρ =

√1−

(γ − 1

2

)ε2 (2.21)

ϕ = atan

ε√

1− 14

(γ + 1

2

)2ε2

1− 12

(γ + 1

2

)ε2

(2.22)

The algorithm will only be stable if ρ ≤ 1, which implies γ ≥ 12. Furthermore, the

characteristic equation (Eq. 2.19) will only have complex eigenvalues if(γ +

1

2

)2

− 4β ≤ 4

ω2h2(2.23)

which holds true for any frequency parameter ωh if

β ≥ 1

4

(γ +

1

2

)2

(2.24)

Numerical damping can be introduced in the Newmark operator by making γ > 1/2by a factor α, γ = 1/2 + α, but it deteriorates the accuracy of the numerical response.This deterioration is quantified next by analyzing the spectral radius of the algorithm, itsperiod error and its numerical damping as a function of the frequency parameter ωh.

Three cases have been considered. The first is the baseline, non-damped, Newmarkalgorithm (α = 0). The second uses α = 0.05. The third uses the factor chosen in the

present work for MSC.Marc, α = 1/3. In all cases γ = 1/2 + α and β = 14

(γ + 1

2

)2.

As α increases, so does the damping in the higher frequencies. In rotor dynamics, onlythe first eight to ten modes are physically meaningful. The highest frequencies of interestare usually around 8/rev. For a rotor rotating at Ω = 1000rev/min ' 100rad/s, thismeans ω = 800rad/s. The time step is constrained by the time-accurate coupling withthe CFD method, which imposes time steps of the order of h ' 2 · 10−4s. Consequently,the frequency parameter ωh takes values around 0.16.

The spectral radius is equal to the modulus of the largest eigenvalue λi. It has anasymptotic behaviour as ωh → ∞, which allows to study the stability of the algo-rithm over the whole frequency domain.

Figure 2.5 shows the spectral radius of the Newmark operator for α = 0, α = 0.05and α = 1/3. In the concerned frequency factors, ωh ' 0.16, the spectral radiusof the damped algorithm with α = 1/3 is still very close to that of the undampedalgorithm. So for the low frequencies the accuracy goes nearly unaffected.

29


Figure 2.5: Spectral radius of the Newmark algorithm.

The period error compares the period of the numerical response to that of the exactone.

∆T

T=ωh

ϕ− 1 (2.25)

Figure 2.6 shows again how at the frequencies of interest in rotor dynamics (aroundωh ' 0.16) the most damped solution is still very close to the undamped one.

Figure 2.6: Period error of the Newmark algorithm.

The numerical damping is obtained from

ξ = − ln(|λ|)ϕ

(2.26)

30

2.3 The CFD code elsA

Figure 2.7 shows the numerical damping. Here lie the starkest differences betweendamping levels at the frequencies of concern. But the absolute value is still reason-able.

Figure 2.7: Numerical damping of the Newmark algorithm.

In conclusion, the amount of numerical damping introduced in the Newmark scheme bytaking α = 1/3 and then γ = 1/2 + α = 5/6 and β = 4/9 is highly suitable for therotor dynamics problems treated here. In the range of frequencies that dominate theblade response, the accuracy of the algorithm is nearly as good as that of the undampedscheme. Yet the undesirable higher-frequencies, such as those triggered by the centrifugalloading onset or initial blade pitch adjustment, are strongly damped.

2.3 The CFD code elsA

The development of elsA software was initiated by ONERA in 1997. This multiapplicationCFD software solves the Euler or Reynolds-Averaged Navier-Stokes equations for all theaerospace configurations from the low subsonic regime to hypersonic, including fixed wing,rotary wing, turbomachinery, space launcher and missile configurations. It uses cell-centered finite volume discretisation for multi-block meshes, including overset and patchedgrid capabilities. It has a wide range of numerical techniques available for space and timeresolution, as well as for turbulence modeling.

For rotor applications, the resolution in space is achieved by using a 2nd order centereddiscretisation in space with Jameson’s artificial viscosity. For the resolution in time, thedual time-stepping method or the Gear implicit sub-iterative method are available toconverge towards a 2nd order accurate solution. These techniques allow the use of largeazimuthal steps: at least ∆Ψ = 1.2deg for the present simulations.

31


The mesh deformation technique for the unsteady rotor aerodynamics is presented forconvenience later in Section 4.3.1.

Reference [47] is a recent paper by Renaud et al. where the state-of-the-art capabilitiesof elsA for rotor simulation are exploited to analyze blade-vortex interaction phenomena.[13] is another recent paper summarizing the status of elsA for all kind of applications.

2.4 The rotorcraft comprehensive analysis code HOST

2.4.1 Introduction

Why comprehensive codes for rotorcraft analysis

The design and analysis of rotary-wing aircraft demands numerical tools capable of mul-tidisciplinary analysis. In fact, the modelling of the in-flight behaviour of a helicopterinvolves at least three disciplines: (1) aerodynamics; (2) structural dynamics; and (3)flight controls. Furthermore, three types of analyses are required: (1) stabilized periodicresponse; (2) transient response in the time domain; and (3) stability in the frequencydomain.

Comprehensive codes owe their name to their capacity to encompass the three afore-mentioned technical disciplines and the three types of analysis. Thus, these codes canperform a wide variety of analyses including vehicle performance, aerodynamics and ro-tor loads, vibration, control system dynamics, aeroelastic stability, and flight dynamics,among others.

The rotorcraft comprehensive analysis code HOST

The rotorcraft comprehensive code HOST (Helicopter Overall Simulation Tool) is devel-oped and used by the helicopter manufacturer Eurocopter. It serves to address a widevariety of rotorcraft analyses and design problems, such as vibration, aerodynamics, hubloads, aeroelastic stability or flight mechanics. It can model a large variety of rotorcraftsystems, from an isolated rotor to the complete aircraft, including helicopters, tiltrotorsand alike. In the industry environment these problems need to be addressed in a practicalfashion, with a design-oriented usability and minimal computational costs.

HOST can perform three types of analyses: simulation, equilibrium and stability. Thefirst two work by solving a set of nonlinear equations of motion, whereas stability problemssolve linearized equations of motion around a state previously obtained with the nonlinearequations of motion. This work will only address time-simulation analyses.

HOST describes a rotorcraft system by picking and assembling models from a modelpool. Each model is characterised by a specific set of degrees of freedom, the equationshandling these degrees of freedom and their control options. Among the most basic modelsone can cite the blade model, the rotor hub model, the swashplate model, the lead-lagdamper model, the fuselage model or the aerodynamics model.

32


The models are hierarchically organized following a parent-child scheme. A parentmodel can have several child nodes, e.g., the rotor hub parent model can have n blade childmodels. Each model has as many input/output nodes as attachments to other models.It is through these input/output nodes that forces and motions are communicated to theimmediate neighbouring model(s), as explained shortly.

Irrespectively of the problem category -i.e., simulation, equilibrium or stability-, HOSTapplies the same basic algorithm. This algorithm is used to obtain the second time-derivative of the state vector -i.e., state acceleration- from the values of the degrees offreedom composing the state vector and their first time-derivatives.

Once HOST has calculated the state vector acceleration, it then proceeds to the specifictreatment of the problem under study. In the case of a simulation, acceleration is usedto integrate the solution in time. This HOST basic algorithm is presented in the nextsection.

A more exhaustive presentation of HOST can be found in [9].

2.4.2 The basic HOST algorithm

The basic HOST algorithm consists in sweeping up and down the model hierarchy, seeFigure 2.8. The first sweep is called the kinematics sweep. Motions are passed fromparent to child models by stepping up the hierarchy. The second sweep is called the loadssweep. Forces are passed from child to parent models by stepping down the hierarchy.Forces include aerodynamic, inertial and elastic loading.

Figure 2.8: The basic HOST algorithm.

Once the state vector acceleration has been obtained, the simulation analysis integratesthe state vector from time t to time t + ∆t. Several methods spanning from 1st to 4th

order are available for this integration: Euler, Gill, Runge-Kutta, Adams-Bashforth. Allof them are explicit integration methods.

33


2.4.3 Aerodynamics

Rotor aerodynamics. The aerodynamics model for rotor blades in HOST is based onblade element theory. The blade is discretized into a series of spanstations. For eachspanstation, its angle of attack and Mach number are deduced from the kinematics of theblade. Table lookup is then used to obtain the aerodynamic coefficients as a function ofangle of attack, Mach number and profile type. The tables containing the aerodynamiccoefficients were obtained experimentally by testing 2D profiles in a wind tunnel.

If the unsteady aerodynamics option is being used, HOST will also use the sectionalaccelerations in feathering and heaving motion to add to the pitching moment a termfrom Theodorsen’s unsteady theory.

Additionally, corrections are available in order to take into account, among others,sweep angle, 3D rotational effects, Reynolds number effects or dynamic stall.

Induced velocity. The flow velocity with respect to the profile due to the structuralmotion is only a component of the profile’s total velocity with respect to the air. In ad-dition, it is necessary to calculate the flow velocity due to rotor inflow, which is stronglyaffected by the rotor wake, before making use of the table lookup methods. There areseveral models in HOST for the induced velocity, including analytic, free-wake and pre-scribed wake methods. Their details are out of the scope of this document and can befound in [16]. But it is important to outline the fundamental variables that determinethe induced velocity because these variables will be exchanged when HOST airloads arecoupled to an external structure solver.

Induced velocity is related to the quantity of air convected through the rotor. Fora given rotor disk area and air density, the flow momentum induced by the rotor isproportional to the induced velocity. Given that momentum dictates the generated thrust,it follows that the induced velocity can be deduced from thrust. And since thrust iscorrelated with the flapping of the blades, flapping can be used for the calculation ofinduced velocity.

This work used the Meijer-Drees induced velocity model [24] because its simplicitymakes it the fastest -and default- method in HOST. The Meijer-Drees model assumesthat the induced velocities vi are a linear function of the spanwise position r/R and aharmonic function (with only the 1st harmonic) of the azimuth Ψ,

vi(Ψ, r) = vi0 +r

R(vi1CcosΨ + vi1SsinΨ) (2.27)

where vi0, vi1C and vi1S are functions of the rotor thrust. More particularly, the cycliccomponents vi1C and vi1S depend on the first harmonics of the flapping angle of the blade,β1C and β1S. Consequently, HOST needs the blade flapping motion in order to correctlycalculate the induced velocity.

Should the prescribed-wake (METAR) or free-wake (MESIR) methods be used, in-duced velocity would be then calculated by lifting line theory, equilibrating the bladecirculation with that of the wake by means of the Biot-Savart law. In those cases, the

34


flapping angle should not be any longer necessary. Instead, blade position and velocity areused. These methods have not been used in this work because they are time-consumingand the extra-accuracy in the airloads was not required.

Summarizing, HOST needs the following variables to calculate the airloads on a rotorblade: quarter-chord velocity and acceleration and blade flapping angle.

2.4.4 Structural dynamics

This section summarizes the method implemented in HOST to calculate the dynamics ofthe rotor blades. It will be seen that beam theory is used, but not with finite elementtheory. Instead of the classical assemblies of mass and stiffness matrices using shapefunctions, HOST has a practical engineering-approach that is described next.

There are two methods in HOST for the modeling of the blade dynamics: the rigidblade and the soft blade methods.

What both have in common is that the blade is represented as an assembly of rigidsegments connected by virtual joints and distributed along a straight axis. This straightaxis is assumed to be colinear with the blade’s axis, which is defined as the pitch axis.Center of gravity and elastic axis offsets are still enabled.

During the kinematics sweep, the motion of the j elements constituting a blade isreconstructed from the rotor head and sweeping the blade segments until the blade tip.

V j+1 = V j + Ωj ∧ drj+1 (2.28)

Ωj+1 = Ωj + ωj+1 (2.29)

Γj+1 = Γj + Ωj ∧ drj+1 + Ωj ∧ (Ωj ∧ drj+1) (2.30)

Ωj+1 = Ωj + ωj+1 + Ωj ∧ ωj+1 (2.31)

where V j and Ωj are the j-th joint’s velocity and angular velocity, respectively, drj+1 isthe position of the joint j+1 with respect to the joint j. ω is the relative angular velocitybetween two segments. The dot denotes time derivative.

The rigid blade method allows no relative motion between segments at the joints.Therefore, ωj, ωj are only non-zero at the flap, lag and pitch mechanical articulations.Hence, the blades hinge rigidly around the flap, lag and pitch articulations.

The soft blade or elastic model is based on beam theory. It models the bending (inboth flap and lag) and torsion deformation, but axial elongation and shear are neglected.Modal decomposition is used to further reduce the number of unknowns. For every jointa lag rotation is first calculated and then followed by a flap rotation. The pitch rotation isnot calculated joint after joint but rather as a single total rotation around the pitch axisoutboard of the pitch bearing. The reconstruction of these rotations provides the defor-mation and displacement of the elastic axis of the blade. Since the torsion deformation isrestricted to a rotation around the elastic axis (which is also the blade’s axis), it followsthat the elastic axis undergoes no displacement due to torsion.

35


Let (δkj , βkj , θ

kj ) denote the rotations in lag, flap and torsion at the j-th joint of the

blade for the k-th mode. The torsion rotation and the incremental bending rotations atthe j-th joint are expressed as

θj =∑k

qkθkj (2.32)

dδj =∑k

qk(δkj − δkj−1

)(2.33)

dβj =∑k

qk(βkj − βkj−1

)(2.34)

where qk is the generalized coordinate of the k-th mode. The modal decomposition

notation is also used for the velocity vector(θj, dδj, dβj

)and the acceleration vector(

θj, dδj, dβj

).

The above angles are used in the soft blade approach to compose the angular velocity

ωj =(θj, dβj, dδj

)Tand the angular acceleration ωj =

(θj, dβj, dδj

)Tof the j-th joint

rotations, where θj = 0 except in the pitch bearing (for there is no incremental torsionbetween two elements), the motion of the articulations is calculated during the kinematicssweep using (2.28)-(2.31).

During the loads sweep, the aerodynamic, inertial and elastic loads that follow thedeformation of the blade during the kinematics sweep are calculated. Then, for everymode k, the generalized coordinates of the acceleration are calculated by enforcing theLagrange equation

d

dt

(∂T

∂qk

)− ∂T

∂qk+∂U

∂qk= W k

G, (2.35)

where T is the kinetic energy, U the potential or elastic energy and W kG the work of the

external loads on the k-th mode. The kinetic energy and its time derivative are calculatedfrom the terms (2.28)-(2.31) and the elements’ masses and inertias. The strains (or rather,curvatures) in bending and torsion to calculate the elastic energy are obtained by derivingthe rotation matrices between articulations.

2.4.5 Review of the start point: the HOST/CFD coupling

The CHANCE project. The CHANCE project took place between 1999 and 2005.It was an international cooperation between ONERA, DLR, IAG1 and Eurocopter. Theobjectives of the CHANCE programme were to develop and validate CFD tools for sim-ulating the flowfield around the complete helicopter. An important part of the projectconsisted in performing CFD analyses of isolated rotors taking into account the blades’elasticity. This was achieved by coupling HOST’s blade dynamics model with an externalCFD solver.

1Institut fur Aerodynamik und Gasdynamik, University of Stuttgart.

36


The CFD software was elsA in France and FLOWer in Germany. HOST predictionaccuracy in aeroelastic simulations was improved as a result of the coupling with the CFD.Overviews of the results obtained during the CHANCE project are available in [17] and[19].

The HOST/CFD coupling. The HOST/CFD coupling developed during the CHANCEproject is available in two forms: weak (or loose) and strong (or tight) coupling. In theweak coupling, structure motions and CFD airloads are exchanged on a per revolutionbasis, in the form of periodic solutions. In the strong coupling, structure motions andCFD airloads are exchanged at every time step in the form of instantaneous solutions.Both methods are summarized in the next two subsections.

2.4.5.1 HOST/CFD weak coupling.

This method assumes that the rotor response is periodic. Its algorithm is:

• An estimation of the rotor periodic response is initiated by HOST,

• The rotor periodic state over one revolution is sent to the CFD software, whichcalculates the periodic aerodynamic field,

• This aerodynamic field is then used in HOST to correct its internal aerodynamicsand loads to a new periodic response of the rotor,

• Iterations continue until equilibrium is reached.

The success of this method lies on three key elements:

1. HOST can yield periodic responses, including trim (see below), expressed as aFourier series,

2. HOST continues to calculate airloads using its conventional aerodynamics and onlythe difference between the CFD airloads and the HOST airloads from the previousiteration are used to correct the conventional HOST airloads. This can be writtenas

~F n = ~F nHOST +

(~F n−1CFD − ~F n−1

HOST

)(2.36)

As the solution converges, each of the terms in the preceding equation converges aswell, implying that the total airloads converge to the CFD airloads.

3. The periodic response includes the trim. Trim is the in-flight equilibrium of therotor. It consists in calculating the rotor controls generating the aerodynamic loadsthat satisfy equilibrium. For example, the lift must counter the weight, the propul-sive force must counter the drag and both pitch and roll moments must be zero tokeep a straight level flight.

37


The weak coupling has two advantages. The first is that the rotor trim is achieved by theperiodic analysis in HOST. The second one is that convergence requires less iterations,in terms of CFD rotor revolutions, than the strong coupling. On the minus side, simula-tions are limited to steady flight conditions. No transient (e.g. maneuver) flight can besimulated.

2.4.5.2 HOST/CFD strong coupling.

In the strong or time-accurate coupling, structure motions and CFD airloads are ex-changed at every time step in the form of instantaneous solutions.

This method has a wider scope of application than the weak coupling, since timedependent maneuver conditions can also be analyzed in addition to steady-flight condi-tions. The exchange of structure motions and airloads must be done carefully in order tomaximize the solution accuracy.

In order to produce a time-accurate solution, the main idea in the strong couplingalgorithm is to shift the fluid and structure integrations by half a time-step (theoreticalbackground with literature references is given in Section 4.2). This algorithm can givesecond-order time-accuracy in the HOST-elsA coupling. The algorithm is summarizedbelow

• Advance the structure in time from time tn−1/2 to tn+1/2,

• Predict the structural deflections at tn+1,

• Update the fluid grid to the predicted structural deflection,

• Advance the fluid in time from time tn to tn+1. Output airloads at time tn+1,

• Use the airloads at time tn+1 to time-integrate the structure from tn+1/2 to tn+3/2.

The above cycle is run until stabilization with fixed rotor controls. As a result, the strongcoupling does not yield trimmed solutions. The trim is approached ‘manually’ at the end ofeach run by means of a sensitivity matrix giving the dependence ratio of the trim variableswith respect to the rotor controls. This sensitivity matrix is calculated beforehand withautonomous-HOST computations. More details on rotor trim can be found in Section 4.4.

The HOST/CFD strong coupling procedure developed under the CHANCE programmewas a precursor of the present work and is often cited in the remainder of this document.That is why the coupling’s main features are presented below. An in-depth descriptioncan be found in [18].

Scope of application. The original HOST/CFD strong coupling can simulate isolatedrotor configurations. The rotor hub is assumed rigid and only the motion of the blades isaccounted for. Blade motion has three components: (1) pitch prescribed by the pushrod;(2) rigid displacements in lead-lag and flapping (made possible thanks to the articulations

38


between the blade and the hub); and (3) elastic deflections in torsion (twist), in-planebending (lead-lag) and out-of-plane bending (flapping). All of these blade motion com-ponents are modelled by HOST. The fluid grid in the CFD analysis is deformed followingthe blade motion given by HOST.

At ONERA, the CFD code used for the CHANCE coupling was elsA. It was used tosolve both the Euler equations for inviscid flow and the Reynolds-Averaged Navier-Stokesequations for viscous flow. However, changing the free-flow direction during simulationrun-time is not possible yet. Consequently, maneuver flight simulations are not possibleyet either. Therefore, the analyses performed during CHANCE were limited to steadyflight conditions.

New software modules for HOST and elsA were developed for the strong coupling.Their details are omitted here, but the next paragraph introduces briefly the softwarearchitecture on how HOST and elsA communicated.

Software architecture. HOST and elsA run on two separate computers. Communi-cation is established with a TCP/IP socket connection. The cycle of the communicationprocess is summarized next.

After each time-step HOST sends the structural motion via the socket connection toelsA who is waiting for the data. As soon as elsA acknowledges the data reception, HOSTgoes into ‘wait’ mode, waiting for the next airloads, that are to be calculated by elsA.

In this communication procedure there is no actual master of the coupling. Instead,the two codes switch alternatively from ‘send’ to ‘receive’ modes.

Exchanged data. The fluid/structure interface is defined by a line, the blade quarter-chord line.

During a simulation in the time-domain HOST uses a modal projection of its beammodel. The blade in HOST is discretised with a finite number of span-wise nodes.

The fluid grid in elsA represents the true geometry of the blade. The surface grids arehowever deformed following a line description of the blade motion. elsA integrates thepressure distribution -and shear distribution when viscous analysis is used- over strips ofthe blade surface to yield nodal airloads along the quarter-chord. These are specified asthree forces and three moments.

The nodes of the structure in HOST are prescribed so as to match the location of thespan-stations from which elsA deforms its mesh. This avoids interpolating the structuredata. The nodal airloads given by elsA are located at mid element. HOST also expectsthem at mid element -and not on the nodes for which motion is calculated-. So no airloadsinterpolation is done either.

The surface grids defining the blades are deformed from the motion given by HOST.This motion contains all of the pitch control inputs, rigid and elastic displacements of theblade.

HOST sends, for every blade of the rotor and for every spanstation of a blade, thecurrent coordinates of the quarter-chord plus a direction cosine matrix. The direction

39


cosine matrix expresses the rotation of a frame of reference fixed to the cross-section ofthe distorted blade to a non deformation-dependent frame of reference.

Data management. The HOST/elsA strong coupling was a pioneer in the use of stan-dard data models for code coupling purposes in the rotorcraft community. A data modeldefines how data is represented, stored and manipulated. The reason for using a pub-lic standard is that it boosts the interoperability because procedures and results can beeasily exchanged with other users of the same data model. Public data models have alsobeen used -and developed- in this new work. This point is addressed in more detail inSection 3.3.2.

2.4.5.3 Results and references.

The CHANCE programme gave rise to a large host of publications. A non-exhaustive listof references related to CHANCE follows.

Two PhD theses, focused on code coupling, were completed in the frame of theCHANCE project: Servera’s in 2002 [49] and Altmikus’ in 2004 [5]. Servera et al.presented in 2000 [50] an early weak coupling of HOST with the Euler code WAVES.Altmikus et al. presented in 2000 [3] and then in 2002 [2] studies on the time-wise accu-racy of strongly coupled procedures. The same author, still in 2002, presented in [4] anoften-cited work comparing the weak and strong coupling procedures, to find they gavethe same results for the application studied.

Cantaloube and Beaumier documented in 2001 [14] the implementation of the ALEtechnique for mesh deformation in elsA.

Beaumier produced in 2003 [6] and 2005 [8] two comprehensive technical reports onHOST/CFD coupling.

Pomin and Wagner presented in 2002 [44] strong coupling results of the 7A rotor.Pahlke and van der Wall reported first in 2002 [36] and then in 2005 [37] weak couplingsimulations using the S4 rotor simulation code and the CFD code FLOWer.

Summary results of the weak and strong HOST/elsA couplings were presented byBeaumier et al. in 2005 [7]. Still in 2005, Poinot et al. presented the application of CGNSin rotorcraft coupling [42]. Finally, CHANCE results overviews were presented in twoconferences in 2005 by Costes et al. [17][19].

2.4.6 Concurrent comprehensive/CFD couplings

Coupling rotorcraft comprehensive analyses with external CFD methods to improve theaccuracy of the airloads and obtain better wake descriptions has been a thriving researchactivity in the recent years, thanks to the emergence of reliable unsteady Euler/RANSsolvers. Several other research teams working on rotary-wing aeroelasticity have developedcomprehensive/CFD couplings that are similar to the HOST/CFD coupling that has beenjust presented.

40


However, none of the previous or concurrent couplings had attempted yet to use 3Dfinite element based structural dynamics in their aeroelastic simulations. According towhat is published, all aeroelastic simulations developed so far use beam theory for thedynamics modeling. Hence, this work has introduced for the very first time advanced 3DFE methods in rotorcraft aeroelasticity.

Another ‘first’ of this work is its capability to adjust the rotor controls during a time-accurate coupled simulation. Time-accurate couplings -data exchanges done at each timestep- are criticized by the literature for not being able to modify the rotor controls andhence, for not being able to yield solutions that respect a set of target aerodynamic forces.

The publications cited below are given to draw an approximate picture of the globalactivities in code coupling for the aeroelastic analysis of rotorcraft. But they do notconstitute the groundwork of the present study because their methods are already verysimilar to those used for the HOST/CFD couplings.

The coming paragraphs present a selection of concurrent comprehensive/CFD cou-plings on the basis of publication assiduity. In this respect, the most active teams outsideEurope are three: (1) the Aeroflightdynamics Directorate at the US Army Research, De-velopment & Engineering Command at NASA Ames Center; (2) The Georgia Institute ofTechnology; and (3) the University of Maryland. Their publications share two common-alities: aeroelastic analysis is applied to isolated rotor configurations; and the blades aremodeled with beam theory.

Potsdam et al., from NASA Ames, presented in a paper [45] and subsequently in anarticle [46] a loose coupling between the rotorcraft comprehensive analysis CAMRAD IIand the CFD solver OVERFLOW-D. In that work the coupling was of loose (or weak)type, this is, the codes are coupled on a per revolution, periodic basis. The RANS CFDgrid contained 26.1 million points. By comparison, the present work used RANS gridsof 2.1 million points. Results are calculated for the UH-60A rotor (for which it existsan extensive flight-test database) for three flight conditions: high-speed, low-speed andhigh-thrust. The agreement of the results with experimental data is very good for thefirst two flight conditions (though the mean is systematically removed from the pitchingmoment comparisons); for the high-thrust configuration agreement is still respectable.

Lim et al., from NASA Ames, used comprehensive/CFD loose coupling in [33] tostudy blade-vortex interaction (BVI) airloads that dominate descending forward flight.The coupled codes were CAMRAD II and OVERFLOW-2. Since BVI requires a fineconservation of the vortices shed by the blade tips, refined RANS grids of up to 107million points were used.

Nygaard et al., again from NASA Ames, presented a detailed paper [35] documentingthe implementation of both weak and strong coupling between the comprehensive codeRCAS and the CFD solver OVERFLOW-2. Nygaard’s work is noteworthy in that itstudied the convergence properties of the loose coupling.

Datta et al., from the University of Maryland, presented in [23] and [20] an implemen-tation of weak coupling between the comprehensive code UMARC and the CFD method

41


TURNS. A remarkable feature of Datta’s work is that the CFD grid contained only oneblade and the far wake inflow was obtained from a free wake model.

More recently, in 2007, Bhagwat et al., from NASA Ames, applied the RCAS /OVERFLOW-2 coupling in both its tight and weak form to the analysis of a pull-upmaneuver [11]. The idea behind the use of weak coupling for maneuver analysis is to con-sider the maneuver as a series of quasi-steady solutions. Thus, weak coupling was usedto simulate 3 out of the 40 rotor revolutions that made up the maneuver using averagerotor controls (known from flight testing). The CFD grid contained 4.4 million points.The airloads and structural loads using the weak coupling were almost as good as thosewith a time-accurate strong coupling. Bhagwat pursued the work and presented in 2008,with Ormiston [10], an even more detailed analysis of the UH-60 pull up maneuver afterhaving corrected a mistake in the off-grid speeds found in the first release of the study.

Publications by researchers at the Georgia Institute of Technology include those ofAbras and Smith [1] and Smith [52]. The first one discusses methodological aspects of thecomprehensive/CFD couplings when using unstructured grids in CFD solvers, somethingrelatively uncommon: none of the publications cited in this section used unstructuredgrids. The second publication discusses conservation issues in the transfer of structuremotion and airloads between non-matching meshes. It is probably the first publicationto address properly the issue of energy conservation at the fluid/structure for rotorcraftapplications.

Finally, a more detailed review of code coupling for rotorcraft aeroelasticity by Dattaand Johnson can be found in [21]. That publication includes, in addition, a comparisonof the state-of-the-art partitioned analyses for fixed wing aircraft, turbomachines androtorcraft aeroelasticity.

42

Chapter 3

Development of a Framework forCode Coupling

The framework for code coupling is the software infrastructure that allows the two ormore individual applications of a partitioned procedure to exchange information.

The purpose of this chapter is to present the framework for code coupling that wasdeveloped as a necessary first step of the present work. The introduction describes brieflythe problems and modus operandi. The second section introduces the specifications thathad to be observed by the new coupling. Then a third section justifies the solutionsadopted to meet the specifications and how these solutions were implemented. Finally,the here developed software interfaces of MSC.Marc and HOST are documented.

3.1 Introduction

What is the problem. Getting two or more codes, originally conceived as separateapplications, to exchange data at every time-step of their respective simulations is achallenging task; not only with respect to the physics of the global solution, but also froma software point of view.

When this work started there was no software infrastructure upon which HOST, theCSM and the CFD could share their solutions. Hence, it was necessary to set up a softwareinfrastructure, independent of the physics of the problem, into which the different codeswould be plugged.

Modus operandi. The idea of connecting two or more codes to simulate a complexsystem is not new. However, most of these kind of works have been traditionally developedby their would-be users using their own tools and starting from scratch. Nowadays, thesurging popularity of partitioned approaches due to increasingly cheaper computationalpower is bringing standard practices for code coupling within the scientific community.

The objective of these emerging best practices for code coupling is to ensure that soft-ware developments are lasting, general and portable. It is also very important to minimize

43

3. DEVELOPMENT OF A FRAMEWORK FOR CODE COUPLING

the complexity of the coupling environment in order to ease software maintenance andupdating. These practices will be reviewed in more detail shortly.

The development of the new coupling framework adopted these emerging practices,all while observing other requirements agreed by the SHANEL partners. The comingsections detail the development constraints and the adopted solutions to build a solid andreliable coupling framework.

3.2 Coupling specification

The coupling framework must fulfill three major requirements. The first one is thatHOST’s autonomous analysis capacity shall be preserved. The couplings are only ac-tivated when extra-accuracy is needed. The second requirement concerns modularity;HOST may be coupled with the CFD, the CSM or both. External models may representa subcomponent of the HOST model (e.g., HOST performs a complete helicopter simula-tion yet only the main rotor is CSM/CFD modeled). The third requirement is to producea general coupling that is not dependent on the CFD and CSM codes used for this work.The coupling framework shall be compatible with other similar codes and interoperabilityis highly desirable.

3.3 Adopted solutions

In order to match the coupling specification requirements presented in the previous sec-tion, the development of the coupling framework was based on three axes of action: (1)choice and implementation of a programming model; (2) choice and implementation of adata model; and (3) choice of a programming language for the framework. Each of theseaxes of action is explained in a dedicated subsection below.

It will be shown that the chosen options took advantage of existing tools becausethis minimized the amount of new developments. In addition, the choices led to a leansoftware architecture that maximizes modularity and facilitates code maintenance andupdates. Finally, the choices were also done trying to adopt solutions that boosted theinteroperability with other partners.

3.3.1 Programming model

The programming model of the coupling framework is based on component architecture.Component architecture designates an architecture in which applications are neatly sep-arated according to their function. The key idea is that each application participatingin the coupling shall be regarded as a component independent of the other applications.Components are designed with standard, clearly defined interfaces which tend to protectthem from changes in the software environment outside their boundaries. In a simulationinvolving several components, the modification of one of them or even its replacement

44

3.3 Adopted solutions

by a similar one should not affect the other components. In the present developments,in which up to three applications run coupled -HOST, CFD and CSM-, each of theseapplications is considered as a component.

3.3.2 Data model

A data model is an abstract model that describes how data is represented and accessed.The use of a public, common and standardized data model boosts interoperability withother users of that data model. This becomes especially advantageous as the amount ofdata increases.

An example of data model is the CFD General Notation System (CGNS). The CGNSstandard provides a public data model for CFD related data: meshes, solutions,etc. TheCGNS standard is backed by many of the leading industry and research aeronauticsinstitutions worldwide. It consists of a collection of conventions, and free and open soft-ware implementing those conventions. It is self-descriptive, machine-independent, well-documented, and administered by an international steering committee. It is also anAmerican Institute of Aeronautics and Astronautics (AIAA) Recommended Practice.

This work took advantage of the previous experience in the CHANCE project in usingthe CGNS standard for rotorcraft applications [43][42].

Since the CGNS standard was not conceived to describe structure data, a temporaryCGNS-like data model was created for the structure data given by HOST.

Nonetheless, the restriction of CGNS to CFD-related data only is likely to be over soon.An action of the SHANEL project (in which this thesis participated) was to submit anextension proposal of the CGNS norm to describe structure data to the CGNS SteeringCommittee1 [41]. The acceptance of the proposal was under good way at the time ofwriting this document.

This work used the CGNS notation for fluid data and the CGNS-like notation forthe structure data first used in the CHANCE project. The layout of the CGNS fluiddata structures used in this work is shown in Table 3.1. It can be seen that the data isstructured in a tree-like fashion, with parent and children nodes. For instance, a bladeof the rotor constitutes a region. The fluid data attached to a type of region follows apattern. The layout of the CGNS-like structure data is shown in Table 3.2.

In conclusion, the advantages of using CGNS are numerous: it provides a common,consistent and precise specification of the fluid data, together with the libraries and toolsneeded for manipulation. Furthermore, the CGNS standard is portable because it isplatform independent. By having a standard interface, the replacement of the CFD bysimilar, CGNS-compliant ones is greatly facilitated.

1ONERA belongs to the Steering Committee of the CGNS organisation, www.cgns.org.

45


CGNSLibraryVersionStrongCoupling-v01 Blade-1 ZoneType bladeNumber

FlowSolutionGlobal GridLocationIterationNumberRadialPositionChordLengthBladeAzimuthBladeSectionForceXSquareMachBladeSectionForceYSquareMachBladeSectionForceZSquareMachBladeSectionTorqueXSquareMachBladeSectionTorqueYSquareMachBladeSectionTorqueZSquareMach

Blade-2Blade-3Blade-4

Table 3.1: CGNS fluid data tree

CGNSLibraryVersionStrongCoupling-v01 Blade-1 Host bladeNumber

sumTimebladePsipointMotionpointSpeedMotionangularMotionangularSpeedMotionframeTransformMatrix

Blade-2Blade-3Blade-4

Table 3.2: Structure data tree sent by HOST or MSC.Marc

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3.4 Architecture of the framework

3.3.3 Programming language for the framework

The scripting language Python was chosen as programming language for the frameworkfor two reasons. The first is that Python is a powerful tool for data handling. It includesa network server-client communication system, based in the XML-RPC protocol, whichprovides an easy and high-level (i.e., user-friendly) means to perform remote procedurecalls on distant machines. This feature was adapted in this work to distribute the couplingcomponents over a network of computers, as shown later.

In addition, Python is an open-source software, well established and documented andwith a strong support of worldwide developers. Python is a popular choice among prac-titioners of partitioned procedures; and since it is open-source software, many scientistspost the Python applications that they develop on the Internet.

3.4 Architecture of the framework

The architecture of the coupling framework, schematically described by Figure 3.1, illus-trates the application of the concepts of component architecture approach, data modeland distributed computing.

Figure 3.1: Coupling architecture

The architecture of the proposed approach consists of a scaleable, highly-modular setof software tools distributed over various processors. HOST is the keystone of this soft-ware assembly. It keeps its current analysis autonomy, based on simple, computationallyinexpensive aerodynamics and structural models. But when a high-fidelity representationof a particular system is needed (e.g., the main rotor in a complete helicopter simulation)couplings with the CFD and/or CSM models of that system are activated. In all casesHOST provides the aircraft flight control. The coupling is open to any CSM or CFDsolver that can handle the CGNS standard.

Each component has a Python interface that links it to the coupling framework. ThisPython interface also acts as a CGNS translator. The components do not communicate

47


directly with each other but rather, they are all clients of a single server. This serveris a passive application and acts as a data ‘hub’. It simply stores and delivers CGNS-formatted data upon request from a client. Typically, the server stores only the n lastiterations in order to bound memory consumption. The server is always activated andavailable. The user can access the server during the simulation, for debugging purposesfor example.

3.4.1 Network distributed computing

The various codes used in this work (HOST, CFD, CSM) are installed and run on distanthardware platforms with heterogeneous architectures; there are Sun 32-bit, HP-UX 64-bitand other. Distributed computing is an advantage because the simultaneous use of severalprocessors reduces the overall computation time.

The type of couplings herein developed exchange information at each time-step. Upto thousands of data exchanges are performed in the course of a simulation. Exchangingthis information via files would be more time consuming than via network and that iswhy the network option was preferred. Yet a network-based exchange system had to befound.

The Python built-in xmlrpc library provides the network communication. It is basedon remote procedure call technology. This procedure consists in that the functions definedin the data server are executed locally in the server upon remote request from the clients.The functioning of the server is simple: it receives requests to either store data or deliverdata. When one of the coupled codes produces a solution, it requests the server, viathe code interface, to store this solution. The same code will then request to the serverinformation calculated by another code. If that information has not been made availableto the server yet, the request is repeated after a short lapse of time.

3.4.2 Properties of the new coupling framework

As a result of the adopted solutions, the coupling framework developed in this workexhibits the following properties:

Modularity and Generality Applications are independent. For instance, the fluidsolver is unaware whether structure motion is given by HOST or the CSM.

Distributed computing The applications run on different computers and consequently,can run in parallel.

Network communication The applications participating in the coupling communicatevia network up to hundreds of times during a simulation.

Freedom of staggered algorithm The coupling framework sets no constraints on theorder in which data exchanges between the codes are performed. Changing the typeof staggered algorithm is straightforward.

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3.5 An interface for MSC.Marc


This section presents the interface that has been developed for the nonlinear finite elementsolver MSC.Marc. This interface serves to couple MSC.Marc with external codes by ac-complishing two roles. The first role is to input and output data into and from MSC.Marcduring a simulation. The input data can be boundary conditions of multiple types: air-loads, prescribed displacements or velocities, and more. Output data include nodal dis-placements, nodal velocities or reaction forces. The types of accepted input/output dataare defined by MSC.Marc and one must conform to them, since MSC.Marc is a com-mercial off-the-shelf software and thereby modifying the source code is not possible. Thesecond role of the interface is to convert the format of the data between the MSC.Marcformat and a common or standardized format used by all the codes participating in thecoupling.

3.5.1 Coupling regions and the user subroutines

Coupling Regions in MSC.Marc. Coupling regions are that part of the finite ele-ment model where the interaction with the external solver takes places. The couplingregion can consist of elements, surfaces or volumes. On coupling regions, the basic me-chanical quantities (e.g.: coordinates, reaction forces, external forces, displacements) canbe exchanged with an external solver via calls to the user subroutines. Figure 3.2 shows a3D model of a blade in which the leading and trailing edge have been defined as couplingregions and can be seen highlighted.

Figure 3.2: The highlighted leading and trailing edges are the coupling regions.

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User subroutines in MSC.Marc. MSC.Marc has a set of user subroutines that pro-vide an application programming interface (API) to couple MSC.Marc with external nu-merical solvers. These user subroutines are written in Fortran. The user subroutines avail-able in the MSC.Marc code coupling interface allow to get and set information from/tothe coupling regions that have been defined on the FE model. The user subroutines areclassified into three groups:

• CPLREG INIT; Initialize coupling regions for a coupled analysis with an externalsolver. This subroutine is called once at the start of the simulation. It is used forchecking general coupling parameters as well as for extracting the structure interfacemesh data.

• CPLREG EXCHANGE; Exchange data on coupling regions. This subroutine iscalled at the start and at the end of each coupling step. Typically, the call at thestart will set the values of the prescribed quantities for this step and the call at theend will extract the new values of quantities -deformation, velocity, etc- computedduring the time-step.

• CPLREG FINALIZE; Finalize coupled analysis with an external solver. This sub-routine is called once, at the end of the analysis.

MSC.Marc’s user subroutines in Fortran constitute the low-level layer of the interface.In this case low-level designates the fact that the Fortran code is independent of the finiteelement model and that the user rarely needs to enter and modify the Fortran sources.The high-level layer of the interface, in which the user sets the particular parametersneeded for every simulation, is based on Python and detailed in the next section.

3.5.2 The python interface

The second role of the interface, the format converter role, is based on a Python script.This is the high-level part of the interface, the one with which the user interacts byspecifying the necessary options for the coupling.

The Python interface collects the structure solutions from MSC.Marc -e.g., structuredeformations- and converts these solutions into a standard format. For example, if thefinite element model output is expressed in mm and in the rotating rotor frame, the usermay order the conversion of the output into S.I. units and perform a frame change inorder to prepare the data for the standardized coupling format. Another task typicallydone in the Python interface is the prediction of the structure state at time tn+1 from thelatest CSM solutions up to time tn. The Python script is also responsible for receivingthe fluid data in CGNS format and interpreting it (regions, frames, units) before sendingthe actual airloads to the FE solver through the Fortran user subroutines.

The advantages of using Python in the interface are numerous. Python is a powerfultool for data handling, partly thanks to the fact that integers, floats and characters canbe easily mixed in lists and sorted. Furthermore, Python includes libraries for network

50


communication that are extremely simple and useful in the frame of code coupling, wherethe codes often run in distant platforms.

Communication between Fortran and Python is provided by the Message PassingInterface (MPI) protocol. This communication consists in the exchange, via memory, oftables of integers, floats and characters between the two programming languages. It ishere acknowledged that this is a particular use of the MPI protocol that deviates fromthe standard MPI practices. MPI is meant to be a protocol to allow parallel computingof a single application over several processors and thus reduce computation time. In theMSC.Marc interface, both Fortran and Python run in the same processor, and MPI ispurely used as a means to exchange data between two different programming languages.There exist other approaches to link Fortran and Python, usually referred as Fortranwrapping for Python. The wrapping approach makes of the Fortran code a library thatcan be called from Python. However, in the present interface, the Fortran subroutinesare not called exclusively by the Python script, but also by the MSC.Marc kernel. In thiscase the wrapping task requires superior computer technology skills, whereas the choiceof MPI allowed a quick setup of the data share scheme between Fortran and Python.

The different layers constituting the MSC.Marc interface are shown in Figure 3.3.The upper part of this figure shows the MSC.Marc application and its user subroutinesin Fortran. Then the MPI application makes it possible to pass data between Fortranand Python. Finally the high-level layer, the Python interface, performs the operationsdetailed in the previous paragraphs and provides as well the network connection for dataexchange with other solvers.

Figure 3.3: MSC.Marc interface for code coupling

Having introduced MSC.Marc’s user subroutines, their accompanying coupling regions,the MPI protocol and the Python script, it will be next shown the algorithms that control

51


the data exchanges during a simulation.Table 3.3 summarizes the data exchanges between the user subroutines in Fortran and

the Python script using the MPI protocol. These data exchanges were done using MPIblocking calls. This means that the program execution will be suspended until the messagebuffer is safe to use. The MPI standards specify that a blocking SEND or RECV does notreturn until the send buffer is safe to reuse (for MPI SEND), or the receive buffer is readyto use (for MPI RECV). Thereby, communication calls between the two executables musttake place in pairs of send/receive functions. This constraint is addressed by means of”matching functions”. For every function in executable A there is a matching functionin executable B. The general interface layout presented in Table 3.3 is divided into threesections, corresponding to the three user subroutines available in the MSC.Marc codecoupling interface.

MSC.Marc API Python scriptCPLREG INIT MPI Init

For every coupling region:send cplreg info recv cplreg info

send cplreg mesh recv cplreg meshCPLREG EXCHANGE For every coupling region:

recv cplreg input send cplreg input(MSC.Marc integrates in time)

send cplreg output recv cplreg outputCPLREG FINALIZE MPI Finalize

Table 3.3: Interface functions

A typical sequence of commands from the Python script is shown in Algorithm 1:

Algorithm 1 (Typical layout of the Python interface)

Coupling Initialization

– initialize coupling parameters: number of iterations, names of the couplingregions, switch on/off options, etc

– get the initial finite element mesh in order to calculate interpolation coefficientswith the CFD mesh

Coupling exchangeDo for each iteration

Set Input

- get, via network connection, airloads in CGNS format

- get, via network connection, rotor control data

- convert CGNS-formatted data into MSC.Marc format (units, frames)

52

3.6 An interface for HOST

- prescribe airloads onto the blades

- prescribe pitch angles onto the blades via the pushrods

Get Output

- get blade deformation

- get pushrod loads

- get rotor hub loads

- if necessary, change units and frame

- if requested, predict structure state at t+ ∆t

- convert output data into the standardized coupling format

- broadcast, via network connection, the structure output

Coupling End

The development of the interface for the CSM solver was done with a progressiveapproach; the early developments used a simplified finite element model of the 7A rotor,made of beam elements only. Later, a 3D finite element model of a blade with simplegeometry (ADM blade) was used to push farther the interface capabilities.

3.6 An interface for HOST

The HOST code, as opposed to MSC.Marc, does not include any application for couplingwith external codes. In the CHANCE project (2000-2005), during which HOST wascoupled to a CFD software, communication was achieved thanks to a socket connectionbetween HOST’s subroutines in Fortran and the CFD software. This socket connectionprovided a bidirectional process-to-process communication and was customized to thatfirst coupling, which implies that it lacked the modularity and generality required for thiswork. Thus, a new interface for HOST had to be developed.

The role of the new HOST interface developed in this work is to provide an access fordata input-output. Following the coupling specifications given in Sec. 3.2, the interfacecan handle standard data models that maximize the interoperability.

The layout of the HOST interface is identical to that of the MSC.Marc interfacedescribed in Sec. 3.5; there is a part of the interface, closer to the HOST kernel, madeof Fortran subroutines (HOST itself is in Fortran). The other part of the interface isa Python script. Like in the MSC.Marc interface, data exchange between Fortran andPython is provided by the MPI protocol. The Python script is in charge of reading theincoming external data and feeding it to HOST, via MPI, into HOST’s format. Conversely,the Python script receives HOST output data, converts it into the standard format anduploads it to the server, via network, for the other codes to download.

During a HOST simulation, data is exchanged once per time-step or increment. Thecall to the interface, schematically described by Figure 3.4, is placed between the kinemat-ics sweep and the loads sweep (refer to Sec. 2.4 for HOST details). In Figure 3.4, ‘pscin’

53


stands for ‘parcours simulation cinematique’ and is the name of the subroutine doing thekinematics sweep. ‘pseff’ stands for ‘parcours simulation efforts’ and is the name of thesubroutine doing the loads sweep.

The coupling does not change the normal HOST workflow. HOST continues to usethe same functions and algorithms that it uses on autonomous simulations. But whenthe coupling is activated, HOST fetches the external data instead of using its own. Forexample, in the loads sweep, the external hub loads given by the CSM are taken insteadof calling the HOST subroutine that calculates them.

tn tn+1

pscin data exchange pseff −→ pscin data exchange pseff

Figure 3.4: HOST interface

A typical sequence of commands from the Python script is shown in Algorithm 2:

Algorithm 2 (Typical layout of the Python interface of HOST)

Coupling Start

Do for each iteration

Get HOST Output

- get rotor controls

- if requested, get HOST airloads and rotate them into the right frame

- convert output data into the standardized coupling format

- broadcast, via network connection, HOST’s output

Set HOST Input

- if requested, get, via network connection, structure motion in CGNS formatand calculate absolute velocities

- get, via network connection, rotor state data

- prescribe structure motion

- prescribe rotor state

Coupling End

The HOST interface presented in this section has some limitations. HOST is a largeand complex program and producing a general interface able to handle any random he-licopter configuration was out of the scope of this work. It is planned in the SHANELproject to develop a more general and consistent coupling interface for HOST.

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Chapter 4

Fluid-Structure Interaction in aTime-Accurate Coupling

In a time-accurate simulation of a partitioned fluid-structure interaction problem the fluidand structure subsystems are time-integrated by different schemes and their solutions areexchanged at each time-step. In other words, time-accuracy refers to the capability tosolve transient problems whose solution is continuously varying as a function of time.

Time-accurate partitioned problems are solved using a staggered algorithm. A stag-gered algorithm regulates the timing of the data exchanges between the subsystems inthe partitioned procedure.

This chapter is organized in three sections following an introduction. The introductionhas a triple role: (1) to motivate the use of time-accurate procedures in this work; (2) topresent the context in which the transfer of motions and loads between the structure andfluid meshes were developed; and (3) to underline an important difference of rotorcraftaeroelasticity with respect to fixed-wing and turbomachinery aeroelasticity. The first sec-tion after the introduction summarizes the state-of-the-art theory in staggered algorithmsfor computational aeroelasticity. It then describes the application of the theory to thepresent coupling. The second section is dedicated to the transfers of structure motionand airloads between the non-matching meshes of the fluid and structure subsystems.The third and last section presents a discipline that is unique to rotorcraft aeroelasticityproblems: rotor control and trim.

4.1 Introduction

Why time-accurate coupling. Previous works in rotor simulation faced two choicesfor the coupling of a CFD solver and a rotorcraft comprehensive analysis: to use either aweak coupling approach or a strong (time-accurate) one.

It has been seen in Section 2.4.5.1 that the weak coupling assumes that the solu-tion is periodic over a rotor revolution and hence the structure motions and airloads are

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4. FLUID-STRUCTURE INTERACTION IN A TIME-ACCURATECOUPLING

exchanged on a per revolution, periodic basis.

Yet in this work there was no possibility to use the weak coupling due to the use ofa finite element solver. A finite element solver, compared to a rotorcraft comprehensiveanalysis, lacks a fundamental feature for weak coupling: the capacity to perform analysesthat yield a periodic solution of the rotor dynamics over one rotor revolution. This type ofanalyses are called ‘equilibrium’ analyses within the rotorcraft community. They assumethat the solution is periodic and therefore the unknowns of the problem become thecoefficients of the Fourier series of the state variables. These Fourier series describe theaeroelastic response of the rotor over a revolution.

The only means to obtain a periodic solution with a finite element structural solver isto integrate the solution in time until it becomes periodic. And for this operation the FEsolver has to be coupled anyway with an aerodynamics solver to get motion-dependentairloads and the ensuing aerodynamic damping. Furthermore, reaching periodicity bytime-integration is lengthy, mostly due to the slow damping of the low frequency lagmotion.

In conclusion, weak coupling and its underlying concept of periodic responses makesno sense when solving the structural dynamics by a finite element method.

The unavailability of weak coupling should not be regarded as constraining though.On the contrary, weak coupling is less general than strong coupling because it cannot beused for transient simulations, given that it assumes periodic solutions. And, in addition,this work has developed a method that eliminates the -up to now- main and long criticizeddrawback of strong coupling: the impossibility to yield solutions that respect rotor trim.

Rotor trim. Rotor trim, or in other words, the need to predict the rotor controls inaddition to the fluid/structure interaction problem, makes one of the largest differencesof rotary-wing aeroelasticity when compared to fixed-wing and turbomachinery aeroelas-ticity. This issue is addressed in the last section of this chapter.

Transfers of structure motion and airloads between non-matching meshes.The coordinates of the nodes of the finite element model do not match those of the fluidsolver. As a result, interpolations must be done each time data transfers are done betweenthe two discretisations.

One advantage of having a highly modular software architecture is the ability to ex-ploit external modules that are dedicated to interpolation operations. However, suchinterpolation modules were not available yet. In the meantime, this work developed sim-ple methods to interpolate across the fluid/structure interface. Transfer operations areaddressed in Section 4.3.

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4.2 Staggered algorithms


The concept ‘staggered algorithm’ refers to the scheme used to synchronize the databetween the fluid and structure subsystems during a coupled simulation.

The research work of C. Farhat, a professor at Stanford University, in the field ofstaggered -and iteration-free- schemes for aeroelastic computations is commonly cited asa well-founded theoretical basis. The present work, within the limits of its particularconstraints, has tried to apply the published theory. The terminology used in this sectionhas been borrowed from Farhat’s publications.

4.2.1 The conventional serial collocated scheme

The most basic staggered scheme is the conventional serial collocated. The term ‘collo-cated’ refers to the fact that both the fluid and structure subsystems stop at the sametime-stations when advancing their solutions in time. The generic cycle of this scheme, de-scribed below, is shown in Figure 4.1, where Q denotes the structure state vector, W thefluid state vector, the superscript n designates the n-th time-station and the superscriptP denotes ‘prediction’. The steps of the cycle are described as follows:

Step 1. Transfer the flow state W n into a structural load.

Step 2. Time-integrate the structure subsystem to advance it from tn to tn+1.

Step 3. Transfer the motion of the structure Qn+1 to the fluid.

Step 4. Update the position of the fluid grid. Then, time-integrate the fluid subsystemfrom tn to tn+1.

The step numbers appear in green in Figure 4.1 and in the remainder of the figures ofthis section.

Figure 4.1: Calculation order of the conventional serial collocated scheme.

57


However, as mathematically demonstrated in [40], the serial collocated scheme is onlyfirst-order time-accurate, irrespectively of the accuracy of the individual flow and struc-tural solvers. For the same computational cost, modified versions of the conventionalserial collocated scheme can give better accuracy.

4.2.2 The parallel collocated scheme

It is sometimes desirable to reduce the total aeroelastic simulation time by using a parallelscheme in which the fluid and structure integrations are performed simultaneously. Thisis especially valuable in the uncommon cases -at least in rotorcraft aeroelasticity- wherethe computation time of the structural solution is of a similar order of magnitude thanthat of the fluid. The parallel collocated scheme is depicted in Figure 4.2. Its genericcycle is summarized next:

Step 1. Predict the structural displacement at time tn+1. Transfer the flow state W n

into a structural load.

Step 2. Update the position of the fluid grid to match the predicted structural displace-ment. Then, time-integrate in parallel the fluid and structure subsystems from tn

to tn+1.

Figure 4.2: Calculation order of the conventional parallel scheme.

The parallel collocated scheme is not optimal either in conserving the energy at thefluid/structure interface. Piperno and Farhat demonstrated both mathematically andwith an example in [39] that the parallel scheme is only first-order energy-accurate, mean-ing that the sum of the works performed by the temporal discretisation of the fluid andstructure subsystems at their interface is of the order ∆t. The conservation of energy atthe fluid/structure interface is critical to the accuracy of the aeroelastic solution.

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4.2.3 The serial non-collocated scheme

Lesoinne and Farhat proposed in [32] an iteration-free staggered scheme for the solution oftransient aeroelastic problems that is second-order time-accurate. The basic idea behindthis scheme is to offset the fluid and structure integrations by half-a-time-step, renderingthe scheme ‘non-collocated’. This offsetting requires the use of a structural predictor.The serial non-collocated scheme is shown in Figure 4.3. Its cycle can be described as:

Step 1. Transfer the flow state W n+1/2 into a structural loading Fn+1/2S .

Step 2. Time-integrate the structure subsystem from tn to tn+1.

Step 3. Predict the structural displacement at time tn+3/2.

Step 4. Update the position of the fluid grid to match the predicted structural displace-ment. Then, time-integrate the fluid subsystem from tn+1/2 to tn+3/2.

Figure 4.3: Calculation order of the serial non-collocated scheme.

In [26], Farhat et al. demonstrated mathematically and by simulation of a complete F-16 configuration that the serial non-collocated scheme can be second-order time-accurate.It achieves this accuracy by a superior conservation of the energy at the fluid/structureinterface (and provided that a few other conditions later detailed are met).

However, second-order time-accuracy does not necessarily require noncollocating thesubsystems’ integrations. The publication above proposes another second-order time-accurate scheme that is collocated. In that case second-order time-accuracy is achievedby: (1) predict the structural displacement at time tn+1; (2) update the position of thefluid grid to match the predicted structural displacement and then time-integrate the fluidfrom tn to tn+1; and (3) use the flow state W n+1 to build the structural loading Fn+1

S forintegrating the structure subsystem from tn to tn+1.

The results and mathematical proofs referenced to in the citations of this section arebased on certain assumptions:

• Linear model of the structure, with equations of motion Mu(t)+Cu(t)+Ku(t) =F (t).

59


• Nonlinear model of the fluid system (i.e., the Euler or Navier-Stokes equations). Yetthe airloads must be a smooth function of the position of the structure.

And the staggered schemes that qualify for achieving second-order time-accuracy do so ifand only if:

• The structure time-integrator is second-order time-accurate (which the Newmarkmethod is),

• The fluid time-integrator is second-order time-accurate,

• The structural prediction is at least second-order time-accurate,

• The flow-induced structural loading Fn is evaluated from the flow field W n, thisis, the aerodynamic load vector is constructed from the latest time instance of theflow field.

Assuming linear structural dynamics is acceptable for most fixed-wing problems. Butrotorcraft dynamics often involve nonlinear phenomena. And the typical unsteady aero-dynamics of a rotor include phenomena such as stall, resulting in airloads that are nota smooth function of the structure position. However, around a given stabilized flightcondition, rotorcraft dynamics can be assumed linear and the nonlinear airloads may beassumed to elapse over a too little span of time as to have an important effect.

4.2.4 Staggered algorithms in rotorcraft aeroelasticity

Staggered schemes for rotorcraft aeroelasticity problems have rarely been a research topicof its own, for there were other more pressing matters, such as fluid grid deformation ordata transfers, the latter including interpolation aspects between non-matching meshes.

Hierholz and Wagner [29] and then Altmikus et al. [2][3] were the first, to the author’sbest knowledge, to start addressing the issue of time-accuracy in coupled simulationsof helicopter rotors. Their work drew from the theory published by Farhat, Piperno etal. that has been presented in Section 4.2. The subsequent HOST/elsA strong couplingdeveloped during the CHANCE programme took also advantage of that theory.

Nygaard et al., from NASA Ames, report in [35] satisfyingly accurate airloads pre-dictions of the UH-60 rotor in a high speed forward flight using a first-order time-accurate coupling scheme (and despite having implemented as well the more-accuratenon-collocated version of Section 4.2.3). However, the time-step in that work is small,equivalent to an azimuthal step of ∆Ψ = 0.25deg. By comparison, the minimal azimuthalstep used in the present work is ∆Ψ = 1.2deg.

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4.2.4.1 Available staggered schemes in the MSC.Marc/HOST/elsA coupling

All of the three staggered schemes presented in Section 4.2 are available in the newcoupling. The first one, the conventional serial staggered scheme (see Figure 4.1), hasa straightforward implementation because no structural predictor is required. It offersthough the lowest accuracy and at no lower computational cost. It was therefore decidedto implement a structural predictor based on the three-step, third-order explicit Adams-Bashforth formula

yn+1 = yn + ∆t

(23

12yn − 4

3yn−1 +

5

12yn−2

)(4.1)

Thus, at the end of each time-step, the nodal displacements and velocities just givenby the CSM plus those of the two previous increments are used to make a structuralprediction. The computational cost of this prediction is negligible, but it allows to usethe more accurate staggered schemes as described next.

The parallel collocated scheme, graphically depicted in Figure 4.4, is summarizedas follows. Both the fluid and structure solvers start at time t0 = 0. Yet no structuralprediction for the fluid grid deformation can be done until the 3 steps of the structuralpredictor, (4.1), have been initialized. Thereby, the first 3 or more coupling iterationsare actually done with a serial scheme: the structure is time-integrated from time tn totn+1. Structural deflections at tn+1 are used to integrate the fluid from tn to tn+1. Fluidoutput at tn+1 is used to integrate the structure from tn+1 to tn+2. This serial patternis pursued until a structural prediction can be accurately done. Then, the CSM outputstwo structural deflections simultaneously: un and the prediction un+1P

(step 11, shownin green in Figure 4.4). Once the fluid has used un to integrate from time tn−1 to tn,the parallel stencil begins. From then on, the fluid grid is deformed using predicteddeflections.

Figure 4.4: Launching the parallel collocated scheme.

The serial non-collocated scheme, shown in Figure 4.5, is operated as follows. elsAstarts not at Ψ = 0, but Ψ = ∆Ψ/2. The start azimuth is a parameter of the elsA

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simulation and is here used to offset the fluid and structure integrations. Structuralpredictions cannot be done until at least the first 3 steps of the Adams-Bashforth predictor(4.1) have been initialized. During the initialization period, the fluid grid is deformed witha trivial structural prediction, un+1/2P

= un. When eventually a structural predictioncan be accurately done, the consistent non-collocated scheme begins effectively.

Figure 4.5: Launching the serial non-collocated scheme.

Switching from one type of staggered scheme to another is immediate thanks to themodularity of the coupling framework.

4.2.4.2 Sources of deterioration of the accuracy of the coupling

It has been seen that the conservation of the energy at the fluid/structure interface is im-portant because it prevents the introduction of artificial numerical damping in the aeroe-lastic simulation, which would otherwise deteriorate its accuracy. Piperno and Farhatidentified in [39] three factors undermining the energy conservation at the fluid/structureinterface:

• The structural predictor does not guess exactly the position of the structure at tn+1;therefore (

xn+1 − xn)6=(un+1 − un

)(4.2)

where u denotes the structure position and x denotes the fluid grid position.

• The motion scheme of the dynamic fluid mesh

• The time-discretisation of the transfer during [tn, tn+1] of the aerodynamic datafrom the fluid to the structure, that is, at which precise time has the flowfield beenevaluated to output the airloads.

The last point is of particular concern in the current status of elsA. The airloads thatelsA outputs at time tn correspond actually to tn−1. This is so because, to output airloadsat tn, elsA evaluates the flux at tn−1, even though it has already calculated the flux attn. This issue may deteriorate the accuracy of the coupling and it is to be corrected in acoming delivery of elsA.

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4.3 The fluid/structure interface


The ‘fluid structure interface’ title of this section refers to the contact region betweenfluid and structure through which structure motion and airloads are exchanged.

The need for a fluid structure interface arises from the fact that the fluid and struc-ture governing equations are solved on separate domains or grids with different spatialdiscretisations. As a result, the transfer of quantities between the non-matching spatialdiscretisations requires some kind of interpolation.

Context

It has been said that one of the objectives of this work is to produce a general coupling inwhich the sophistication of the models is scaleable. Hence, the structural models of therotor blades may be based on beam representations or 3D representations. Similarly, theaerodynamic loads may be given by HOST and its semi-analytic methods or by a full 3DCFD method. And all combinations of structure model and aerodynamics model shouldbe feasible.

The consequence of this scaleability is that the data transfer across the fluid/structureinterface should be able to handle any combination of the two types of structure interfacewith the two types of aerodynamics interface.

Nevertheless, it is later shown that elsA, despite using a 3D mesh representing the truegeometry of the blade, uses a line-based interface. This means that elsA expects structuremotion to be expressed as a line deformation and that elsA gives the aerodynamic loadingas a set of concentrated loads along this line. The developments done during this workwere adapted to this constraint of elsA.

The consequences are that both beam models and 3D FE models may be coupled tothe CFD, but in both cases the deformation of the blade has to be projected onto a line.This is anyway the natural option for the beam model, but the 3D FE model requiressome more manipulation, as described later in Section 4.3.7. Table 4.1 summarizes thecombinations of fluid and structure geometries that were handled in this work.

As the SHANEL project continues after this thesis, it is planned to develop an inter-polation toolbox that would handle any combination of fluid and structure models. Thistoolbox would be yet another module plugged into the coupling architecture. It wouldread the CGNS trees coming from the fluid and from the structure, recognize the type ofdata and manipulate it accordingly.

Aerodynamic interfaceStructure interface line surface

line√

×surface

√×

Table 4.1: Summary of available combinations of fluid and structure geometries.

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Fluid/structure interface of a 3D FE model with a CFD method. When ageometrically rigorous FE structural model of the blade is available, the most directapproach would be to deform the wet surface of the CFD grid following the structuremotion and to apply the pressure -and shear stress when viscous flow is used- distributiongiven by the CFD over the surface of the FE model. This approach, already used in fixed-wing and turbomachinery aeroelasticity, has never been done in rotary-wing aeroelasticitybecause, so far, all the couplings with CFD used structural beam models for the blades.Besides, this approach presents the following disadvantages in terms of added complexity:

• the surface of the 3D FE model has to respect the real geometry of the blade to beable to receive a pressure distribution.

• the interface being a surface, large chunks of data must be handled

• interpolations between two non-matching surface discretisations are costly

• difficult interpretation of the transferred quantities and thereby difficult softwaredebugging

This increased complexity should however pay back in terms of superior accuracy instructure motion and airloads transfers, but the gains are hardly worth the extra cost.This is suggested by concurrent CSM/CFD analyses like [46] that, by coupling structuralbeam models to advanced CFD methods, have obtained results in very good agreementwith the experimental data all while using simple fluid-structure interfaces defined bythe quarter-chord line. In the current state of the things, priority shall be given to theindividual accuracy of the structure and flowfield models.

4.3.1 CFD grid deformation technique

The grid deformation technique for rotor applications in elsA was developed during theCHANCE project (1999-2005) by Cantaloube [15][14]. The present work exploited thiscapability of elsA without adding new developments.

The grid deformation technique is limited to structured meshes with C or O topologyin the chordwise direction and H topology in the spanwise direction, noted j. A numberof blade j-profiles is defined, which are the intersection of the blade with the topologicalmesh j-planes, as shown in Figure 4.6.

First, the surface grids defining the blades are deformed by letting each j-profile (air-foil) undergo a rigid body motion, composed of translations and rotations. Thus, profiledeformation cannot be modeled. In order to minimize the fluid grid deformation, theblades already have some pitch in the initial fluid grid.

Second, the volume grid is adjusted to account for the motion of the surface of theblade. The volume grid is deformed by topological planes. Each j-profile of the bladeinduces the deformation of the volume grid points in the same j-plane. The volume grid

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Figure 4.6: Grid deformation by spanwise topological planes.

planes outboard of the blade follow the deformation of the blade tip profile. Deformationsare damped to a null value at the external boundary of the grid.

The fluid grid is deformed to take into account the rigid-body movement of the bladesand their elastic deflection. Rigid-body movements include collective and cyclic pitchcontrols. The structure solver provides information that comprises both rigid-body andelastic deflections. Structural data is given at time tn+1 in a tree like the one shown inTable 3.2 so that elsA can update the position of the fluid mesh to integrate in time fromtn to tn+1.

The structure motions must be specified as the coordinates of the quarter-chord lineplus rotation matrices defining the orientation of the profiles. The rotation matrix is thematrix relating the section frame, which is attached to the distorted blade section, tothe rotating rotor frame. These six motions contain the control inputs and elastics. Thegeometric twist of the blade is already included in the CFD grid.

The forward tilt angle of the rotor shaft, αq (see Figure 4.16), is fixed at the simulationstart and cannot be modified during run-time. This was a problem because the rotor shaftangle is one of the four trim control variables. As a temporary solution before this problemwas corrected by the elsA developers, one of the four trim targets had to be dropped, asexplained in the trim control section 4.4.

4.3.2 CFD airloads

elsA integrates the calculated pressure distribution and the shear forces for viscous flowsover the blade surface in order to provide nodal loads along the quarter-chord line, see

65


Figure 4.7. These nodal loads consist of the three force components and the three momentcomponents. elsA provides the airloads contained in a CGNS tree like the one shown inTable 3.1).

Figure 4.7: Locations for structural input and airloads output on the CFD grid.

4.3.3 HOST airloads for an external structure

The advantages of using a partitioned approach include the possibility to choose howmuch fidelity is needed for a given model. Lower fidelity models are usually faster andsmaller, and, depending on the study case, their accuracy can be satisfying. Rotorcraftcomprehensive methods, contrarily to CFD analyses, for example, offer the ability toisolate and examine individual parts of the problem and thus help provide insight andfundamental understanding.

In this context, it might sometimes be interesting to use comprehensive aerodynamicsrather than a computationally intensive CFD analysis. It is for this reason that a couplingof MSC.Marc with HOST airloads was developed.

Recalling what has been presented in the Section 2.4.3, when no wake is modeled,HOST calculates the blade airloads from three variables: (1) velocity; (2) acceleration;and (3) blade flapping angle. In a coupled analysis these variables are provided by theCSM.

Yet MSC.Marc was used in a non-inertial frame (rotating rotor frame, shown in Fig-ure A.3), and, as a consequence, the structural displacements, velocities and accelerationsare relative to that frame. Velocity terms due to the rotation of the rotor and its transla-tion in space -the rotorcraft flight velocity- must be added to the MSC.Marc results. Thisis done by the coupling interface of HOST, as detailed next.

Calculating absolute velocities. The velocity terms due to flight speed are first cal-culated in the fixed rotor frame (FRF), see Figure A.3, from the free flow velocity, U∞

66


and the rotor shaft angle, αq (see Figure 4.16). Then they are converted into the rotatingrotor frame (RRF), Figure A.3, using the blade azimuth, Ψ.

V RRFflight =

cosΨ sinΨ 0−sinΨ cosΨ 0

0 0 1

︸︷︷︸

FRF → RRF

−U∞cosαq0−U∞sinαq

(4.3)

The velocity terms due to rotor rotation are simply the vectorial product of the rotorangular velocity, Ω and the coordinates of the ith section of the blade in the rotating rotorframe, ri,

ViRRFrotation = Ω ∧ ri

RRF (4.4)

Discretisation. Problems related to non-matching fluid/structure discretisations whencoupling MSC.Marc with HOST airloads can be easily bypassed. This is thanks to theflexibility with which the blade discretisation in HOST can be changed. It suffices toimpose a list of radii matching that of the finite element model. As a result, there is noneed to interpolate the structure motion nor the airloads.

4.3.4 Definition of a fluid-structure interface for the beam mod-els

Generally, a beam model of a blade is assimilated to the quarter chord line of the bladebecause that is the approximate location of the elastic axis. There is not much choice todefine a fluid/structure interface for the structural beam models; the nodes of the beamelements are used to read the blade motion and to input the external airloads.

Furthermore, it is later shown in the results for the 7A rotor that the radial locationof the nodes along the blade was enforced to match that of the CFD spanstations, so thatinterpolations in the structure transfer were avoided.

4.3.5 Transfer of the structure motion of the beam FE models

Reading the displacements and translation velocities of a beam blade from its nodes issimple. However, when it comes to the rotations in flap, lag and pitch (which are neededfor the aerodynamics) new difficulties arise.

Unlike comprehensive structural models, finite element models do not work directlywith variables such as the blade flapping, lead-lag or feathering angles. It is up to theuser to evaluate these quantities from the nodal displacements and rotations -the latteronly available when beams and shells are used- of the nodes.

In the case of the beam model, displacements alone are not enough to define theorientation of the cross-sections. And interpreting the orientations from the nodal x, yand z rotations is ambiguous because the order and projections in which MSC.Marc givesthe angles are unknown.

67


4.3.5.1 Dummy nodes

In order to determine the rotation angles of the blade, each of the nodes of the beam modelfor which its orientations were to be extracted had a pair of dummy nodes associated.These dummy weightless nodes were attached with a weightless rigid link to the beamnode, so that structural properties were not affected. Both of the links lay in the planedefined by the cross-section, at a user-defined distance from the blade. One link belongedto the horizontal plane, the other to the vertical. This configuration is shown in Figure 4.8.

Figure 4.8: Dummy nodes for cross-section orientation on beam finite element models.

The dummy nodes, being fixed to the cross-section, constitute a section-fixed frame(Figure A.5). Then, for each cross-section, the rotation matrix from the section frame tothe rotating rotor frame (Figure A.3) is directly given by Eq. 4.5.

TRRFSF =

| | |x y z| | |

(4.5)

where y is the unit chord-wise vector from the dummy node in the horizontal plane to thebeam node in the blade, z is the unit thickness-wise vector from the beam node to thedummy node in the vertical plane and x is the vectorial product of z and y, x = z ∧ y.

4.3.6 Definition of a fluid-structure interface for the 3D models

The definition of a fluid-structure interface means declaring the nodes or elements of the3D finite element model that will be used for exchanging quantities (motion, airloads)with the fluid.

68


The most natural option would be to use the entire surface of the blade and map thefluid grid deformation on it. Yet Section 4.3.1 exposed that the fluid grid deformationtechnique in elsA requires as input a projection of the blade motion onto the quarterchord line. So this work conformed to this constraint by projecting the deformation ofthe 3D blades into a line.

Assuming that the cross-sections do not deform significantly, a practical choice is thenodes along the leading and trailing edges. Yet due to a limitation in the MSC.Marc’scoupling features, which do not accept declaring a coupling region from a set of nodes,the declared coupling regions are the upper and lower shells composing both leading andtrailing edges, as shown in Figure 4.9. The Python interface of MSC.Marc, at the simu-lation start, collects all the nodes of these shells and picks the nodes at the intersection.From then on, these nodes are used for reading the structural motion and inputing theexternal airloads.

Figure 4.9: Intersection of the upper and lower shells constituting the leading edge.

Example: the ERATO rotor. The ERATO rotor is one of the test applicationspresented in the next chapter. The interface of the ERATO rotor blades with the fluid iscomposed by the nodes of the leading and trailing edges, as shown in Figure 4.10(b). Themotion of the quarter chord line that the CFD expects is constructed from the motion ofthe leading and trailing edges, as described later. Conversely, the airloads given by theCFD along the quarter chord line are distributed between the leading and trailing edgesof the finite element model.

4.3.7 Transfer of the structure motion of the 3D FE models

Figure 4.11 shows the CFD surface mesh of a generic blade in which a cut has been doneon the plane defined by the chords. The green and orange nodes at the leading and trailing

69


(a) 3D finite element model of the ERATO blade. (b) Leading and trailing edges.

Figure 4.10: Fluid-structure interface of the ERATO blade.

edges, respectively, represent the structure nodes for which deformation is known. Theblue line represents the quarter-chord line, and the nodes along this line are the sectionsat which the CFD expects the motion input. Motion at the blue nodes is interpolated asa function of blade radius from the structure nodes in two steps; first structural motionis interpolated along the leading and trailing edges at the radii of CFD input-stations;second the motion of the quarter-chord is constructed from the motion of the leading andtrailing edges.

In the 3D finite element model, as opposite to the beam models, no dummy nodesare needed any longer to determine cross-section orientation. The technique consists inusing the coordinates of the leading and trailing edges to obtain the orthogonal right-handed triad of vectors (~x′, ~y′ and ~z′) fixed to the cross-section -the so called sectionframe, Figure A.5-. The technique is illustrated in Figure 4.12. A chord-wise vector, ~y′,is defined from the trailing edge to the leading edge. A second vector, ~q′, is defined fromthe trailing edge to the quarter-chord of the contiguous section. Their cross product givesa vector normal to the blade chord plane, ~z′, which is in turn cross multiplied with thechord-wise vector ~y′ to obtain a span-wise vector normal to the cross-section, ~x′.

The unit vectors ~x′, ~y′ and ~z′ can be written as the direction cosine matrix T , whichis the rotation matrix from the section frame (SF) to the rotating rotor frame (RRF,Figure A.3)

TRRFSF =

x′1 y′1 z′1x′2 y′2 z′2x′3 y′3 z′3

(4.6)

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Figure 4.11: Structure motion transfer to fluid grid.

Figure 4.12: Obtaining the coordinate system fixed to the cross-section.

71


xyz

RRF

= TRRFSF

xyz

SF

(4.7)

4.3.7.1 Offsetting rotation terms due to geometric twist

In addition, the twist of the blade must also be taken into account when constructingthe cross-section orientation matrices. Indeed, the aerodynamics solver (either HOSTor the CFD) already integrate geometric built-in twist information in their calculations.The CSM shall send information regarding only structure displacement and deformation,but not geometric twist. Thereby, in order to avoid sending to the CFD or to HOSTorientation matrices that include rotation terms due to blade geometric twist, these termshave to be systematically corrected upon the matrices calculation at each iteration.

In order to offset the rotation terms due to geometrical twist, see Figure 4.13, thefollowing procedure was implemented:

Figure 4.13: Taking into account twist distribution.

• at the simulation start, when the CSM outputs data for the yet undeformed struc-ture, direction cosine matrices TRRF

SFiare calculated for every section.

• The rotation matrix from the ith section to the 0th section (the innermost section,for which grid pitch is defined in the CFD) is given by

T SF0SFi

=[TRRFSF0

]−1TRRFSFi

(4.8)

The rotation matrices between sections are computed once for all at the simulationstart and stored.

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• From then on, the rotation terms due to geometrical twist are offset from the rotationmatrices

TRRFSFi

= TRRFSFi

[T SF0SFi

]−1(4.9)

where the TRRFSFi

is the orientation matrix computed at each new time-step and

the inverse of T SF0SFi

(transpose for these orthogonal rotation matrices) had beencomputed at the simulation start.

4.3.8 Transfer of airloads from the CFD to the 3D FE model

The CFD airloads are given as three force components and three moment componentsalong the quarter-chord. But the structural interface nodes are located along the leadingand trailing edges of the blade, as shown in Figure 4.14. Therefore, some kind of transferwas required to convert the CFD airloads into a structural forcing.

Figure 4.14: ERATO blade. Non-matching fluid and structure discretisations.

4.3.8.1 Conservation of total loads and energy

When the fluid and structure discretisations are not equal, an algorithm is used to dis-tribute the fluid loads onto the structure. This transfer in ‘space’ must satisfy two require-ments. The first one is that the net structural forcing must be equal to the net pressureand shear distribution on the cell surfaces of the fluid interface. Since in this work thestructure was loaded by concentrated forces on the nodes, this can be expressed as∑

i

fSi=

∫SF

(pn+ σn) dS, (4.10)

where i are the nodes constituting the interface of the structure model, fS are the structurenodal loads, p is the pressure distribution, σ is the viscous shear distribution, SF is theblade surface on the fluid grid and n is the normal to the surface.

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There is however an infinite number of nodal load sets that can satisfy the conservationof the total loads. A second requirement is used to determine the correct one, stated asthe conservation of energy. For this purpose, the structural virtual work performed by thedisplacement r of the concentrated loads must be equal to that of the original distributedloads over a displacement u of the fluid grid∑

i

fSiδri =

∫SF

(pn+ σn) δudS. (4.11)

Thus, in an energy conservative load transfer, loads are transferred across the fluid/structureinterface without artificial dissipation or creation of energy. This is important because itmeans that no positive or negative damping is artificially applied onto the structure.

4.3.8.2 Application of a conservative load transfer

In the present study the CFD integrates the pressure and shear distribution to yielddirectly the equivalent nodal loads over the quarter-chord line

fFj=

∫SF

(pn+ σn)DjdS, (4.12)

where Dj is some function with a local or global support on SF . This integration isassumed to be load and energy conservative because it is done internally in the CFDsoftware using the fluid grid [25].

In order to apply nodal loads on the structure, it was assumed that the blade sectionsare rigid, which is indeed a good hypothesis because a blade section is very stiff. Then,the CFD forces are distributed on the structure nodes of the leading and trailing edgeson a 75% and 25% proportion, respectively. This guarantees respecting the loads tensor.Moments are integrally applied on the leading edge, which also respects the loads tensorbecause the leading edge and the quarter-chord are parallel.

The CFD airloads, expressed as a piece-wise distribution along the blade span, arethen integrated over the CSM segments -a CSM segment being defined as the length ofspan surrounding each structure node-. The integration consists in performing a fractionalweight based on the fraction of the length of the fluid segment present in each structuresegment

fnSi=

N∑j

fnFjdcj (4.13)

where fSiis the total load on the structure segment i at time step n, fFj

is the total loadon fluid segment j at time step n and dc is the fraction of segment j that lies on segmenti. This process is described by Figure 4.15, where a distribution of CFD lift is integratedover the CSM segments of a generic blade.

74

4.4 Rotor control and trim

The CFD airloads integration is used for all six airloads components. The bladeleading edge discretisation is not necessarily equal to that of the trailing edge. In suchcases, the CFD airloads distribution is integrated separately.

Figure 4.15: Integration to transfer CFD airloads over structure discretisation.

A small investigation was carried on the ERATO application to evaluate how conserva-tive were the load transfers. The results of this investigation are presented in Section 5.2.3.


The aeroelastic simulation of a rotor requires, in addition to the modeling of the fluid-structure interaction, a method to control the rotor operating state. The isolated rotoroperating state is defined by the rotor pitch control angles -collective θ0 and cyclic θ1C andθ1S- and the rotor shaft attitude αq shown in Figure 4.16. These parameters determinein turn the rotor thrust, which can be decomposed into lift and propulsive force, and theblade flapping, which is used to tilt the rotor disk and thus steer the aircraft.

Trim designates the equilibrium of the aircraft in steady flight. For example, in orderto hold a level forward flight, the lift must counter the weight, and the propulsive force,the drag, while the rotor disk must remain laterally level. The trimmed condition isachieved by rotor control.

The importance of trim lies in the fact that numerical results are better comparedto wind tunnel measurements if the simulation matches the experimental trim. The

75


comparison would not be fair if, for example, the numerical rotor did not generate thesame thrust than the test rotor. Unlike turbomachinery and fixed-wings, rotary-wingaeroelasticity must comprise flight mechanics in addition to the coupled fluid-structureanalysis.

In previous works [7][35], trimmed solutions in strong coupling were obtained by re-peating manually an iterative process made of three steps. This consisted in running firsta simulation with constant controls until convergence. Then compare the trim variablesto their targets. Correct consequently the controls and relaunch the simulation. This pro-cess had to be repeated three or four times. It is a tedious operations that increases thecomputation time, not only because the simulation has to be stopped, but also becausetrim can only be evaluated at best once per revolution.

This work has introduced a new approach, called ‘active trim’, which consists in cor-recting the rotor controls during the time integration, at each time-step. In order to betterpresent the active trim method, the trim variables are introduced first.

Controls of the isolated rotor. The isolated rotor has four control variables. Three ofthem are the blade pitch control angles (collective θ0 and cyclic θ1C and θ1S). The fourthone is the rotor shaft angle, pictured in Figure 4.16. The shaft angle is not actuateddirectly by the pilot in real flight, but it is controlled in the wind tunnel.

Figure 4.16: Definition of the rotor shaft angle αq.

Trim variables. Since the isolated rotor has four controls, it follows that the trim mustbe limited to four targets. The trim variables used in this work are those of the windtunnel tests and are listed next.

1. Lift coefficient, Zb. HOST calculates this coefficient with the following expression

Zb = 100Fz

1/2ρSbV 2tip

, (4.14)

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where Fz is the lift force, ρ the air density, Vtip the blade tip speed (due torotation only) and Sb, the total surface of the blades. The total surface of theblades is the number of blades N times the blade chord c times the blade spanb, Sb = Nbc.

2. Propulsion coefficient, CXSP. HOST calculates this coefficient with the following

expression

CXSP=

Fx1/2ρSbU2

∞, (4.15)

where Fx is the propulsive force and U∞, the free-flow speed. Note that theSP in CXSP

stands for ‘surface pales’.

3. First longitudinal flapping harmonic, β1C. Coefficient multiplying the cosineof the azimuth in the blade flapping expression (Eq. 4.16). It is the longitudinaltilt of the rotor disk.

β = β0 + β1CcosΨ + β1SsinΨ (4.16)

4. First lateral flapping harmonic, β1S. Coefficient multiplying the sine of the az-imuth in the blade flapping expression (Eq. 4.16). It is the lateral tilt of therotor disk.

Flapping laws. A flapping law describes the flapping motion of the rotor blades insteady flight. Flapping laws are used in wind tunnel campaigns when testing isolatedrotors. For example, if a straight flight condition is being reproduced, the rotor mustnot generate lateral force. In short, the rotor disk must not be tilted sideways, whichcorresponds to a null lateral flapping angle, β1S = 0. For the longitudinal flapping angleβ1C there are two approaches:

1. The Modane flapping law. The first lateral flapping harmonic is null, β1S = 0.The first longitudinal harmonic is equal and of opposite sign to the longitudinalcyclic control pitch angle, β1C = −θ1S. This identity was observed to beapproximately true in real helicopter flight and it was then decided to enforceit on the isolated rotor in the wind tunnel.

2. The American flapping law. Both first flapping harmonics are null. β1S = 0and β1C = 0. This law minimizes the bending efforts on the rotor shaft.

The wind tunnel measurements shown in this work were done with the Modane flappinglaw for the 7A rotor and the American law for the ERATO rotor.

Just as in the case of manual trims, the active trim option is based on the knowledgeof the sensitivity ∂O/∂C of the trim objectives O with respect to the rotor controls C.The effect of rotor controls depends on the flight condition and hence, the sensitivities orgradient matrix must be calculated for every different flight condition that is analyzed.

77


The method to calculate the gradient assumes that the rotor behaviour is linear aroundthe flight condition and it consists of the following steps: first an autonomous HOST trimcomputation for the studied flight condition is done. This gives the set of controls thatrespects trim. Then the sensitivities of the trim variables with respect to rotor controlsare derived by introducing a small perturbation in one of the controls and looking atthe variations in the trim variables. Repeating this procedure for each control gives thematrix of sensitivities of the trim with respect to the controls,

J =

∂O1

∂C1· · · ∂O1

∂C4...

. . ....

∂O4

∂C1· · · ∂O4

∂C4

. (4.17)

For a given trim objectives vectorO0 there is a corresponding control vector C0, whichcan be found by

C0 = C + J−1 (O0 −O) , (4.18)

where C is the current control vector and (O0 −O) is the difference between the objectivetrim and the actual trim. In order to have a continuous adjustment of the trim, it isproposed to use a continuous control correction of the type

dC

dt=

1

a(C0 − C) , (4.19)

which is an ordinary differential equation whose solution is C = C0 +(C0−Ci)e−1at, where

Ci is the initial control value and a is a time constant representing the delay after which63% (for t = a, e−1 = 0.36) of the solution is attained. For t = 0.69a (ln(0.5)), 50% ofthe solution is attained, and for t = 4.6a (ln(0.01)), 99% of the solution is attained.

Substituting the term (C0 − C) of (4.18) in that of (4.19) yields

dC

dt= −∂C

∂O(O −O0)

1

a. (4.20)

At each time-step HOST evaluates the difference between the current trim and thetarget trim and updates the rotor controls.

θ0

θ1C

θ1S

αq

n+1

=

θ0

θ1C

θ1S

αq

n

−∆t1

aJ−1

(O −O0

)(4.21)

where the bar in O indicates that the variable has been averaged over the latest rotorrevolution.

Convenient values of a were around 1.5 revolution periods. The value of a can beadjusted in HOST through the expression

a =3T

k, (4.22)

78


where T is the period of a rotor revolution and k a user parameter detailed later. Thegreater the value of k is, the greater the amplification of the rotor controls. If k is too big,the controls will tend to overshoot the solution and convergence may be delayed. Thenext section reports a small investigation to optimize the value of k and examples of theactive trim.

4.4.1 Amplification of the rotor controls

In order to find a convenient value for the amplification factor k in (4.22), tests were donewith autonomous HOST simulations. The simulations started from a perturbed trim stateand had to reach a target trim. The 7A rotor was used for this exercise.

The target trim was that of a test performed in the wind tunnel for the 7A rotor inhigh-speed level flight, defined by a Zb = 12.5, Modane flapping law and CXSP

= 0.1.The perturbed start trim was Zb = 16.0, β1s = 1deg and β1c + θ1s = 1deg. CXSP

was notperturbed and thus was equal to 0.1. The rotor shaft angle αq was not piloted duringthe simulation in order to imitate the CFD aerodynamics, in which the αq angle cannotbe modified during run-time yet. The propulsive coefficient CXSP

was left out of thetrim targets because this variable, if initialized properly with the right αq, remains quiteindependent of the variations in the rotor’s collective and cyclic pitch controls.

Two exercises were done. The first one compared the effect of different time constantson the convergence of the trim. Two k values were tested, k = 3.0 → a = T andk = 1.0 → a = 3T . The convergence of the lift coefficient Zb and the collective pitchcontrol are shown in Figure 4.17. Please note that the lift coefficient is not exclusively afunction of the collective control, but nevertheless these two variables are plotted togetherbecause they are strongly correlated. Similarly, the cyclic pitch controls have a dominanteffect on the cyclic flapping of the blades. The convergence of the first flapping harmonicsand the cyclic controls is shown in Figure 4.18. As expected, the smaller time constant ayields faster response but also tends to oscillate more.

The second exercise compared two simulations with equal time constants but differentgradients. The objective of this exercise was to prove that the accuracy of the gradientis not critical to the convergence of the active trim problem. One gradient was obtainedfrom HOST trim computations in which the blades were rigid, this is, their only motionwas rigid motion around the articulations. The second gradient was obtained by usingflexible blade computations. Therefore, the second gradient was derived from a higher-fidelity simulation. The k factor used for these simulations was equal to 1.5 (a = 2T ).In the convergence of the lift coefficient and collective control, see Figure 4.19, onlyminor differences can be observed. Figure 4.20 reveals that with the rigid blade gradientthe longitudinal control overshoots more starkly than with the flexible gradient. Thesedifferences are due to the fact that the simulations used the first eight elastic modeshapesof the blades and thereby the blade behaviour was closer to that predicted in the flexibleblade gradient. After ten revolutions, the rotor controls by the two gradients are nearlyequal and have practically converged to their final value. The important point is that the

79


Figure 4.17: Lift coefficient and collective control convergence for k=3.0 and k=1.0.

Figure 4.18: First flapping harmonics and cyclic controls convergence for k=3.0 and k=1.0.

80


quality of the gradient is not critical to the convergence of the active trim.

Figure 4.19: Lift coefficient and collective control convergence.

From the previous exercises it seems that a time constant between a = 2T (k = 1.5)and a = T (k = 3.0) is optimal with respect to the trade-off between rate of convergenceand oscillatory behaviour of the rotor controls. When using a = 3T (k = 1.0), theconvergence was too slow. k = 1.5 gives a sharper reaction while remaining stable. Thiswas checked in Figure 4.21, which shows the trim convergence over 25 revolutions fork = 1.5 and proves that the solution remains stable.

The here found optimal value for the time constant can only be regarded though asa rule of thumb. The ideal value will vary with the particular aerodynamic behaviour ofevery rotor. The optimal value for an autonomous HOST simulation will not necessarilybe the same for a coupled CFD/CSM analysis.

81


Figure 4.20: First flapping harmonics and cyclic controls convergence.

Figure 4.21: Full trim convergence.

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Chapter 5

Applications

The results are presented in chronological order, reflecting the evolution of the develop-ments in an increasing degree of complexity.

The early developments were done using a simple structural beam model of the 7Arotor. This served not only to get acquainted with MSC.Marc and its built-in tools forexternal couplings, but also as a support to develop the software framework.

Then work moved on to the first application using 3D FE modeled blades. The ADMrotor was used for this purpose because it has a very simple geometry and a 3D FE modelwas available from another study. The simple geometry of the ADM rotor constituted agood development support to upgrade the MSC.Marc interface from beam to 3D models.However, the lack of experimental data for this rotor motivated a quick transition toanother rotor for which there were experimental measurements.

The ERATO rotor was chosen as an application because a 3D FE model of its bladeswas available at ONERA from a previous dynamics study [55]. The ERATO bladesfeature a complex geometry. In this case a 3D structural model is interesting because thecurrent beam model in HOST fails to predict accurately the modeshapes. Moreover, thesimulation of the ERATO rotor in high-speed level flight constitutes a validation test-caseof the SHANEL project.

5.1 7A rotor

The 7A rotor was chosen as the first application because it has been thoroughly studied atONERA and more particularly, it had been the study rotor of the coupled HOST/CFDanalyses of the CHANCE project back in 2005. The same CFD model could be used.Numerical results from that first coupling and wind tunnel measurements were availablefor comparison and hence, validation.

The 7A rotor is a four-bladed rotor tested in ONERA’s S1MA wind tunnel. The blade,shown in Figure 5.1, has an aspect ratio equal to 15 and rectangular planform. It has achord of 0.14m and a radius of 2.1m. The ONERA profile OA213 is used from the bladeroot to r/R=0.75. Then the profile evolves until r/R=0.90 to become an OA209. Each of

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5. APPLICATIONS

the three spanwise regions defined by the profiles has a different linear twist. Maximumtwist is found at r/R=0.75 and is equal to -4.54deg.

Figure 5.1: 7A blade definition.

Flight condition. It was chosen to simulate the same flight condition than in theCHANCE project with the HOST/CFD coupling to be able to compare the results. Thisflight condition has been tested in the wind tunnel. It reproduces a high-speed level flight,corresponding to the wind tunnel counter 312. This counter is defined by an advanceratio1 of µ = 0.4 (316.65 km/h), a lift coefficient Zb = 12.5, a propulsive coefficientCXSP

= 0.10 and the Modane flapping law, this is, zero lateral rotor tilt (β1S = 0) and afirst longitudinal flapping angle harmonic equal and of opposite sign to the longitudinalcyclic control (β1C = −θ1S).

5.1.1 Structural model

The finite element model of the 7A rotor, shown in Figure 5.2, was made ad hoc for thiswork. The objective was to have a simple and computationally inexpensive model thatwould help in the developments. That is why only beam elements were used. Each bladeconsists of 44 beam elements. The blades are clamped separately and thereby assumedmechanically independent.

The beam element (number 98 in the MSC.Marc element library [34]) is an elasticbeam with transverse shear. Each of its two nodes has the three displacements plus thethree rotations as degrees of freedom. Linear interpolation is used for displacements androtations. Since the type of analysis here used assumed large displacements but smallstrains, large curvature changes in the beam element were neglected.

The location of the beam nodes was enforced to match the radial discretisation atwhich the fluid grid expects structure motion, thus avoiding interpolations. Nevertheless,the CFD airloads, which are given at the center of the segments enclosed by the inputradii, do require interpolation.

1The advance ratio µ is defined as the ratio of flight speed Vflight over blade tip speed Vtip (due torotation only), µ = Vflight/Vtip.

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5.1 7A rotor

Figure 5.2: 7A rotor finite element model with close-up view on the pitch control system.

5.1.2 Fluid grid

The CFD model of the 7A rotor used in the present study was obtained from a paststudy by Beaumier et al. [7], in which it had been coupled to HOST. It is a viscous flow(Navier-Stokes) model of the isolated 7A rotor.

The grid consists of four blocks, one per blade, as shown in Figure 5.3. Noting theblade radius by R, the block outer boundaries lie at 2R in the span-wise direction and±1.5R in the thickness-wise direction. Each block, of C-H topology, includes 189 nodeschord-wise (of which 132 wrapped around the profile), 57 nodes span-wise (of which 32on the blade) and 49 nodes thickness-wise, resulting in a total number of grid points forthe complete rotor of 2.1 · 106 approximately.

Figure 5.3: The multi-block mesh of the 7A rotor.

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5. APPLICATIONS

A 2nd order centered discretisation in space with Jameson’s artificial viscosity wasused. The resolution in time was achieved by the dual time stepping method in the firsttests. Later the Gear method was used, which improved substantially the prediction of thesectional pitching moments. The computationally inexpensive algebraic model of Michelwas used for turbulence.

Other concurrent rotorcraft CSD/CFD studies like [10], [35], or [46] do not use lessthan 4.4 ·106 points, so the present grid is relatively coarse. This is not however a concern,because in a first time the priority was to develop the coupling, whereas high-accuracy,detailed analysis comes second.

5.1.3 Results

5.1.3.1 Validation by comparison with HOST/CFD results

Nine rotor revolutions were simulated. During the first three revolutions the airloads weregiven by HOST. They were used to initialize the rotor dynamics at a low computationalcost. After the third revolution the airloads were provided by elsA. The rotor controlswere generated by HOST during all the revolutions. Results are first presented for thesectional airloads, which are compared to experiments and to previous HOST/CFD strongcoupling results from the CHANCE programme back in 2005. Then the convergence ofthe trim and rotor controls is presented.

Figure 5.4 shows the sectional airloads at two blade spanstations: r/R=0.70 andr/R=0.975. The results obtained by coupling MSC.Marc/HOST/elsA are plotted in blue,whereas the 2005 HOST/elsA results are plotted in green. It can be seen that, as ex-pected, the new coupling reproduces very closely the previous results. This similarity wasexpected because the CFD method is the same and the beam model in MSC.Marc doesnot provide significant advantages over the beam model in HOST. The 2005 HOST/elsAresults were obtained after 9 revolutions, during which the simulation was stopped sixtimes in order to evaluate the trim and adjust the rotor controls. The new results wereobtained after 9 revolutions as well, although only the last 6 of them were done with CFDairloads. Rotor controls were automatically corrected at each time-step throughout thenine revolutions.

The convergence of the non dimensional lift coefficient Zb and of the collective controlis shown in Figure 5.5. The convergence of the flapping trim variables and cyclic controlsis shown in Figure 5.6. In both figures a perturbation of the trim variables is observableat the start of the fourth revolution. This is due to the switch from HOST airloadsto CFD airloads. A fair convergence to the trim objectives is attained after six rotorrevolutions. Another revolution is yet necessary for the controls to start stabilizing. Inthe last revolution, the relative error of the rotor lift coefficient, Zb, with respect to itstrim objective is 0.77%. The absolute error of the Modane flapping trim variables inFigure 5.6 is 0.027deg for the first lateral flapping harmonic, β1S, and 0.023deg for thevariable θ1S + β1C . The absolute variation of collective control during the last revolution

86

5.1 7A rotor

(a) Pitching moment r/R=0.70 (b) Normal force. r/R=0.70

(c) Pitching moment. r/R=0.975 (d) Normal force. r/R=0.975

Figure 5.4: 7A sectional airloads.

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5. APPLICATIONS

is -0.008deg. For the lateral control this variation is of 0.003deg and for the longitudinalcontrol, -0.004deg. The rotor propulsive force coefficient, CXSP

, (not plotted), althoughleft out of the trim, stays satisfyingly close to the trim target.

Figure 5.5: 7A rotor. Collective control and rotor lift coefficient.

In conclusion, the successful reproduction of the HOST/CFD results validates the newHOST/MSC.Marc/elsA coupling. The active trim is also validated. The next sectioncompares the results to experimental measurements.

5.1.3.2 Comparison with experimental results

The agreement of the numerical results with the experiment was first discussed afterthe HOST/CFD results in 2005 [7]. Experimental airloads are especially elusive in theadvancing blade area (Ψ < 180deg), where the negative peaks in normal force and pitchingmoment are largely underpredicted.

Figure 5.7 shows the sectional airloads at two blade spanstations: r/R=0.70 andr/R=0.975. The results obtained by coupling MSC.Marc/HOST/elsA are plotted in blue.The 2005 HOST/elsA results are plotted in green and the experimental results, in red.

The negative peaks in pitching moment at Ψ ≈ 90deg are due to the well-knowndownstream shift of the center of pressure of an airfoil past transonic flow. This negativepitching moment induces an important nose-down twist of the blade tip, which reinforcesin turn the already negative lift, see Figure 5.7(d). Note that lift at the tip of the

88

5.1 7A rotor

Figure 5.6: 7A rotor. Cyclic control and flapping trim.

advancing blade would be negative even in a rigid blade. What makes it negative isthe strong geometric twist of the blade combined to a low pitch setting. A second, local,minimum in lifting force is found at the retreating blade (Ψ ≈ 270deg), as the flow velocityslows down to a minimum.

There is a second interesting feature in Figure 5.7(d): the experimental lift remainsroughly constant during a short lapse comprised between Ψ = 40deg and Ψ = 80deg.This is probably due to blade-wake interaction. In high-speed level flight, blade-vortexinteraction is not a dominant phenomenon because the wake is quickly convected past therotor, which is strongly tilted forward. But there is limited wake interaction in the firstquadrant, which could explain the experimental lift perturbation. This phenomenon hasbeen studied by Potsdam et al. on the UH-60 rotor in [46] and by Pahlke and van derWall for the 7A rotor in [36] and [37]. None of these investigations succeeded in capturingproperly the sectional lift of the blade tip in the lower vicinity of Ψ = 90deg.

Another important aspect is that the wind-tunnel conditions were not included in theCFD model. The rotor control angles provide evidence of some of the consequences of thissimplification. Table 5.1 summarizes the values of the rotor control angles as obtainedby HOST, the HOST/CFD coupling and the new coupling. The largest deviation withrespect to experiment occurs for the lateral control angle, θ1C , which is underpredicted bymore than 2 degrees. Rodriguez proved in [48] that this is a consequence of the fuselage-like support on which the 7A rotor was mounted in the wind tunnel (see Figure 5.8),

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5. APPLICATIONS



Figure 5.7: 7A sectional airloads.

90

5.1 7A rotor

which was not included in the present CFD model. The support induces an upwash inthe upstream part of the rotor, increasing thus the local incidence in that region. Theopposite is true on the rear rotor. Longitudinal variations of the angle of attack induce,via gyroscopic effect, a lateral tilt of the rotor, and that is why more lateral control inputis needed in the experiments.

θ0 θ1C θ1S αqExperiment 14.54 3.43 -3.70 -13.75

HOST 14.14 0.65 -3.29 -13.18HOST/elsA 14.76 1.35 -3.78 -13.17

HOST/MSC.Marc/elsA 14.01 1.53 -3.69 -13.18

Table 5.1: 7A rotor pitch control angles (deg).

Figure 5.8: 7A rotor in S1MA wind tunnel.

The coupling method may be held responsible for some of the discrepancy with theexperimental data. But, for the 7A rotor, the largest potential for improvement lies inthe CFD analysis. This is illustrated in the next section.

5.1.3.3 Improving the 7A rotor results

The results presented above used the Dual Time Stepping method as time-integrationscheme for the CFD. The accuracy of the airloads can be significantly improved by simplyswitching to the Gear method, leaving all other things equal.

The simulations of the previous sections were re-run with the Gear method. Figure 5.9shows the sectional airloads at two blade spanstations: r/R=0.70 and r/R=0.975. It canbe seen, compared to Figure 5.7, that the amplitude of the negative pitching moments issubstantially improved, especially near the blade tip, at r/R=0.975.

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5. APPLICATIONS



Figure 5.9: Improved 7A sectional airloads using the Gear method.

92

5.2 ERATO rotor

5.2 ERATO rotor

The ERATO (Etude d’un Rotor Acoustiquement Optimise) rotor is the outcome of aresearch project seeking to design a blade with minimum acoustic signature. Each of itsfour blades (see Figure 5.10) features a complex chord and twist distribution, forward andbackward sweep and a profile evolution including up to four different profiles. Like the7A rotor, it has a rotor diameter equal to 4.2m.

Figure 5.10: ERATO blade definition.

Flight condition. The simulations presented here reproduce a high-speed level flight(advance ratio equal to µ = 0.423) tested in wind tunnel. The trim condition in the windtunnel is defined by a non dimensional rotor lift coefficient, Zb = 12.48, a non dimen-sional propulsive force coefficient of 1.6 and null pitching and roll moments around thehub, which is equivalent to a null cycling flapping (β1S = β1C = 0, American flapping law).

The use of advanced structure models for this blade is interesting because HOST’sstructure model underpredicts the torsional response, which is critical to an accurateaeroelastic analysis. This fact is illustrated in Figure 5.11, where the measured torsioncomponent of the third rotating flap mode is compared to a prediction done with the 3Dfinite element model and with HOST’s model. The latter underpredicts the amplitude ofthe torsion component.

The high content of torsion in the flap mode is explained by the flap-torsion couplingintroduced by the backward sweep of the blade tip.

5.2.1 Structural model

The 3D finite element model of the ERATO blade is shown in Figure 5.13. The modelwas built by K.V. Truong with the minimum number of elements necessary to reproducethe measured modal frequencies, see below. Still, each blade consists of around 13,000elements, including beams, shells and cubic hexahedra. The corresponding number ofdegrees of freedom per blade is roughly estimated at around 40,000.

Eigenfrequencies of the ERATO blade. The FE model of the ERATO blade repro-duces accurately the experimental eigenfrequencies that were measured in two conditions:clamped blade and free-free conditions (suspended blade). As Truong reported in [54],

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5. APPLICATIONS

Figure 5.11: Torsion deflection of the third rotating flap mode of the ERATO blade.

the maximum relative error of the predicted eigenfrequencies for the first 6 modes withrespect to the measurements is equal to 4%. Truong’s report contains as well the pre-dicted Campbell diagram of the ERATO blade that is shown in Figure 5.12.

Each blade of the FE model is equipped with a pitch control system and a lead-lagdamper. In order to trim the computation costs of the four bladed rotor, two MSC.Marcapplications run in parallel, each containing two blades, see Figure 5.14. This is possiblebecause the blades are assumed mechanically independent (rigid hub). Running parallelCSM applications is possible and easy thanks to the modularity of the coupling frameworkdeveloped in this work.

The two parallel applications run the same input file. Despite containing only twoblades, this file is already over 56,000 lines long (in which the user often has to navigate),so its half size compared to that of the entire rotor makes it more practical. This comesas a second advantage after the reduced computing time.

It would have been technically possible to run four MSC.Marc applications -a bladeeach- in parallel, but there were not as many MSC.Marc software licenses available.

5.2.2 Fluid grid

Inviscid model. The first simulations were carried out with an inviscid Euler model,for lower computational cost. Multi-block grids (one C-H block per blade) were used forthe simulation of the isolated rotor, see Figure 5.15. Each block has 141 nodes along thechord direction, 40 nodes in the spanwise direction (of which 26 over the blade) and 26nodes in the direction normal to the rotor plane, resulting in a total number of nodes forthe complete rotor over 5.8 · 105.

A 2nd order centered discretisation in space was used. The time integration was donewith the implicit Gear method. The azimuthal step was equal to ∆Ψ = 1.2deg.

94

5.2 ERATO rotor

Figure 5.12: Predicted Campbell diagram for the ERATO blade.

Figure 5.13: 3D finite element model of the ERATO blade.

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5. APPLICATIONS

(a) (b)

Figure 5.14: Two finite element models run in parallel.

Figure 5.15: Multi-block grid used for inviscid simulations of the isolated ERATO rotor.

96

5.2 ERATO rotor

Viscous model. A second set of simulations was done using a viscous RANS flowmodel. The grid is a refined version of the Euler one. The complete rotor grid has around2.1 · 106 points. There is one C-H block per blade. Each block has 189 nodes along thechord direction, 57 nodes in the spanwise direction (of which 32 over the blade) and 49nodes in the direction normal to the rotor plane.

A 2nd order centered discretisation in space with Jameson’s artificial viscosity wasused. The resolution in time was achieved by the Gear method. The algebraic model ofMichel was used for turbulence.

5.2.3 Conservation of the energy in the loads and motion trans-fers

A small investigation was carried on the ERATO application to evaluate how conservativewere the load transfers between the CFD and the 3D FE model.

It consisted in comparing the energy on the fluid and structure interfaces, W F andW S, respectively. This was done by evaluating the work done by the flow-induced forceF with the displacement vector r plus that of the moments M with the rotation vectorα during a given time-step,

W =N∑i=1

(F i · ri +M i ·αi) , (5.1)

where the i subscripts denote the discrete nodes of the fluid or structure systems. Thefollowing procedure was repeated at each time-step:

Step 1. Receive airloads FF and transfer them to structural loading FS with the methodexposed in Section 4.3.8,

Step 2. Advance the structure in time from tn to tn+1,

Step 3. Evaluate structural energy,

W S =

NS∑i=1

(FS i ·

(rn+1i − rni

)+MS i ·

(αn+1i −αni

)), (5.2)

where NS is the number of structural nodes in the fluid/structure interface and rni

is the position of the i-th node at time tn.

Step 4. Interpolate structure motion to fluid discretisation as explained in Section 4.3.7,but without using structural prediction,

Step 5. Evaluate fluid energy. Since fluid loads are given at the cell center but fluidmesh deformation is prescribed at the cell nodes, displacements are averaged at the

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5. APPLICATIONS

cell center

W F =

NF−1∑j=1

(FF j ·

(r∆tj+1 − r∆t

j

2

)+MF j ·

(α∆tj+1 −α∆t

j

2

)), (5.3)

where NF is the number of fluid nodes in the fluid/structure interface and r∆tj is

the displacement of the j-th node during a time-step.

The reason for not using structural prediction in the above scheme is that the pointhere was to evaluate the spatial energy conservation in the loads transfers within a giventime-step. The predictor is used to improve the energy conservation in the temporaldiscretisation of the staggered scheme. But it does not guess exactly the position of thestructure at tn+1. So the fluid grid deformation xn+1, based on this prediction, will not beequal to that of the structure, un+1, when it eventually reaches tn+1. In consequence, thestructure predictor is, albeit modestly, a factor undermining spatial energy conservation.

The computation simulated a hover flight, with constant rotor control angles. Only asixth of a revolution was simulated because there was no need for a stabilized response;the spatial transfer algorithms work the same in transient or stabilized regime.

The results here shown contain only the energy due to the vertical displacement timesvertical force, W = Fzrz, for one blade, because there were uncertainties in the calcu-lation of the angles α and an error related to the frames of reference was found in themanipulation of the x and y displacements. This is however not a problem, because thealgorithm for spatial transfers is the same for all displacement and rotation components.If it works for one it works for all.

Figure 5.16 shows the fluid and structure energies as calculated by the expressions(5.2) and (5.3). The right vertical axis of the same figure shows the relative work error,calculated as (W F −W S)/W S)100. It can be seen that the relative error is bound under0.5%. This result proves that the motion and loads transfer algorithms between the 3DFE model and the CFD model of the ERATO blade preserve well the energy at thefluid/structure interface.

5.2.4 Results

This section presents the results of the simulation of the high-speed flight condition of theERATO rotor. Results are first shown for a simulation that used an inviscid flow model.Then a second set of results is shown corresponding to a viscous flow model.

5.2.4.1 Euler CFD, inviscid flowfield

Eight rotor revolutions were simulated with the following parameters:

• Common time-step for the CFD, the CSM and HOST equivalent to an azimuthalstep of ∆Ψ = 1.2deg.

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5.2 ERATO rotor

Figure 5.16: Conservation of the energy in the transfers at the fluid/structure interface.

• Half a time-step offset between the fluid and structure integrations (staggered schemepresented in Section 4.2.3).

• The damping coefficient of the spring/dash-pot system was increased by a factor of10 during revolutions 1, 2 and then linearly ramped down to a factor of 1 duringrevolutions 3,4.

• Active trim constant k = 1.5, yielding an amplification factor of the controls a = 2T .

Sectional airloads for the CnM2 and CmM

2 at two blade spanstations, r/R=0.75 andr/R=0.975, are shown in Figure 5.17. Airloads are compared against autonomous-HOSTresults and experimental measurements. Unfortunately, no coupled HOST/CFD resultswere available for the ERATO rotor, and this work did not have the time to performthem.

Similarly to the 7A rotor results, the amplitude of the negative peaks in lift andpitching moment in the advancing blade (Ψ ≈ 90deg) are underpredicted. There isa small phase shift as well between simulation and experiment. More generally, thefrequency content of the numerical airloads is lower than that of the experiment.

The HOST results were obtained by performing a trim computation with HOST ver-sion 10, trying to get as much accuracy as possible, using the unsteady aerodynamicsoption plus the prescribed-wake method METAR. The first 7 modes were used for themodal projection. HOST airloads are notable in that they have a higher frequency con-tent, closer to experiment. The amplitude of the negative peaks in lift in the advancing

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5. APPLICATIONS

blade are better predicted by HOST than by the CFD, especially near the blade tip, atr/R=0.975, see Figure 5.17(d). Note however that, for this spanstation, HOST’s negativepeaks in lift and pitching moment on the advancing blade have a larger phase shift withrespect to experiment than the CFD airloads. Finally, pitching moments are substantiallybetter predicted by the CFD than HOST, although both CFD and HOST fail to capturethe dynamic stall in the retreating blade, observable around the azimuth Ψ ≈ 300deg inFigure 5.17(c).

Figure 5.18 compares the blade tip torsion against HOST’s results and experimentalmeasurements. The backward sweep of the ERATO blade leads to minimum torsion whenthe blade tip of the advancing blade (Ψ ≈ 90deg) undergoes transonic speeds. The aft-swept blade tip, pulled down by negative lift, acts as a lever introducing pull-up pitchto the rest of the blade. The frequency of the measured torsion is 5/rev, whereas bothHOST and the coupling results yield a 4/rev frequency. This is understandable, given thelow frequency content of the pitching moment excitations.

The convergence of the non dimensional lift coefficient Zb and of the collective controlis shown in Figure 5.19. The convergence of the cyclic flapping angles (trim targets)and cyclic controls is shown in Figure 5.20. The rotor propulsive force coefficient (notplotted), although left out of the trim, stays satisfyingly close to the trim target. There isa mild overshooting tendency in the rotor controls, but in the last four revolutions theirvariation is satisfyingly confined within a range of 0.20deg.

5.2.4.2 Reynolds-Averaged Navier-Stokes CFD, viscous flowfield

Next, results are shown for the ERATO high-speed flight condition, but using a viscousflowfield modeling. These results are actually older than the inviscid ones presented justabove. In fact, after this simulation it was found that the spring definition of the lead-lag dampers of blades 2 and 4 was wrong due a mistake in the frames of reference. Itwas then decided to repeat the simulation with correct spring settings but using the lesscomputationally intensive inviscid modeling.

Another small issue emerged after the simulation. One of the inputs of the CFDanalysis is the Reynolds number at the blade tip. As a result of copying the CFD cardof the 7A rotor analysis, the Reynolds number prescribed for the ERATO rotor was thatof the 7A, which is equal to 2 · 106. Although both rotors have a similar airspeed at theblade tip, the chord of the ERATO blade tip is half that of the 7A blade. Therefore, theReynolds number should have been halved.

Eight rotor revolutions were simulated with the following parameters:

• Common time-step for the CFD, the CSM and HOST equivalent to an azimuthalstep of ∆Ψ = 1.2deg.

• Parallel collocated staggered scheme (staggered scheme presented in Section 4.2.2).

• The damping coefficient of the spring/dash-pot system was increased by a factor of100 during revolutions 1 and 2, by a factor 50 during revolution 3 and then linearly

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5.2 ERATO rotor



Figure 5.17: ERATO sectional airloads by Euler CFD.

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5. APPLICATIONS

Figure 5.18: ERATO rotor, blade tip torsion. Euler CFD.

Figure 5.19: ERATO rotor. Collective control and rotor lift coefficient.

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5.2 ERATO rotor

Figure 5.20: ERATO rotor. Cyclic control and flapping trim.

ramped down to a factor of 10 during revolution 4 and finally ramped linearly downagain to a factor 1 during revolution 5. Later it was deemed, by looking at the lagdisplacements and on an engineering judgment basis, that damping was too stark.Consequently, the damping coefficient was lowered in the subsequent simulations(i.e., the simulation with inviscid Euler CFD).

• Active trim constant k = 2.0, yielding an amplification factor of the controls a =3T/k = 1.5T .

Sectional airloads for the CnM2 and CmM

2 at two blade spanstations, r/R=0.75 andr/R=0.975, are shown in Figure 5.21. The viscous airloads provide a very substantial im-provement compared to the inviscid results in Figure 5.17, particularly in terms of pitchingmoment. The amplitude of the negative peak in pitching moment of the advancing bladeis now better predicted, both for r/R=0.75 and r/R=0.975.

The convergence of the non dimensional lift coefficient Zb and of the collective controlis shown in Figure 5.22. The convergence of the cyclic flapping angles (trim targets) andcyclic controls is shown in Figure 5.23. The rotor propulsive force coefficient (not plotted),although left out of the trim, stays satisfyingly close to the trim target.

Most remarkably, trim convergence is not as fast and smooth as for the previousresults. This is in good part due to the use of a greater amplification of the rotor controlcorrections (k = 2.0 here against k = 1.5 in the inviscid results). There is an overshootingtendency in the rotor controls and trim variables. And it seems that the collective control

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5. APPLICATIONS

(a) Pitching moment. r/R=0.75 (b) Normal force. r/R=0.75


Figure 5.21: ERATO sectional airloads by Reynolds-Averaged Navier-Stokes CFD.

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5.2 ERATO rotor

will converge to a value higher than in the inviscid case, which is reasonable, consideringthe greater drag generated by a viscous flow.

Figure 5.22: ERATO rotor. Collective control and rotor lift coefficient.

5.2.5 Discussion of the results

Assessing rotor aeroelastic solutions is a bit of a chicken and egg dilemma. It is not easy totell if the shortcomings of the coupled solution arise from the flow modeling, the structuremodeling, or both. Generally, capturing accurately the highly unsteady 3D flowfield ismore challenging than simulating the structural dynamics. But the coupling method canalso be a source of inaccuracy. This section contains some comments on problems relatedto the individual disciplines. The next section studies in more detail the consequences ofthe coupling method.

When comparing results to experimental data it must be kept in mind that the ERATOsimulations were more of a test exercise than a high-fidelity analysis. Indeed, simulationruns were often repeated as new errors emerged. And, given the high computational costof the solutions (see Section 5.2.6), this adjustment process was lengthy, with further workstill being necessary.

Compared to experiment, the main shortcomings of the predicted airloads -both forinviscid and viscous flowfields- are a too low frequency content and difficulties in capturing

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5. APPLICATIONS

Figure 5.23: ERATO rotor. Cyclic control and flapping trim.

the negative peaks in lift and pitching moment in the advancing blade: their amplitudesare underpredicted and their location is shifted to an azimuth a few degrees lower thanexperiments.

The dynamic stall in the retreating blade is not captured at all by the CnM2 and

CmM2 coefficients of the viscous airloads. In the retreating blade the Mach number M

is at its lowest, which may further dampen any Cn or Cm perturbation. A check wasdone by comparing the bare normal coefficient Cn and pitching moment coefficient Cm tothose of experiment. These coefficients should reflect better a small perturbation on thepressure distribution and thus be more sensitive to the dynamic stall. In Figure 5.24(b) itcan be seen, at Ψ ≈ 290deg, a sharp but short rise in the experimental lift, denoting thestart of the dynamic stall. The CFD lift does not capture this. Similarly, Figure 5.24(a)shows a smooth CFD pitching moment curve where the experiment undergoes a sharpand deep negative peak. A plausible argument for the lack of stall capturing could lie inthe turbulence model. Note that this work used the simplest turbulence model availablein elsA, namely the algebraic model of Michel. More sophisticated turbulence modelscould improve the prediction of flow separation phenomena.

Another flow feature that remains elusive takes place near the blade tip, at ther/R = 0.975 section, in the first quadrant, close to Ψ = 90deg. The pitching moment,Figure 5.21(c), can be seen to surge before plunging into a deep negative peak. Simulta-neously, the lift force, Figure 5.21(d), presents in the same azimuthal location not a surgebut a modest lift gain that tends to counter an otherwise fast reduction in lift. Arguably,

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5.2 ERATO rotor

(a) Pitching moment Cm. r/R=0.975 (b) Normal force Cn. r/R=0.975

Figure 5.24: ERATO. RANS CFD sectional coefficients, r/R=0.975.

these phenomena are due to blade-wake interaction. This hypothesis has already beenadvanced in the 7A rotor results, with references to literature, but is recalled next.

In high-speed level flight, blade-vortex interaction is not a dominant phenomenonbecause the wake is quickly convected past the rotor, which is strongly tilted forward.But there is limited wake interaction in the first quadrant, which could account for theperturbations in the airloads. Indeed, a vortex shed by a preceding blade could rise thelocal incidence angle of the advancing blade’s tip, provided the vortex passes close enough.This claim is further supported by the observation that the convergence of the airloadswas slowest in the first quadrant, as shown in Figure 5.25.

The fluid grid for the viscous RANS analysis of this study contains 2.1 · 106 points,which is far less than many other concurrent studies, and is not conceived for optimal wakeconservation. It would be interesting to use a refined grid to see its impact on the solution.

As a last remark on the viscous results, it is here recalled that the Reynolds numberwas accidentally set to a value double of the right one (2 ·106 instead of 1 ·106). Althoughthis may not have a significant impact, it does nonetheless push in the wrong direction:it removes viscosity. Greater viscous forces, as compared to inertial, may help in improv-ing stall prediction, for example. Furthermore, in the viscous simulation, the springs ofblades 2 and 4 were defined in a wrong frame of reference. In addition, the springs ofall four blades had an excessive damping, which did not let the structure ‘breath’ normally.

Like in the 7A rotor, the support rig of the rotor in the wind tunnel was not in-

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5. APPLICATIONS


Figure 5.25: ERATO. Convergence of the RANS sectional airloads, r/R=0.975.

cluded in the CFD models. The primary effect of the support is to induce an upwash onthe inboard part of the rotor blade, near 180deg azimuth, thereby increasing the normalforce in this region. No sectional airloads have been shown for such inboard blade sections.

The blade tip torsion (obtained with inviscid airloads) has been shown in Figure 5.18.The shape of the coupled solution reproduces fairly well that of the experiment, but thefollowing differences can be observed: there is a constant phase lag between solution andexperiment, the solution being in advance. The solution does not twist as much as theexperiment in the Ψ = 200deg and Ψ = 270deg regions. More generally, the amplitudeof the solution oscillations is too low. And the first torsional peak at Ψ = 30deg isnot captured at all. This makes both HOST and the coupled solution to have a 4/revfrequency in torsion, instead of the experimental 5/rev.

A look at the predicted Campbell diagram for the ERATO blade, Figure 5.12, revealsthat there are no modes in the 5/rev frequency when the normalized rotation velocity isequal to 1 (i.e., the nominal rotor rotation speed Ω). By contrast, the Campbell diagramdoes show the third flap mode to cross the 4/rev harmonic at the nominal rotation speed.It has been seen in Figure 5.11 that the third flap mode contains a lot of torsion deflectiondue to the aft-swept tip of the ERATO blade. Therefore, it is coherent that the predictedtorsion in Figure 5.18 does not exhibit 5/rev harmonics but 4/rev, which correspondsactually to a flap mode.

Furthermore, an analysis of the frequency content of the inviscid airloads shows thatthe amplitude of the 4/rev and 5/rev -and nearly all other harmonics- is largely under-

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5.2 ERATO rotor

predicted with respect to experiment, see Figure 5.26; together with the lack of 5/revmodes, it is natural that the structure does not respond at that frequency. And, giventhe lower amplitude with respect to experiment of the inviscid loads, the underpredictionof the twist amplitude appears coherent as well.

The presence of a 5/rev component in the measured torsion response remains unclear,given that no modeshapes are predicted at that frequency. And the finite element model isbelieved to be accurate because the discrepancies between measured and predicted eigen-frequencies of the non-rotating blade differ at maximum by 4%.

Figure 5.26: Harmonic analysis of the inviscid airloads on the ERATO rotor.

Summarizing, there is no strong evidence of flaws in the mechanical model, save forthe Coriolis forces, which are not modeled at all by MSC.Marc, as Section 2.2.1.6 proved.The lack of Coriolis forces is not expected to deteriorate much the solution though.

An interesting exercise to test the quality of the 3D FE mechanical model would beto prescribe the experimental airloads. This was not done due to lack of time. And suchtests introduce problems of their own, since the experimental airloads are only available at5 or 6 spanstations and therefore extensive interpolations and extrapolations are required.

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5. APPLICATIONS

A second and important exercise would be to compare the here presented results forthe ERATO rotor obtained by the HOST/MSC.Marc/CFD coupling with the predictionsof a HOST/CFD coupling. This exercise would allow to use the same aerodynamic modelon both simulations and thus focus on the comparison of the dynamics predictions by the3D FE model and by the beam model in HOST. Again, this comparison was not done dueto lack of time to prepare the HOST/CFD coupling for the ERATO rotor. It is plannedto perform this simulation shortly.

5.2.6 Computational cost

All simulations used five processors, scattered over three computers, simultaneously. MSCMarc runs were done on a HP-Itanium cluster with 1.4GHz processors, using one processorfor each of the two parallel two-bladed finite element models. A third processor on thiscluster hosted the coupling server. The server hardly requires computing power and canbe run anywhere. HOST, which consumes very little computing power as well, was run ona SUN computer. The CFD was run on a NEC SX-8 vectorial supercomputer with 2 GHzprocessors. Due to a limitation in the grid deformation tool, the four blocks of the rotorgrid had to be run on a single processor. For the CFD options used here, each time-steptook approximately 2 wallclock minutes for RANS simulations and less than 30 secondsfor Euler simulations. A time-step in MSC.Marc took approximately 2 wallclock minutes.Therefore, a rotor revolution made of 300 increments with RANS CFD can be obtained inapproximately 10 wallclock hours if a parallel staggered scheme is used. For a convergedsolution of 8 rotor revolutions, 80 wallclock hours are necessary. This computation timedoubles if a serial staggered scheme and RANS CFD are used. Computations times areexpected to diminish significantly in the near future as CFD accepts parallelization andMSC.Marc is run on the supercomputer.

5.3 Comparison of coupling staggered algorithms

Three staggered schemes are compared in this section. The objective is to confirm theliterature claims on the different versions of coupling algorithms.

The procedure consisted in simulating the three initial ERATO rotor revolutions inthe high-speed flight condition of the previous sections, after starting from scratch eachtime. The time-step was increased by a factor of 3 with respect to the time-step used forthe results of the previous sections. The new time step was equivalent to ∆Ψ = 3.6deg.This three-fold increase of the time-step was done in order to put a greater strain on thetime-accuracy of the staggered schemes.

The inviscid Euler CFD model was used because flow accuracy was not a priority andit would save computational resources. For further simplification, the active trim was notused initially. Instead, fixed rotor controls were used.

The following three staggered schemes were tested: the serial collocated (SC), theparallel collocated (PC) and the serial non-collocated (SNC).

110


The comparison criterion was the work performed by the fluid and structure subsys-tems on the fluid/structure interface.

Figures 5.27 – 5.29 show the time-histories of the integrated works WF and WS per-formed by the fluid and structure subsystems on the fluid/structure interface of a sin-gle blade and computed using the serial collocated, parallel collocated and serial non-collocated schemes. These works are evaluated as

WF =n∑1

W nF , WS =

n∑1

W nS , (5.4)

where W nF and W n

S are the works performed by the fluid and structure subsystems duringthe time interval [tn−1, tn]. The evaluations of W n

F and W nS are done as follows

WS =1

2

(FnS + Fn+1

S

)T (un+1S − unS

), (5.5)

WF =1

2

(FnF + Fn+1

F

)T (xn+1F − xnF

)(for SC and PC), (5.6)

WF =1

2

(W

n+1/2F +W

n+3/2F

)(5.7)

=1

4

(Fn−1/2F + Fn+1/2

F

)T (xn+1/2F − xn−1/2

F

)(5.8)

+1

4

(Fn+1/2F + Fn+3/2

F

)T (xn+3/2F − xn+1/2

F

)(for SNC), (5.9)

where F are the airloads, including three forces and three moments. uS and xF representthe position of the structure and fluid nodes at the fluid/structure interface, respectively.

Due to a mishandling of the frames of reference, the x and y components of the workwere badly calculated. The works plotted in the figures below correspond only to the workperformed by the vertical force and vertical displacements along the z direction. This isanyway a good indicator because lift and flapping account for most of the work at thefluid/structure interface.

Figures 5.27 – 5.29 show that the serial non-collocated (SNC) scheme is the one violat-ing in the least the conservation of the energy at the fluid/structure interface. The worksperformed by the fluid and structure subsystems using the SNC scheme look coincidentin Figure 5.29. With the SNC, the relative variation (WF −WS) /WS oscillates between0.15% and 3% (apart from two offshots at the very start of the simulation). Thereby,the serial non-collocated scheme does preserve the energy at the fluid/structure interfacebetter than the parallel and serial collocated schemes, in agreement with what has beenpublished in [26].

Despite the evident differences in energy conservation at the fluid/structure inter-face of the different staggered schemes, when the aeroelastic solutions (i.e., airloads andstructural displacements) are compared, differences between the schemes become negligi-ble. Only the sectional airloads using the parallel scheme can be observed to oscillate at

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5. APPLICATIONS

Figure 5.27: Integrated work at the fluid structure interface using the serial collocatedscheme.

Figure 5.28: Integrated work at the fluid structure interface using the parallel collocatedscheme.

112


Figure 5.29: Integrated work at the fluid structure interface using the serial non-collocatedscheme.

higher frequency during the first revolution. This is shown in Figure 5.30, where the pitch-ing moment obtained with the parallel collocated and serial non-collocated schemes arecompared. Figure 5.30(a) shows the greater oscillations in pitching moment produced bythe parallel scheme, although only during the first revolution. The second and third revo-lutions are indistinguishable to those of the serial non-collocated scheme (Figure 5.30(b)).The normal force, not plotted, was also observed to oscillate with the parallel scheme,but not as much as the pitching moment.

The serial collocated and non-collocated schemes gave equal airloads evolutions.

A second series of algorithm comparisons was performed by including the active trim.The objective was to observe the effects, if any, of the inclusion of varying rotor controlsin the comparison of the staggered schemes. The rest of the simulation conditions werethe same than in the previous exercise: ∆Ψ = 3.6deg. The serial collocated scheme, beingless accurate than the non-collocated but equally computationally costly, was left out ofthe comparisons.

The rotor controls variations in the two simulations are very similar and are shown inFigure 5.31.

The evolution of the sectional airloads at r/R=0.975 with the parallel scheme is shownin Figure 5.32. The evolution of the sectional airloads at r/R=0.975 with the serial non-collocated scheme results is shown in Figure 5.33. It can be readily observed that the

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5. APPLICATIONS

(a) Pitching moment using parallel scheme. (b) Pitching moment using serial non-collocatedscheme.

Figure 5.30: Comparison of the pitching moments at r/R=0.975 using parallel and serialschemes.

Figure 5.31: Rotor controls evolution with the parallel collocated and serial non-collocatedschemes.

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parallel scheme leads to non-physical oscillating airloads, even more markedly for thepitching moment, see Figure 5.32(a). On the other hand, the serial non-collocated scheme,all other things equal, yields a smooth evolution of the airloads.

These results illustrate again the inferior accuracy of the parallel scheme as comparedto the serial non-collocated scheme. However, it seems likely that the airloads oscillationsobserved with the parallel scheme would disappear if the simulation ran for a few morerevolutions and rotor controls began to stabilize.

5.3.1 Remarks on the comparison of staggered schemes

In view of the obtained results and in agreement with theory, there are two optimalstaggered algorithms. If accuracy is to be maximized, the serial non-collocated schemepreserves best the energy at the fluid/structure interface for the same computationalcost than the serial collocated. However, for those applications where both the fluid andstructure subsystems have equal computation times, the parallel scheme also proves to bea very interesting choice.

In the applications of this work no significant differences were observed between thedifferent schemes, probably because of too small time-steps. In such case the parallelscheme becomes the best option because it minimises the overall simulation time.

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5. APPLICATIONS


Figure 5.32: Parallel collocated scheme, with active trim.


Figure 5.33: Serial non-collocated scheme, with active trim.

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Conclusions and Perspectives

This research has introduced for the first time three-dimensional finite element models ofthe rotor blades in the aeroelastic analysis of a helicopter rotor. A finite element solverhas been successfully coupled to a three-dimensional unsteady fluid solver and to a rotor-craft comprehensive analysis. The coupled procedure involves dynamic simulations usinga time-accurate coupling procedure, where solutions are exchanged at each time-step.

The basis of a general coupling framework open to any CFD or CSM solver compliantwith the CGNS public data model has been developed. This framework is noteworthy forits modularity, which permits to plug a variable number of applications. In addition, thenew framework allows to combine fluid and structure models of varying complexity, thusproviding a means to tailor the numerical modeling to the specific problem under study.

This study has solved a long criticized drawback of time-accurate couplings in rotor-craft aeroelasticity: their incapacity to yield trimmed solutions of steady flight conditionswhile respecting a set of target aerodynamic forces necessary for in-flight equilibrium. Thesolution consisted in implementing a continuous rotor control modification in proportionto the offset of the actual rotor generated aerodynamic forces with respect to the targetvalues. This new method has been successfully applied to both autonomous HOST sim-ulations and to the coupled analyses of 3D unsteady CFD with 3D finite element baseddynamics. Rotor controls become reasonably stabilized after 6 rotor revolutions.

The capabilities of the new coupling are demonstrated by applying it to investigatethe aeroelastic response of two different isolated rotors: the 7A rotor and the ERATOrotor. The numerical results are compared to measurements.

The 7A rotor has been used to validate the HOST/MSC.Marc/elsA simulation bysuccessfully reproducing the airloads predicted by an equivalent HOST/elsA coupling,yet at a lower computational cost because less CFD rotor revolutions were performed andtrim convergence was performed automatically.

The ERATO rotor has been used as a demonstrator for the first ever application of3D finite element based models of the blades in the aeroelastic analysis of an isolatedrotor. Simple yet effective and conservative methods to transfer structure motion andloads between spatially non-matching meshes have been implemented. The conservationof the energy at the fluid/structure interface has been used to validate the accuracy of

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Conclusions and Perspectives

the transfers. The early results are in fair agreement to experiment; sectional airloadsand blade tip torsion reproduce the experimental trends, but the frequency contents andthe amplitude of the experiment are underpredicted. It is not possible yet to comparethe prediction of the dynamic response of the blades by HOST with that of the 3D FEmodel because different aerodynamics models were used. Further investigation is required.

Three different iteration-free staggered schemes for time-accurate couplings have beentested: the serial collocated, the parallel collocated and the serial non-collocated. The con-servation of the energy at the fluid/structure interface has been used to ascertain that themost accurate method is the serial non-collocated, in agreement with literature. However,the three staggered schemes did not yield significantly different solutions. Consequently,the best choice is the parallel collocated scheme because it minimises the computationtime.

This work has contributed to the initial phase of an international collaboration involv-ing the research centers ONERA and DLR and a helicopter manufacturer. The develop-ments here undertaken will be pursued until the end of the project in 2011.

Future axes of research will include:

• To continue the analyses of the isolated rotors here undertaken. New simulationswith the Coriolis forces taken into account properly and a more refined fluid grid.

• To assess the beam model in HOST by comparing HOST/CFD aeroelastic predic-tions with HOST/MSC.Marc/CFD ones.

• To implement to a full extent the CGNS data model to describe structure data onceit is approved by the international CGNS Steering Committee.

• To perform simulations in which the fluid grid surface is deformed following thesurface deformation of the 3D FE model and the structural loads are prescribed as adistribution of pressure and shear. More generally, to develop a general interpolationtoolbox that can handle all combinations of geometries.

• To apply the new 3D finite element capability to model non-beam like structures,such as bearingless rotor hubs.

• To enable coupled simulations in which the structure is modeled in HOST and only acritical component of the structure is modeled by external 3D CSM methods. Thisoption is very appealling because it limits the use of computationally expensive3D models to a minimum, yet at the same time the partition of a single structurebetween two structural solvers is challenging and requires investigation.

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The work to come will stand on the bases here settled. As the rotorcraft couplingbecomes more mature, both research and industry are to exploit the versatility and high-accuracy potential of the coupled analysis to gain better understanding of rotorcraftaeroelastic phenomena and eventually, design the rotorcraft of tomorrow.

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Appendix A

Frames of reference

The frames of reference given in this appendix were extracted from the document “CHANCE- Definition of Output Quantities”, Revision 1.2, October 16 2001.

Figure A.1: Helicopter frame and fixed rotor frame. Side view.

The rotating rotor frame, shown in Figure A.3, is a rotor-fixed frame, as opposed tothe fixed rotor frame, which does not rotate with the rotor angular speed Ω.

125

A. FRAMES OF REFERENCE

Figure A.2: Helicopter frame and fixed rotor frame. Top View.

Figure A.3: Fixed rotor frame and rotating rotor frame.

126

Figure A.4: Pitch, flap and lag angles of the blade.

Figure A.5: Section frame.

127

CSM/CFD Coupling for the Dynamic Analysis of Helicopter Rotors

This work has developed a method to perform aeroelastic simulations of helicopter rotors by coupling athree-dimensional finite element structural (CSM) solver to a computational fluid dynamics (CFD) solver and to aflight mechanics code. The objective is to develop a numerical analysis tool able to predict accurately theaeroelastic behaviour of arbitrary rotorcraft configurations and geometries. This is not possible with today’sanalyses for rotorcraft aeroelasticity because they are all constrained by the use of beam theory in their structuralmodels.The basic idea is to use a partitioned approach in which existing specialized software for fluid, structure and flightmechanics are coupled in a time-accurate fashion. The solvers exchange their solutions at each time-step with aniteration-free staggered scheme. For this purpose, a software environment for coupling codes distributed over anetwork was developed, together with a new method to control the rotor and obtain aeroelastic predictionsrespecting in-flight equilibrium. The new coupling was applied to the simulation of isolated rotors in levelhigh-speed flight and results were compared to wind tunnel measurements.The coupling can solve both steady and transient flight conditions, including nonlinear phenomena. It can be usedfor the analysis of complex, non slender, structures such as bearingless rotor hubs. The partitioned approachenables the exploitation of state-of-the-art methods in each subdiscipline while conserving the energy at thefluid/structure interface, and the iteration-free staggered schemes can yield second-order time-accuracy. Theaeroelastic predictions match the flight mechanics targets. The coupling is successfully applied to the aeroelasticsimulation of a rotor with 3D finite element modelled blades. Future work will pursue the developments hereengaged in view of an industrial deployment.

Mots-clés : AEROELASTICITY ; ROTORCRAFT ; FLUID-STRUCTURE INTERACTION ; PARTITIONED PROCEDURE ;

CFD/CSM COUPLING ; TIME-ACCURACY

Couplage CSM/CFD pour l’Analyse Dynamique des Rotors d’Hélicoptère

Cette thèse a développé une méthode pour réaliser des simulations aéroélastiques de rotors d´hélicoptère encouplant un code tridimensionnel d’éléments finis (CSM) avec un code de mécanique de fluides (CFD) et un codede mécanique de vol. L’objectif a été de développer un outil d’analyse numérique capable de prédire avecprécision le comportement aéroélastique d’une configuration a voilure tournante de géométrie arbitraire. Cecin’est pas possible avec les outils actuels d’aéroélasticité hélicoptère car ils sont tous limités par l’utilisation demodèles structuraux de type poutre.L’idée principale est d’utiliser une approche partitionnée qui permet de coupler les logiciels spécialisés existantspour la structure, le fluide et la mécanique de vol. Ces codes échangent leurs solutions à chaque pas de tempsavec une méthode alternée sans sous itérations. Pour cela, un environnement logiciel pour le couplage de codesdistribués sur le réseau a été développé, ainsi qu’une méthode pour contrôler le rotor et obtenir des prédictionsaéroélastiques respectant l’équilibre en vol. Le nouveau couplage a été appliqué à la simulation de rotors isolésen vol horizontal d’avancement à grande vitesse et les résultats ont été comparés aux mesures en soufflerie.Le couplage peut également calculer des réponses en vol de manœuvre et prendre en compte des effets nonlinéaires, ainsi que les moyeux complexes modernes de type ‘bearingless’. L’approche partitionnée permetl’exploitation de l’état de l’art dans chacune des disciplines tout en conservant l’énergie à l’interfacefluide/structure, et le schéma de couplage sans sous itérations peut être précis au deuxième ordre dans le temps.Les prédictions aéroélastiques respectent les objectifs de la mécanique de vol. Le couplage a été appliqué avecsuccès à la simulation aéroélastique d’un rotor isolé dont les pales ont été modélisées par éléments finis 3D. Lesdéveloppements ici entamés seront poursuivis et inscrits au sein de la gamme d’outils utilisés dans l’industrie.

Keywords : AEROELASTICITE ; HELICOPTERE ; INTERACTION FLUIDE-STRUCTURE ; METHODE PARTITIONNEE ;

COUPLAGE CFD/CSM ; PRECISION TEMPORELLE

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