A fluid-structure interaction partitioned algorithm applied to flexible flapping wing propulsion

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A Fluid-Structure Interaction Partitioned

Algorithm Applied to Flexible Flapping Wing

Propulsion

Thèse

Mathieu Olivier

Doctorat en génie mécanique

Philosophiæ doctor (Ph.D.)

Québec, Canada

© Mathieu Olivier, 2014

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Résumé

Cette thèse concerne l’étude des ailes oscillantes flexibles et des méthodes numériques qui s’y rattachent. De ce fait, la thèse est divisée en deux parties. La première contribution concerne le développement d’un algorithme de couplage fluide-structure qui prend en charge les interac-tions entre un solide élastique en grands déplacements et un fluide incompressible. L’algorithme est basé sur une approche partitionnée et permet d’utiliser des codes numériques de mécanique des fluides et de mécanique des solides existants. L’utilisation d’un terme de compressibilité artificiel dans l’équation de continuité du fluide combinée à des choix algorithmiques judi-cieux permet d’utiliser cette méthode de couplage efficacement avec un code de mécanique des fluides utilisant une méthode de projection de type SIMPLE ou PISO. La seconde contribu-tion est l’étude de l’effet de flexibilité des ailes sur le vol à ailes battantes. Deux principaux régimes de vol sont mis en évidence concernant la déformation de l’aile : déformation causée par la pression et déformation causée par l’inertie. Les effets de ces régimes sur la topologie de l’écoulement et sur les performance de l’aile en propulsion sont discutés. Il est montré que les cas avec des déformations causées par la pression présentent généralement des efficacités plus élevées avec une flexibilité modérée. Il en est de même pour la force de poussée lorsque l’amplitude de tangage est faible. D’autre part, lorsque les déformations sont causées par l’inertie, les performances de l’aile sont généralement réduites. Certains cas montrent une aug-mentation marginale des performances lorsque le synchronisme des déformations est optimal, mais ces cas représentent davantage une exception que la norme. Il est également démontré que la flexibilité peut être utilisée comme mécanisme de tangage passif tout en conservant des per-formances intéressantes. Enfin, un modèle d’aile oscillante flexible non contraint est présenté. Il est démontré que le mouvement de déviation observé dans la nature est une conséquence d’un phénomène aérodynamique de mise en drapeau.

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Abstract

This thesis concerns the study of flexible flapping wings and the related numerical methods. It thus contains two distinct themes. The first contribution is the implementation of an efficient fluid-structure interaction algorithm that handles the interaction of an elastic solid undergoing large displacement with an incompressible fluid. The algorithm is based on the partitioned approach and allows state-of-the-art fluid and structural solvers to be used. Stabilization with artificial compressibility in the fluid continuity equation along with judicious algorithmic choices make the method suitable to be used with SIMPLE or PISO projection fluid solvers. The second contribution is the study of the effects of wing flexibility in flapping flight. The different regimes, namely inertia-driven and pressure-driven wing deformations are presented along with their effects on the topology of the flow and, eventually, on the performance of the flapping wing in propulsion regime. It is found that pressure-driven deformations can increase the thrust efficiency if a suitable amount of flexibility is used. Thrust increases are also observed when small pitching amplitude cases are considered. On the other hand, inertia-driven deformations generally deteriorate aerodynamic performances of flapping wings unless meticulous timing is respected, making them less practical. It is also shown that wing flexibility can act as a passive pitching mechanism while keeping decent thrust and efficiency. Lastly, a freely-moving flexible flapping wing model is presented. It is shown that the deviation motion found in natural flyers is a consequence of a feathering mechanism.

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List of Tables

3.1 x-component of the beam tip displacement, beam solver. . . 46

3.2 y-component of the beam tip displacement, beam solver. . . 46

3.3 x-component of the beam tip displacement, finite-volume solver. . . 47

3.4 y-component of the beam tip displacement, finite-volume solver.. . . 47

4.1 Average number of outer-iterations for the first 20 time-steps with the beam finite-element solver. . . 80

4.2 Average number of outer-iterations for the first 20 time-steps with the finite-volume solver. . . 81

4.3 Summary of the stability of the possible solver combinations. . . 81

4.4 Flow and structure parameters. . . 84

4.5 Levels of resolution used for the Heathcote problem. . . 110

4.6 Simulations using different FSI schemes and solver combinations. . . 119

4.7 Relative simulation times for Case D . . . 123

6.1 Dimensionless parameters for the flexible flapping wing problem. . . 157

6.2 Flight characteristics of some animal species. . . 161

7.1 Dimensionless parameters for the freely-moving flexible flapping wing problem. . . 199

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List of Figures

3.1 Force coefficients comparison using OpenFOAM, Fluent, and the Vortex method. . 40

3.2 Normalized vorticity (ωz/f) at ft = 1.75. . . 41

3.3 Undeformed and deformed configuration of the beam undergoing a nodal transver-sal force.. . . 48

3.4 Undeformed and deformed configuration of the beam undergoing a 2π normalized bending moment. . . 48

3.5 Instantaneous views of the deformed beam at various instants. . . 49

3.6 Relative transverse displacement of the beam tip. . . 50

3.7 Vectors used to calculate surface deformation. . . 53

3.8 Illustration of control points selection. . . 54

3.9 Mesh of a NACA0015 after a rotation of 90◦_{. IDW interpolation is used with all} boundary points. . . 58

3.10 Mesh of a NACA0015 after a rotation of 90◦_{. IDW interpolation is used with radial} damping. . . 59

3.11 Illustration of cell to point interpolation failure when a finite-volume differential equation based mesh motion algorithm is used. . . 60

3.12 Rigid body rotation of a three-dimensional plate. . . 62

3.13 Initial mesh of a block. . . 63

3.14 Parabola-shaped deformation of the block. . . 65

3.15 Rigid-body rotation and translation of the block. . . 66

3.16 Rotation and elastic deformation of a 2D flat plate. . . 67

4.1 Fluid mesh used for numerical tests also showing the boundary conditions. . . 78

4.2 Instantaneous velocity field at different times with ρ∗_{= 50.} _{. . . 79}

4.3 Geometry of the benchmark problem. . . 85

4.4 Near-body mesh used for the coarse-resolution simulations. . . 85

4.5 Force coefficients for the FSI2 test case. . . 86

4.6 Instantaneous vorticity field for the FSI2 test case. . . 87

4.7 Instantaneous pressure coefficient field for the FSI2 test case. . . 88

4.8 Force coefficients for the FSI3 test case. . . 89

4.9 Instantaneous vorticity field for the FSI3 test case. . . 90

4.10 Instantaneous pressure coefficient field for the FSI3 test case. . . 91

4.11 Temporal evolution of the net volumetric influx. . . 93

4.12 Meshes used in the flexible tube simulations . . . 94

4.13 Number of outer-iterations for the coarse and fine simulations. . . 95

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4.15 Pressure wave propagation in the flexible tube with a lower normalized rigidity E∗_. ₉₇

4.16 Pressure residual evolution in two arbitrary time-steps in the simulation. . . 98

4.17 Speed-up obtained with multiple processor compared to ideal speed-up. . . 99

4.18 Pressure wave propagation in the long flexible tube. . . 100

4.19 Pressure probes at various position on the center line of the tube. Comparison of the different solution methods. . . 101

4.20 Grid independence verification using the pressure probes on the center line for the partitioned FVM solution (current study). . . 102

4.21 Grid independence verification using the pressure probes on the center line for the monolithic FEM solution, axisymmetric solutions with linear elements (P1-P1 in the fluid region). . . 103

4.22 Grid independence verification using the pressure probes on the center line for the monolithic FEM solution, axisymmetric solutions with quadratic elements (P2-P1 in the fluid region). . . 104

4.23 Domain and boundary conditions used for the numerical simulation of the Heath-cote experiment. . . 105

4.24 Mesh used for the numerical simulation of the Heathcote experiment.. . . 106

4.25 Pressure coefficient, vorticity, and eddy viscosity fields obtained using the 2D Navier-Stokes and the 2D Spalart-Allmaras models with f∗ _{= 0.876} _{and e}∗ ₌

5.6× 10−4_. _{. . . 107}

1.41× 10−3_. _{. . . 108}

5.6_{× 10}−4_. _{. . . 109}

4.28 y-component of the trailing edge relative displacement for the reference Heathcote problem with e∗ _{= 5.6}_{× 10}−4 _{and f}∗ _{= 0.876}_{using different resolutions.} _{. . . 111}

4.29 Instantaneous vorticity field at ft = 0.75 and ft = 1 with f∗ _{= 0.876} _{and e}∗ ₌

5.6_{× 10}−4 _{for three different numerical resolutions.} _{. . . 112}

4.30 y-component of the trailing edge relative displacement for the reference Heathcote problem. . . 113

4.31 Instantaneous vorticity field at ft = 0.75 and with f∗ _{= 1.22}_{and e}∗_{= 8.5}_{× 10}−4_. ₁₁₃

4.32 Number of outer-iterations required to reach convergence in a single oscillation cycle. . . 114

4.33 Medium mesh used for the fluid region. . . 118

4.34 Number of outer-iterations for the three simulations used for the resolution verifi-cation. . . 119

4.35 Snapshots of the vorticity field at different times with the coarse, medium, and fine resolution (mesh and time-step). . . 120

4.36 Comparison of force coefficients with different resolutions. . . 121

4.37 Pressure coefficient at ft = 0.125.. . . 122

4.38 Far-field pressure coefficient near the beginning of the simulation (ft = 0.025). . . 124

4.39 Pressure coefficient a long time after the start-up (ft = 5.75). . . 124

4.40 Aerodynamic thrust coefficient of the whole plate for the last simulation cycle using different models and parameters. . . 125

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4.41 Comparison between the structural reaction on the LE and the aerodynamic force

on the FSI interface. . . 126

4.42 Evolution of the pressure coefficient field. . . 127

4.43 Vorticity field in the wake. . . 128

5.1 Three-dimensional wing motion. . . 136

5.2 Two-dimensional wing motion types. . . 136

6.1 Geometry of the rounded-edge flat plate cross-section used as a flapping wing. The light gray region is rigid while the dark gray region is flexible. . . 149

6.2 Imposed motion on the rounded-edge flat plate. . . 150

6.3 Illustration of the effective pitching angle θEff. . . 151

6.4 Initial and deformed configurations of the wing section. . . 155

6.6 Efficiency contours. . . 163

6.7 Mean thrust coefficient contours. . . 164

6.8 Mean thrust coefficient normalized standard deviation contours. . . 165

6.9 Comparison of the instantaneous thrust coefficient for two flapping plates with different thicknesses. . . 166

6.10 Normalized vorticity field for a high-efficiency case. . . 168

6.11 Normalized vorticity field for a high-thrust case. . . 169

6.12 Normalized vorticity field for a moderate-thrust case. . . 170

6.13 Efficiency of the flexible flapping wing as a function of the flexibility. . . 172

6.14 Mean thrust coefficient of the flexible flapping wing as a function of the flexibility. 173 6.15 Mean power coefficient of the flexible flapping wing as a function of the flexibility. 174 6.16 Vorticity field for various flexibilities involving strong interaction. . . 175

6.17 Pressure coefficient field for various flexibilities involving strong interaction. . . . 175

6.18 Configuration exhibiting the second deformation mode. . . 176

6.19 Force coefficients acting on the driving mechanism for the pressure driven defor-mation cases (Σ = 50) . . . 177

6.20 Aerodynamic force coefficients for the pressure driven deformation cases (Σ = 50). 178 6.21 Instantaneous effective pitching angle (Σ = 50). . . 179

6.22 Comparison of the effective pitching angle between the optimal flexible wing (Σ = 50 and δ∗ = 1.30), the rigid wing, and the modified pitching motion. . . 179

6.23 Aerodynamic thrust coefficient of the rigid oscillating wing with a modified pitching motion. . . 180

6.24 Vorticity field for various flexibilities involving weak interaction. . . 182

6.25 Pressure coefficient field for various flexibilities involving weak interaction. . . 182

6.26 Aerodynamic force coefficients for inertia driven deformation cases (Σ = 0.10). . . 183

6.27 Instantaneous effective pitching angle (Σ = 0.1). . . 184

6.28 Thrust coefficients acting on the driving mechanism for an inertia driven deforma-tion case. . . 184

6.29 Force coefficients acting on the driving mechanism for inertia driven deformation cases (Σ = 0.10). . . 186

6.30 Instantaneous effective pitching angle (Σ = 1.67). . . 188

6.31 Force coefficients acting on the driving mechanism for mixed deformation cases (Σ = 1.67). . . 189

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6.33 Vorticity field for various flexibilities involving moderate interaction. . . 191

6.34 Pressure coefficient field for various flexibilities involving moderate interaction. . . 191

6.35 Propulsion performance metrics for inertia-driven deformation cases with θ0= 0. . 193

6.36 Comparison of the thrust coefficients between an inertia-driven deformation case with θ0 = 0and the equivalent rigid case. . . 194

7.1 Harmonic heaving motion combined with a harmonic deviation. . . 201

7.2 Vorticity field for various flexibilities involving strong interaction. . . 203

7.3 Pressure coefficient field for various flexibilities involving strong interaction. . . . 203

7.4 Rigid wing displacement relative to a point that moves at the wing average velocity for various Σ. . . 204

7.5 Vorticity field for various flexibilities involving weak interaction. . . 205

7.6 Pressure coefficient field for various flexibilities involving weak interaction. . . 205

7.7 Aerodynamic thrust coefficient of rigid wings for various Σ. . . 206

7.8 Wing displacement relative to a point that moves at the wing average velocity for various flexibilities δ∗ _{at Σ = 50.} _{. . . 207}

7.9 Aerodynamic thrust coefficient of flexible wings for Σ = 50. . . 208

7.10 Propulsion performance metrics for rigid freely-moving wings with respect to Σ. . 209

7.11 Efficiency of the freely-moving flexible flapping wing as a function of the flexibility. 210 7.12 Mean normalized velocity of the freely-moving flexible flapping wing as a function of the flexibility. . . 211

7.13 Mean power coefficient of the freely-moving flexible flapping wing as a function of the flexibility. . . 212

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À Geneviève, Édouard et Charlotte.

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Remerciements

Dans un premier temps, je tiens à remercier chaleureusement mon directeur de recherche, le professeur Guy Dumas. Son support, sa passion et son expérience ont été une grande source de motivation et d’inspiration tout au long de mon parcours aux études graduées. Guy est un directeur exceptionnel, tant au niveau professionnel que personnel.

Je tiens également à remercier tous mes collègues, passés et présents, du Laboratoire de Mécanique des Fluides Numérique (LMFN). Depuis mon premier stage au laboratoire en 2006, vous avez toujours fait duLMFNun endroit propice aux discussions et à l’entraide. Plus particulièrement, je tiens à remercier Thomas Kinsey et Steve Julien pour m’avoir introduit aux rudiments de la mécanique des fluides numérique, Julie Lefrançois pour sa disponibilité ainsi que Simon Lapointe et Philippe Bélanger-Vincent pour l’entraide mutuelle avec certains logiciels obscurs... Je salue également Jean-François Morissette et Frédérik Chan avec qui j’ai eu la chance de collaborer plus étroitement. Je voudrais aussi remercier François Lesage et Nicolas Hamel de RDDC Valcartier pour leur collaboration lors de mes deux premières années aux études graduées.

Je remercie également le Conseil de Recherches en Sciences Naturelles et en Génie du Canada (CRSNG) pour l’octroi d’une bourse de doctorat en recherche ainsi que le Fonds de Recherche du Québec – Nature et Technologies (FRQNT) pour l’octroi d’une bourse de doctorat dans le domaine de l’aérospatiale. Ce soutien financier a été une incroyable opportunité qui m’a permis de réaliser l’un de mes objectifs de carrière.

Mon cheminement académique n’aurait sans doute pas été possible sans l’appui inconditionnel de mes parents Johanne et Claude et de mon frère François qui me supportent et m’encouragent depuis le début. Je les remercie chaleureusement.

Enfin, je tiens à offrir mes derniers remerciements à ma conjointe Geneviève. Son amour et son support m’ont permis de réaliser un rêve tout en menant une vie équilibrée. Elle a été une excellente conseillère qui a su trouver les bons mots dans les périodes de doute. Une grande part de mon succès lui revient. Geneviève, merci pour tout ce que nous vivons ensemble et, comme tu le dis si bien, j’aime ce que nous devenons !

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Nomenclature

Symbols

A Beam cross-section

C External moment on a beam, force or power coefficient CP Power coefficient CT Thrust coefficient Cp Pressure coefficient Co Courant number D Drag E Young modulus E∗ _{Normalized rigidity}

I Beam cross-section area moment of inertia I0 Beam cross-section area moment of inertia per

unit span

J Determinant of the deformation gradient tensor L Length, lift

Lref Reference length

M Internal bending moment in a beam, moment M a Mach number

N Internal normal force in a beam P Power

Q Volumetric flux

R Radius, residual functional Re Reynolds number

S Surface, external shear force on a beam St Strouhal number

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T External traction force on a beam, thrust force, period

Tref Reference time

U∗ Normalized velocity U∞ Freestream velocity

V Volume

Vref Reference velocity

Γ Mesh pseudo-diffusion or pseudo-rigidity Σ Interaction strength parameter

α Angle of attack, coefficient in RBF interpola-tion

αEff Effective angle of attack

¯

U Mean velocity

β Coefficient in RBF interpolation

δ

δt Discretized temporal derivative operator

δ∗ Normalized flexibility η Efficiency

δ

δx Discretized spatial derivative operator

λ Second Lamé coefficient µ First viscosity coefficient µ0 _{Second viscosity coefficient}

∇ Differential operator ν Poisson coefficient

φi Finite element shape function, radial function

ψ Finite element test function, compressibility coefficient

ρ Density ρ∗ _{Density ratio}

σeq Equivalent (von Mises) stress field

θ Pitching angle θ0 Pitching amplitude

θEff Effective pitching angle

˜

ψ Artificial compressibility coefficient ˜

τ Lineic shear load on a beam ˜

ξ Curvature of the deformed beam ˜

p Lineic pressure load on a beam υ First Lamé coefficient

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D Strain rate tensor field

E Green-Lagrange strain tensor field F Deformation gradient tensor field

H Linear system right-hand side coefficient of the momentum equation without the pressure con-tribution

P First Piola-Kirchhoff stress tensor field

Q Deformation tensor field used in motion decom-position

S Second Piola-Kirchhoff stress tensor field ˆ

n Unit surface normal vector ˆ

v Control surface velocity ω Vorticity vector field σ Cauchy stress tensor field c Relative convective velocity d Displacement vector field f Arbitrary body force q Heat flux vector field r Position vector

t Displacement pure translation component v Velocity vector field

ξ Coordinate on the reference beam element ζ Virtual efficiency

a Linear system matrix coefficient

b Linear system right-hand side coefficient c Chord length

e Wing thickness, total specific energy field, xx component of the Green-Lagrange strain tensor field

e∗ Normalized wing thickness f Frequency

f∗ _{Reduced frequency}

fref Reference lineic load

h Heaving displacement

h∗ Normalized heaving amplitude h0 Heaving amplitude

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p Pressure field, polynomial pref Reference pressure

r Internal heat generation rate, radial distance s specific entropy field

t Time

t∗ Normalized time w Interpolation weight x∗

P Normalized pitching axis position

xP Pitching axis position (pivot point)

Subscripts

0 Reference state, initial configuration f fluid s solid x x-component of a vector y y-component of a vector z z-component of a vector LE Leading Edge

N Neighbor cell index

P Pivot point, current cell index ref Reference quantity

TE Trailing Edge

Superscripts

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Acronyms

ALE Arbitrary Lagrangian Eulerian CFD Computational Fluid Dynamics FEM Finite-Element Method

FSI Fluid-Structure Interaction FVM Finite-Volume Method

GAMG Geometric-Algebraic Multi-Grid GCL Geometric Conservation Law IDW Inverse Distance Weighting ILES Implicit Large-Eddy Simulation LES Large-Eddy Simulation

LMFN Laboratoire de Mécanique des Fluides Numérique

MAV Micro Air Vehicle NAV Nano Air Vehicle NN Nearest Neighbor

PBiCG Preconditioned Bi-Conjugate Gradient PISO Pressure-Implicit with Splitting of Operators PIV Particle Image Velocimetry

RANS Reynolds-Averaged Navier-Stokes RBF Radial Basis Function

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

SIMPLEC SIMPLE-Corrected SIMPLER SIMPLE-Revised

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TPS Thin Plate Spline WR Weighted Residual

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Introduction

Context

The way insects and birds fly has always been fascinating. It inspired early models of flying machines that have evolved up to modern aviation. Human knowledge has greatly progressed in many related fields since the early flying attempts. Nevertheless, flapping flight, which is commonly found in nature, appears to be a very complex process that only begins to reveal its secrets.

The evolution and the miniaturization of technologies in many fields such as mechanical sys-tems and electronics (guidance and navigation syssys-tems, communications, etc.) brings the possibility to build small scale flying machines generally referred to as Micro Air Vehicle (MAV) or Nano Air Vehicle (NAV). Because of their small size, energy storage is limited and achieving efficient lift and thrust becomes a major concern. However, it is now well known that traditional aerodynamic devices such as fixed and rotary wings are much less efficient at low Reynolds number. As an example, Kunz and Kroo (2001) report that lift-to-drag ratio of a conventional wing profile can drop one order of magnitude when the Reynolds number is largely below what it is in conventional aeronautical applications. More specifically, they present results in which the drag doubles when the Reynolds number drops from 6000 to 2000 while the lift remains mostly unchanged.

Although a lot of progress has been made notably in fluid mechanics, understanding the physics of flapping flight is still very challenging. Indeed, it involves complex transient flows along with moving wings. Furthermore, on most flying species, the wings often undergo significant deformations that interact with the flow field. The first striking observation is that theoretical models commonly used in classical aerodynamics are not usable for such problems. Indeed, these models predict that insects produce insufficient lift to carry their mass. It thus becomes clear that unsteady flow mechanisms play a significant role in the production of aerodynamic forces (seeAnsari et al.(2006);Dickinson et al.(1999), andEllington et al.(1996)). Obviously, such mechanisms introduce new possibilities, but also more complexity to the problem. Among the new possibilities, there are stationary flights and sudden course changes while high speed

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and efficient forward flight still appears to be possible. According toShyy et al.(2008), some bird species undergo accelerations as high as 14 G. Furthermore, many bird species migrate over very long distance, which suggests that they produce lift and thrust efficiently.

Regarding the effect of flexibility on flapping wing, an early study byKatz and Weihs(1978) showed that the chordwise flexibility has the effect of reducing the lift and thrust, but to increase the efficiency. Their study is based on the potential flow theory and linear elastic deformations. Experimental work by Heathcote and Gursul (2007) confirms these trends. They present results of a heaving flexible foil with different thickness corresponding to different flexibilities. They show that moderate flexibility is beneficial on thrust and efficiency. Here again, too much flexibility deteriorates the propulsive performances. Numerical simulations involving a complete Navier-Stokes model coupled to flexible structures have not been achieved until recently. Eldredge et al. (2010) study the effects of chordwise flexibility of a composite wing undergoing hovering kinematics. The wing is made of rigid parts connected with springs and the flow is solved with a Lagrangian Vortex method. Kang et al.(2011) present a general overview of the effects of flexibility on flapping wing. Their study, which is mainly based on numerical results, also presents an interesting analysis involving simplified analytical models. These analyses provide interesting scaling laws that relate flexible flapping wings performances to their flexibility.

From another perspective, the progresses made in computer science and technologies allow large mathematical problems involving both complex flows and mechanical structures to be solved. These numerical methods generate a large amount of easily accessible physical in-formation. It is not a surprise that numerical methods involvingFluid-Structure Interaction (FSI) are also being developed. Such methods are of great interest for the study of flapping wing flexibility effects. However,FSInumerical methods are still widely studied in the scien-tific community and it is not clear if these methods are readily available. Indeed,FSImethods split into two categories, namely monolithic methods and partitioned method. Monolithic methods intend to use a global discretization process that encompasses both the fluid and the solid media. This approach has been used by Hübner et al. (2004), Tezduyar et al. (2006) andWalhorn et al. (2005) among others. The monolithic method has the advantage of being stable, but the resulting linear systems are generally complex and somewhat tedious to solve. On the other hand, partitioned methods are based on existing methods (one for the fluid and another for the solid) that are coupled together in a sequential manner via the boundary con-ditions of each solver. The partitioned method thus relies on the fact that existing numerical solvers in both disciplines are optimized and efficient in order to build an efficientFSIsolver. This strategy has proved to be reliable in classical aeroelastic problems involving aeronautical applications (see Farhat and Lesoinne (2000)). However, in the case of strong interactions involving incompressible flows, the partitioned method is unstable because of the so-called

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added-mass effect described by Causin et al. (2005). To overcome this instability, various methods such as dynamic relaxation, quasi-Newton methods, and artificial compressibility have been proposed (see Küttler and Wall (2008); Gerbeau and Vidrascu (2003); Degroote et al. (2009), and Degroote et al.(2010c)).

Objectives

This doctoral project is the continuation of the author’s master project. During the master project, early investigations of flexible flapping wings were carried out (see Olivier (2009),

Olivier and Dumas (2009a), and Olivier and Dumas (2009b)). At that time, the objective of the project was to characterize the effect of flexibility on flapping wing operating as a lift and thrust device on a NAV. This project was conducted in collaboration with Defence Research and Development Canada Valcartier researchers who were interested inNAVtechnologies (see

Lesage et al. (2008b) and Lesage et al. (2008a)). However, even though interesting ideas were put forward, it appeared that the numerical methods available at that time were not sufficient to make a thorough physical investigation. Therefore, this project aims to improve the numerical methods related to fluid-structure interaction and to provide a deeper analysis of the effects of flexibility on flapping wings. More specifically, the project objectives are:

1. To develop an efficient fluid-structure interaction solver based on existing numerical technologies. That includes :

• the implementation of coupling boundary conditions;

• the development of an efficient and stable partitioned coupling strategy; and • the implementation of efficient and robust moving mesh algorithms.

2. To establish the potential of the proposed numerical method.

3. To determine the effects of wing flexibility on a flapping wing propulsion device that undergoes a combined pitching and heaving motion.

4. To understand the mechanisms that cause flapping wing chordwise deformation and their impact on the propulsive performance.

Thesis outline

This thesis concerns two principal themes: numerical methods in fluid-structure interaction and flexible flapping wings. The thesis is thus separated in two parts. Although the last few paragraphs have presented a succinct overview of the relevant literature, more complete reviews are presented in their respective parts. The first part of the thesis, which was partly

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presented at the Computational Fluid Dynamics Society of Canada (seeOlivier and Dumas

(2012)), focuses on the FSI-related numerical methods and is organized as follows.

Chapter 1 presents the mathematical models that describe nearly incompressible flows and elastic solids. A beam model as well as the FSI coupling conditions are also presented. A thorough literature review on numerical methods used inFSIis then presented in Chapter2. Chapter3 concerns the numerical methods developed and used within this doctoral project. The fluid and solid solvers are described in details. A section that concerns deforming meshes is also proposed. Various mesh motion algorithms are presented. Lastly, the partitionedFSI

coupling algorithm is presented along with the stabilizing strategy and the fluid-solid interface treatment. Chapter4 presents various numerical simulations involvingFSI. The objective of this chapter is to establish the potential of the proposed FSI scheme as well as to validate the whole numerical method. The cases used include comparison with numerical benchmarks and experimental data found in literature. Overall, it is shown that the presented partitioned algorithm stabilized with artificial compressibility is efficient on a wide variety ofFSIproblems. Part II concerns the second theme of the thesis, namely flexible flapping wings. A literature review that focuses on the physics of flapping wings used as a lift and/or thrust device is presented in Chapter5. Papers related to flexibility effects in flapping wing are also reviewed. In Chapter 6, the 2D flexible flapping wing problem definition, the related dimensionless parameters and the performance metrics are introduced. Numerical results and discussions on the effect of wing flexibility are presented. Numerical results show that for a pitching and heaving flapping wing, flexibility increases the efficiency when the interaction is strong (e.g.: light wings in dense fluid) but always reduces the thrust because of a feathering effect. On the other hand, when purely heaving wings are considered, flexibility can be interpreted as a passive pitching motion that may increase both thrust and efficiency whether the interaction is strong or not. The underlying mechanisms are discussed. Then Chapter7 revisits the 2D flexible flapping wing problem, but this time, the wing is free to move in the forward direction. It appears that the resulting deviation motion causes an additional feathering effect that has an effect of leveling the aerodynamic forces.

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Part I

Numerical methods in fluid-structure

interactions

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Chapter 1

Mathematical modeling

High fidelity analysis of aeroelastic phenomena requires appropriate mathematical models that are meant to describe the behavior of elastic structures and fluid flows. Fortunately, the mechanics of continuous media provides such a mathematical background. Furthermore, this field of study is well documented in many textbooks among which Malvern (1969), Aris

(1989), and Ogden (1997) are well established references. Therefore, the goal of this chapter is to establish the mathematical basis for the description of nearly-incompressible fluids and elastic structures.

1.1 Governing equations

The theory governing continuous media is established by recalling major laws of physics such as:

• the mass conservation; • the momentum conservation;

• the angular momentum conservation;

• the energy conservation (first thermodynamic principle); and • the entropy generation (second thermodynamic principle).

For each one of these principles, we can recall a conservation equation. For a continuous medium with a density field ρ moving according to the velocity field v, these equations are

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respectively for mass, momentum, angular momentum1_{, energy, and entropy:} ∂ (ρ) ∂t + ∇ · (ρv) = 0, (1.1) ∂ (ρv) ∂t + ∇ · (ρvv) = f + ∇ · σ, (1.2) ∂ (ρ(r_{× v))} ∂t + ∇ · (ρ(r × v)v) = r × f + r × ∇ · σ, (1.3) ∂ (ρe) ∂t + ∇ · (ρev) = f · v + r + ∇ · (σ · v + q), (1.4) ∂(ρs) ∂t + ∇ · (ρsv) ≥ r θ + ∇ · q θ , (1.5)

where f is an arbitrary body force, σ is the stress field (Cauchy stress tensor), r is the internal heat generation rate, q is the heat flux, θ is the temperature field, e is the total specific energy, and s is the entropy. Note that Eqs. (1.1) to (1.5) are written in their conservative form using an Eulerian description and spatial coordinates.

Eqs. (1.1) to (1.5) are general and can be applied for all problems involving continuous media as long as the problem remains in the range of classical mechanics. This, of course, includes nearly-incompressible flows and elastic solids. What distinguishes a medium from another are the constitutive laws. These laws provide information on the behavior of each medium and close the system of Eqs. (1.1) to (1.5).

1.2 Nearly incompressible Newtonian fluid flows

Since this research focuses on relatively low velocity applications evolving in air or in water, the constitutive law of Newtonian fluids along with the incompressibility hypothesis are suitable to describe fluid flows. However, for numerical reasons that are going to be explained in chapters

3 and 4, a barotropic model, which is suitable for nearly incompressible flows, is presented. This barotropic model is one of the simplest compressible models as it relates linearly the density to the pressure with a compressibility coefficient ψ:

ρ = ρ0 + ψ(p − p0). (1.6)

The continuity equation can thus be re-arranged as: ∇ · v = − ψ ρ0 − ψp0 ∂p ∂t + ∇ · (pv) . (1.7) 1

Eq. (1.3) is, in fact, only the moment of linear momentum contribution to the total angular momentum conservation equation. This simplification is used as non-polar media are exclusively considered in this work. As a consequence, there are no internal couples and the stress field σ is a symmetric tensor field. SeeAris

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In the case of fully incompressible flows, the compressibility coefficient is zero and the conti-nuity equation simply becomes:

∇ · v = 0. (1.8)

The Newtonian fluid constitutive law relates the Cauchy stress tensor field σ to the strain rate tensor field D = 1

2(∇v + ∇v

T₎ _{according to:}

σ = µ0tr (D) _{− p}I + 2µD

= µ0_{∇ · v − p}I + µ _{∇v + (∇v)}T_, (1.9)

where µ and µ0 _{are the first and second viscosity coefficients and p is the pressure field. For}

nearly-incompressible flows, µ0

∇ · v ' 0 and the constitutive law simplifies to:

σ = _{−pI + µ ∇v + (∇v)}T. (1.10) Therefore, the momentum equation becomes2_:

∂ (ρv)

∂t + ∇ · (ρvv) = f − ∇p + µ∇

2_v. _(1.11)

From that point, one can observe that only the mass conservation equation (Eq. (1.1)) and the momentum conservation equation (Eq. (1.11)) are required to define a closed problem. Indeed, since the density is either constant (incompressible model) or directly related to the pressure (barotropic model) this set contains four scalar equations and four unknowns, namely the three velocity components and the pressure. The dimensionless forms of these equations are, for the incompressible model3_:

∇∗· v∗ = 0 (1.12) Lref VrefTref ∂v∗ ∂t∗ + ∇ ∗ · (v∗v∗) = − pref ρV2 ref ∇∗p∗ + µ ρLrefVref ∇∗2v∗, (1.13) where the body force term f is not considered.

In the context of flows around moving geometries, it is generally more convenient to use the integral form of conservation equations which can be integrated over an arbitrary moving volume. Such an approach is sometimes referred to as the Arbitrary Lagrangian Eulerian (ALE) approach (seeDonea et al. (2004)). The system of equation governing fluid flows thus becomes: ∂ ∂t Z V (t) dV − Z S(t) ˆ n· ˆvdS = 0, (1.14) ∂ ∂t Z V (t) ρ dV + Z S(t) ρˆn_{· c dS = 0,} (1.15) 2

Because the µ0∇ · v term is neglected, the momentum equation is the same, whether the slightly com-pressible barotropic model is used or not.

3

The barotropic model is not considered for this dimensional analysis as it will only be used as a numerical tool to stabilizeFSIalgorithms in chapter3and4.

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∂ ∂t Z V (t) ρv dV + Z S(t) ρv (ˆn_{· c) dS =} Z V (t) fdV ₋ Z S(t) p ˆndS + Z S(t) ˆ n_{· µ∇v dS,} (1.16) where ˆv is the velocity of the control surface and c = v − ˆv is the relative convective velocity. The additional equation (Eq. (1.14)) is the so-calledGeometric Conservation Law (GCL)which is described in details byDemirdžić and Perić (1988) and Donea et al. (2004). Although the

GCLdoes not bring any physical information – it simply states that a control volume changes according to the volume swept by its moving surfaces – it is of utmost importance that the numerical schemes used to solve a fluid dynamics problem involving moving boundaries respect theGCL. Otherwise, it can be shown that mass is most likely to be created as it is pointed out byDemirdžić and Perić (1988). Appendix A provides more details on theALE formulation. Here again, the mass conservation equation (Eq. (1.15)) can be completed with the constitutive law given by Eq. (1.6). Rearranging terms and using Eq. (1.14), the following convenient form is obtained: Z S(t) ˆ n· v dS = − 1 (ρ0− ψp0)   _∂t∂ Z V (t) ψp dV + Z S(t) ψpˆn· c dS    , (1.17) which simplifies to: _Z

S(t)

ˆ

n· v dS = 0, (1.18)

when the flow is considered fully incompressible. These two equations are used in Chapter3

to build the pressure equation in the fluid flow solver.

1.3 Elastic structures

Elastic solids undergoing large structural displacements within the limit of small strains are considered in this research. This hypothesis is well suited for thin structures since small transverse loads induce large deflections even if internal strains remain small. Therefore, the constitutive law is expected to be linear. However, a formulation that supports large displace-ments is inherently nonlinear and is generally referred to as being geometrically nonlinear. Since a solid structure is a continuous medium, Eqs. (1.1) to (1.5) also apply. However, in a structural analysis, the whole structure is generally considered (on the opposite, in fluid mechanics, only a portion of the flow is analyzed). Therefore, the Eulerian formulation is not very convenient in such analyses because boundaries are not fixed in space. For this reason, the Lagrangian formulation is often preferred. The Lagrangian momentum conservation equation can be written as (see AppendixA):

ρDv

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where D( )

Dt is the material time derivative. This equation has an inconvenient: since the

deformed configuration of the structure is not known, it is not possible to evaluate directly ∇ · σ. To circumvent this problem, Eq. (1.19) is written in terms of the coordinates of the original configuration:

ρ0

∂2_d

∂t2 = f0 + ∇0· P, (1.20)

where P is the first Piola-Kirchhoff stress tensor and the subscript 0 refers to the original configuration. This approach is often referred to as a total Lagrangian approach. It is worth pointing out that the stress tensor P corresponds to a surface force per unit of undeformed area (see Malvern (1969)). It can be related to the Cauchy stress tensor by:

P = JF−1_{· σ,} (1.21)

where F = I + (∇0d)T is the deformation gradient tensor and J = det (F). The first

Piola-Kirchhoff stress tensor can also be related to the second Piola-Piola-Kirchhoff stress tensor S, which is used to define the constitutive law, by:

P = S_{· F}T_. _(1.22)

The rheology of an elastic solid undergoing large structural displacements is defined by the St. Venant-Kirchhoff constitutive law:

S = 2υE + λtr(E)I, (1.23)

where υ and λ are respectively the first and second Lamé coefficients and E is the Green-Lagrange strain tensor which is a typical strain measure used in large displacement analysis and which is consistent with the total Lagrangian formulation. It is defined by:

E = 1 2 ∇0d + (∇0d) T ₊ ∇0d· (∇0d)T . (1.24)

Therefore, neglecting the body force term f0, the momentum equation becomes:

ρ0 ∂2_d ∂t2 = ∇0· υ_∇0d + υ∇0dT + υ∇0d· ∇0dT + λtr (_∇0d) + 1 2λtr ∇0d· ∇0d T ·I+_∇0d , (1.25) which is a closed system that allows the displacement field of a solid to be solved. The Lamé coefficients are related to the Young modulus E and the Poisson coefficient ν by4_:

υ = E

2(1 + ν), (1.26)

4_{Under the plane stress assumption, λ =} νE (1 + ν)(1 − ν).

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λ = νE

(1 + ν)(1 − 2ν). (1.27)

Recalling that the strain tensor E and the Poisson coefficient are dimensionless, the dimen-sionless stress tensors can easily be defined as:

S = ES∗. (1.28)

P = EP∗. (1.29)

Therefore, the dimensionless momentum equation can be obtained (for the case of uniform rigidity): ∂2_d ∂t2 = ET2 ref ρ0L2_ref ∇∗0· P ∗ , (1.30)

where a single dimensionless parameter appears in the brackets. One must keep in mind that the Poisson coefficient ν represents a second dimensionless parameter that appears in the definition of P∗_.

1.4 Thin elastic structures

In this research, 2D problems involving the transverse cross-section of a thin plate are con-sidered. In those situations, the plate mathematical model degenerates to a beam model. In the case of a straight (not curved) beam moving in the xy-plane with large displacements, the equations of motion are (in a Lagrangian formulation reported in the initial configuration):

ρsA ∂2_d x ∂t2 = ∂ ∂x N 1 +∂dx ∂x + ∂ ∂x M∂ 2_d y ∂x2 + ∂ 2 ∂x2 M∂dy ∂x + fx − ˜p ∂dy ∂x + ˜τ 1 +∂dx ∂x , (1.31) ρsA ∂2_d y ∂t2 = ∂ ∂x N∂dy ∂x − ∂ 2 ∂x2 M 1 +∂dx ∂x − ∂ ∂x M∂ 2_d x ∂x2 + fy − ˜p 1 +∂dx ∂x − ˜τ∂dy ∂x, (1.32) where dx and dy are respectively the x and y components of the displacement field, fx and fy

are the x and y components of the lineic load on the undeformed configuration, ˜p is the lineic pressure on the deformed configuration, and ˜τ is the lineic shear on the deformed configuration. The internal normal force N as well as the internal bending moment M are given by the Euler-Bernouilli beam theory:

N = EAe = EA " ∂dx ∂x + 1 2 ∂dx ∂x 2 + ∂dy ∂x 2!# , (1.33)

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M = EI ˜χ = EI ∂2_d y ∂x2 1 +∂dx ∂x − ∂dy ∂x ∂2_d x ∂x2 , (1.34)

where e is the xx component of the Green-Lagrange strain tensor E and ˜χ is the curvature of the deformed beam reported in the undeformed configuration. Eqs. (1.31) to (1.34) were derived from the variational formulation (virtual work formulation) by Epstein and Murray

(1976).

All nonlinear terms in Eqs. (1.31) to (1.34) account for large displacements. For the sake of completeness and simplified analysis, it is worth pointing out that, in the case of small displacements, the equations of motion simplify to:

ρsA ∂2_d x ∂t2 = ∂ ∂x EA∂dx ∂x + fx, (1.35) ρsA ∂2_d y ∂t2 = − ∂2 ∂x2 EI∂ 2_d y ∂x2 + fy. (1.36)

The associated dimensionless equations are given by: ρsALref frefT_ref2 ∂2_d∗ x ∂t∗2 = EA frefLref ∂2_d∗ x ∂x∗2 + f ∗ x, (1.37) ρsALref frefT_ref2 _∂2_d∗ y ∂t∗2 = − EI frefL3_ref _∂4_d∗ y ∂x∗4 + f ∗ y. (1.38)

Here, the body forces terms f∗

x and fy∗have been left in the equation (as opposed to Eq. (1.30))

as they represent lineic loads corresponding to pressure and viscous stresses in a FSIcontext. It is worth noting that Eqs. (1.31) to (1.34) provide the same set of dimensionless parameters since all additional nonlinear terms are dimensionless.

1.5 Fluid-structure coupling conditions

Since both the fluid and solid problems are handled with the theory of continuous media, the treatment of the fluid-solid interface makes no exception. Therefore, all physical principles stated by Eqs. (1.1) to (1.5) also apply at the interface. In this research, only the mass conservation and the momentum conservation equations are considered as the other physics principles are not required to describe elastic solid and nearly incompressible flows. Therefore, the momentum and mass conservation at the FSIinterface yields the following conditions:

vf = vs (1.39)

ˆ

n· σf = ˆn· σs (1.40)

Apart from these mathematical coupling conditions, it is certainly relevant to qualify the fluid-solid interaction. For the interaction to be important, two conditions must be met:

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• the structural deformation must be large so that its modified geometry has an impact on the flow; and

• the pressure and/or shear forces coming from the interaction with the flow must be significant when compared to the solid inertia.

The first condition is directly related to the flexibility of the structure. In eachFSIproblem, a dimensionless quantity representing the normalized flexibility (or its inverse, the rigidity) exists. The second condition, which will be referred to as the strength of the interaction throughout the text, also has an associated dimensionless parameter that gives the ratio between fluid forces and inertia forces.

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Chapter 2

Review of numerical methods in

Fluid-Structure interaction

In order to establish the scope of this work within the research that has been done in the field of numerical methods in fluid-structure interaction, an overview of numerical methods is presented. This review presents specific methods related to both incompressible fluid flows and elastic solids as well as coupling strategies allowing the solution of coupled fluid-solid systems. Since we are interested in methods that are applicable on general geometries, emphasis is put on the well established Finite-Element Method (FEM) and Finite-Volume Method (FVM). Many textbooks such asFletcher(1991);Bathe (1996), andFerziger and Perić(2002) provide complete explanations of these methods. The applicability of some Lagrangian methods is also discussed.

2.1 Numerical methods in fluid and solid mechanics

Two difficulties are present in the Navier-Stokes equations: the nonlinear convective term and the coupling between velocity and pressure. To assess the first difficulty, an appropriate linearization combined with an iterative solution procedure is generally sufficient. Concerning the second one, the difficulty arises from the fact that the continuity equation does not exhibit any dependence on the pressure field. This has the consequence that a checker-board pressure field may appear in the solution if care is not taken in the development of the numerical method. Furthermore, dealing directly with the continuity equation generally leads to badly conditioned linear systems. Specific procedures to achieve a numerically consistent velocity-pressure coupling will thus be discussed further in relation with each discretization method.

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2.1.1 Finite-volume method

TheFVMis certainly one of the most popular discretization techniques that is used to solve fluid flows numerically. This method is closely related to control volume analyses that are effectively widely used in fluid mechanics. TheFVMpossesses the following advantages:

• It produces conservative schemes.

• It is relatively easy to implement (at least for second order discretizations). • It applies naturally to transport equations such as the Navier-Stokes equations. • It generalizes easily to complex geometries.

In the context of incompressible fluid flows, the FVM usually uses a pressure or pressure-correction equation (instead of the continuity equation) to complete the momentum equation. This pressure equation is obtained by taking the divergence of the momentum equation. In order to remain consistent with the method and to keep its conservative properties, this operation is performed on the discretized equations rather than on its analytic counterpart. Unfortunately, such a procedure generally allows checker-board pressure fields to appear. To avoid these oscillations it is possible to use a staggered grid or a Rhie-Chow coupling scheme (seeRhie and Chow(1983)). The former has the advantage of being perfectly consistent while the latter works on collocated grids, which is a good advantage since consistent boundary conditions and complex geometries are handled more easily on such grids.

From that point, many solution procedures are possible. Algorithms that have gained much popularity over the years are the projection methods such as the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) by Patankar and Spalding (1972), the SIMPLE-Corrected (SIMPLEC)method by Van Doormaal and Raithby (1984), the SIMPLE-Revised (SIMPLER)method byPatankar(1980), and thePressure-Implicit with Splitting of Operators (PISO)by Issa (1986). These algorithms have been used with both staggered and collocated grids. They use a segregated approach to solve the system of coupled equations. The main lines of these algorithms are as follow:

• The components of the momentum equation are solved sequentially. Inter-component coupling terms, if any, are treated explicitly using values from the previous iteration. The convective term is linearized in a fixed-point manner like:

Z

S(t)

ρv(cg· ˆn) dS, (2.1)

where cg is the guessed relative velocity field, typically taken from the previous iteration

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• A pressure or pressure-correction equation is solved using the momentum predictor. Then, the velocity is corrected using the new pressure or pressure-correction field to enforce the incompressibility constraint. This procedure can be repeated a few times, or can involve further correction step, depending on the variant of the projection algorithm. The drawback of these algorithms is that the lack of coupling in the numerical formulation can reduce dramatically the convergence rate. Indeed, as it was shown byDarwish et al.(2009) and

Chen and Przekwas (2010), for steady-state studies, coupled solvers are generally much more efficient than segregated solvers, particularly for large problems. However, these algorithms have some properties that make them attractive, especially for transient analysis. Foremost, the use of Eq. (2.1) as a linearization procedure combined with the segregated approach allows the construction of a single common matrix for all three (two in 2D) components of the momentum equation. This reduces significantly the required memory and the matrix assembly time. Secondly, in transient analysis, the initial guess used for the solution of a single time-step is generally the solution from the previous time-step. Therefore, the smaller the time-step, the better is the initial guess and, as a consequence, the smaller is the nonlinear contribution of convective terms. It follows that the number of iterations within a time-step is generally rather small when the time-step is chosen to achieve good precision.

Another interesting aspect of the finite volume method is the attractive structure of the gen-erated matrices. Indeed, when using simple second order schemes such as linear interpolation for convective fluxes and central differences for face gradients in conjunction with a perfectly orthogonal mesh, the off-diagonal terms of matrices only imply direct neighbors of the cell as-sociated to each matrix line. Furthermore, it appears that these matrices are diagonally domi-nant. Therefore, these matrices are particularly well suited for iterative solvers. In the case of more complex schemes (e.g.: second order upwinding, least-squares approximations etc.) and of non-orthogonal meshes, the discretization involves at least neighbors of neighbors, which enlarges the matrix band. An elegant way to deal with this is to use the so-called deferred-correction approach (see Ferziger and Perić (2002)). This technique proposes to strictly use simple schemes to generate the matrices and to add a correction on the right-hand-side of the linear system in order to respect another specific scheme, thus increasing the explicit character of the algorithm. Corrections associated to mesh non-orthogonality are also directly reported to the right-hand-side of the linear system. It appears that this technique does not increase significantly the number of iterations since the magnitude of the deferred terms is generally not high. On the other hand, the deferred-correction approach allows matrices to keep their elegant diagonal-dominant structure so that fast iterative solvers can be used to solve linear systems.

When iterative projection methods are used (or any other method that is not in the Newton-Raphson family), the use of iterative solvers for the solution of linear systems yields another

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advantage: it is possible to solve partially the linear system at each iteration without com-promising the convergence rate. Indeed, it is not necessary to solve linear systems to tight tolerances since nonlinear terms (and possibly cross-component coupled terms) need to be updated. Such a strategy would not be acceptable in a Newton-Raphson solver as it would deteriorate the quadratic convergence rate.

The finite-volume discretization can also be applied to structural mechanics equations. Al-though finite-elements are generally preferred for structural applications (this is mostly a matter of tradition), some researchers have, nevertheless, studied such practice. One of the first major contributions in this field of research is the work of Demirdžić and Muzaferija

(1994). They present a structural cell-centered finite-volume solver based on the segregated approach that deals with unstructured grids. The linear system construction is done by con-sidering only direct cell neighbors contribution in the matrix. All the other terms are left in the right-hand side vector. This results in diagonally dominant matrices that are solved iteratively.

Fallah et al.(2000) present a comparison between finite-volume and finite-element discretiza-tion procedures. The finite-volume method used is node-centered (called a cell vertex for-mulation in the paper) and, thus, more analogous to the finite-element method as only the choice of weighting functions differs. They present solutions of solids involving geometrically nonlinear deformations and they show that both theFEMand theFVMprovide comparable results and computational efficiency. In both cases, nonlinear effects are taken into account iteratively with a Newton-Raphson method.

Bijelonja et al.(2005) andBijelonja et al.(2006) present a finite-volume structural solver based on a pressure-displacement formulation. The pressure-displacement formulation is closely related to the pressure-velocity coupling algorithm found in classical SIMPLE flow solvers and it is also analogous to the mixed formulation used in FEM. In the paper by Bijelonja et al. (2006), an incompressible solid model involving small displacements is implemented with theFVM. In the paper byBijelonja et al.(2005), the authors present a similar numerical method adapted for the solution of hyperelastic materials. An updated Lagrangian approach is used along with an incremental formulation of the governing equations. Since the mass conservation law is directly satisfied in a Lagrangian formulation, the GCLis used to derive the pressure equation. Furthermore, the authors report in both papers no occurrence of the locking phenomenon and they suggest that the conservative nature of theFVM may lead to locking-free schemes.

Following the same idea,Giannopapa and Papadakis(2008) also present a finite-volume struc-tural solver involving a pressure equation. However, they show how to express the strucstruc-tural problems in terms of velocity. The resulting pressure-velocity formulation can directly be

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generalized to a unified system including a fluid and a solid. Such a method is thus relevant for FSIproblems (more details are provided in Section 2.2). However, the presented solver is based on the PISOalgorithm. As it was discussed in Section2.1, projection methods such as

PISO or algorithms from the SIMPLEfamily are sensitive to the time-step size with respect to the grid size and physical time scales. That may cause some restriction on the time-step size of a coupled fluid-structure problem if time scales in the fluid differ largely from the time scales in the solid as it is often the case.

Maneeratana and Ivanković (1999) use a different mathematical formulation of the govern-ing equations along with the finite-volume method to solve structural mechanics problems involving large displacements. The various formulations used include incremental and non-incremental equations based on the initial or the actual configuration. It is shown that incre-mental equations used in an updated configuration cause numerical error when the increment size is not small enough.

Other studies related to the FVM in structural mechanics include the works of Jasak and Weller (2000) and Tuković and Jasak (2007). The latter presents the implementation of a linear elasticity solver based on the FVM. It is shown how one can maximize the implicit contribution of the ∇ · σ term in a segregated algorithm by using appropriate mathematical identities in order to maximize the contribution of the Laplacian term. The idea is to treat only pure rotation terms explicitly. The same idea is used in Tuković and Jasak (2007) for the implementation of a large-displacement structural solver that uses an updated-Lagrangian formulation.

2.1.2 Finite-element method

Although it was initially devoted to structural mechanics, the FEM is also frequently used in fluid mechanics nowadays. Typically, this method uses a variational formulation of the governing equations as a starting point. Such a formulation introduces weight functions and thus represents a weak formulation of the original set of differential equations.1 _Furthermore,

within each element, the physical field is represented by a set of shape functions that are determined through nodal values. Many variations of this method exist. The next lines provide a brief description of the most popular variations of theFEM.

The Galerkin FEM is probably the most popular. In this method, the weight functions are chosen to be the same as the shape functions. In terms of structural mechanics, the resulting variational formulation physically represents the virtual work of the discretized system. This technique is sometimes referred to as the classical FEMand it is particularly well suited for 1_{The finite volume method can be viewed as a special case of the}_FEM _{where the weight functions are}

unitary. Therefore, the formulation of the former is said to be strong and it provides conservative numerical schemes.

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structural mechanics applications. However, the Galerkin method is known to produce spu-rious oscillations with high Reynolds number flow problems since the convective term is not self-adjoint (see Zienkiewicz and Taylor (2000)). To overcome this difficulty, the Stremline-Upwind Petrov-Galerkin (SUPG) weight functions can be used. Such functions have an addi-tional term that increases the value of the weight in the region of the upcoming flow. This is analogous to the use of upwind schemes in finite-volume or finite-difference methods. Another approach growing in popularity is the discontinuous finite-element method (see Li (2006)). This method, which can be seen as a super-set of the traditional FEM, is effectively well suited for convection-dominant problems, but is found to be inferior to the latter for elliptic problems.

The Petrov-Galerkin FEMcan be applied directly to the conservation equations even in the case of incompressible flows (there is no need to formulate a pressure equation). However, like in the finite volume method, a special treatment is required to avoid pressure oscillations. Indeed, the Brezzi-Babˇuska condition (seeBabˇuska(1973) and Brezzi(1974)), also known as the inf-sup condition, must be respected to ensure the uniqueness of the solution and thus, to obtain a non-oscillating pressure field. This condition leads to two families of elements that are said to be compatible: continuous pressure and discontinuous pressure. Continuous pressure elements (such as the Taylor-Hood element) have pressure degrees of freedom on the boundary of the elements so that the pressure is continuous between elements. On the other hand, discontinuous pressure elements (such as Crouzeix-Raviart elements) possess pressure degrees of freedom only inside the elements which effectively allows discontinuities in the pressure field from one element to another. In both cases, if the velocity uses polynomial shape functions of degree k and if the pressure uses polynomial shape functions of degree k− 1, a kth _{order convergence (in terms of the norm of the corresponding functional space)}

can be achieved.

When compared to the finite volume method theFEMexhibits the following advantages: • The solution is known in the whole domain, not only at the computational nodes, thanks

to the shape functions. This brings some benefits such as the ease of implementation of a grid-refinement algorithm.

• The solid mathematical framework that encompasses the method allows the straightfor-ward generation of higher order approximations with their corresponding convergence analysis procedures.

• According to Fletcher (1991), a finite-element formulation using quadratic shape func-tions for the velocity in conjunction with linear shape funcfunc-tions for the pressure is more precise than a typical second order SIMPLE-like algorithm on a collocated grid, even though the global order of the truncation error is equivalent (linear shape function for the

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pressure yields second order discretization error). This remains true as long as boundary layers of the flow are well resolved.

• TheFEMis often accompanied with the Newton-Raphson method or one of its variants when nonlinear problems occur. This is not surprising since the method naturally pro-duces fully implicit systems of equations, (which is not always the case with finite-volume methods, particularly in complex unstructured meshes). It follows that the method can benefit from quadratic convergence when an appropriate initial guess is given.

On the other hand, there is a price to pay for these advantages. Those drawbacks are: • Again according toFletcher(1991),FEMdiscretization based on quadratic shape

func-tions for the velocity and linear shape funcfunc-tions for the pressure produces denser matri-ces than second orderFVMand, since the pressure approximation relies on linear shape functions, the overall accuracy is of the same order even if the velocity is approximated by quadratic shape functions.

• In addition to being relatively dense,FEMmatrices are generally not diagonally domi-nant. It follows that simple iterative procedures cannot solve the resulting linear systems. The computational cost associated with the linear system solution may thus scale badly for large problems. However, the use of projection methods similar to those used in the

FVMis also possible with theFEMand may help to reduce the effort required in solving linear systems.

2.1.3 Lagrangian particle methods

Although they are not as widespread as previously described grid methods, Lagrangian particle methods have interesting properties that are worth considering. Among the particle methods, the Smooth Particle Hydrodynamics (SPH) has received much attention in the last several years. Indeed, this method is used in many fields such as fluid and solid mechanics, molecular dynamics, computer graphics, astrophysics, etc. Monaghan(1992) presents the fundamentals of the method and its application to the most famous equations of physics, including the classical conservation laws used in mechanics. The time integration schemes introduced are explicit and, therefore, require the fulfillment of the CFL condition. Although the applications presented in this review are related to astrophysics, a more recent review byMonaghan(2012) presents recent progress made in the field of SPH that are related mainly to fluid flows and some fluid-structure interactions. One of the greatest advantages ofSPHover other numerical methods is the ease with which it can handle complex multi-physics problem such asFSI, but also contact and fragmentation problems, multiphase flows, complex boundaries, etc. Some turbulence models are also being developed for that specific method. However, it is reported