Fundamentals of Fluid-Solid Interactions-Analytical and

Fundamentals of Fluid–Solid
Interactions
Analytical and Computational Approaches

MONOGRAPH SERIES ON NONLINEAR
SCIENCE AND COMPLEXITY
SERIES EDITORS
Albert C.J. Luo
Southern Illinois University, Edwardsville, USA
George Zaslavsky
New York University, New York, USA
ADVISORY BOARD
Valentin Afraimovich, San Luis Potosi University, San Luis Potosi, Mexico
Maurice Courbage, Université Paris 7, Paris, France
Ben-Jacob Eshel, School of Physics and Astronomy, Tel Aviv University,
Tel Aviv, Israel
Bernold Fiedler, Freie Universität Berlin, Berlin, Germany
James A. Glazier, Indiana University, Bloomington, USA
Nail Ibragimov, IHN, Blekinge Institute of Technology, Karlskrona, Sweden
Anatoly Neishtadt, Space Research Institute Russian Academy of Sciences,
Moscow, Russia
Leonid Shilnikov, Research Institute for Applied Mathematics & Cybernetics,
Nizhny Novgorod, Russia
Michael Shlesinger, Office of Naval Research, Arlington, USA
Dietrich Stauffer, University of Cologne, Köln, Germany
Jian-Qiao Sun, University of Delaware, Newark, USA
Dimitry Treschev, Moscow State University, Moscow, Russia
Vladimir V. Uchaikin, Ulyanovsk State University, Ulyanovsk, Russia
Angelo Vulpiani, University La Sapienza, Roma, Italy
Pei Yu, The University of Western Ontario, London, Ontario N6A 5B7, Canada

Fundamentals of Fluid–Solid
Interactions
Analytical and Computational Approaches
XIAODONG (SHELDON) WANG
Department of Mathematical Sciences
New Jersey Institute of Technology
323 Martin Luther King, Jr. Blvd.
Newark, NJ 07102, USA
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First edition 2008
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Contents
Preface vii
Chapter 1. Introduction 1
1.1. Historical background 1
1.2. Motivation 3
1.3. Outline 4
Chapter 2. Aspects of Mathematics and Mechanics 5
2.1. Preliminary concepts and notations 5
2.1.1 Logic; set, group, and field; Boolean algebra 5
2.1.2 Mapping and linear space; grad, div, and curl 9
2.1.3 Tensors and curvilinear coordinates; complex variables 15
2.1.4 Eigenvalues and eigenvectors; singular value decomposition 25
2.2. Kinematical descriptions and conservation laws 31
2.2.1 Mass point and rigid body; continuum 32
2.2.2 Vector calculus; transport theorems and conservation laws;
variational principles 37
2.3. Some mathematical tools 52
2.3.1 Fourier series and transform; Laplace transform 52
2.3.2 Contour integral; conformal mapping 58
2.3.3 Sturm–Liouville problems; Frobenius’ methods and special
functions 71
Chapter 3. Dynamical Systems 85
3.1. Single- and multi-degree of freedom systems 86
3.1.1 Basic concepts; limit sets 86
3.1.2 Bifurcations; Lagrangian dynamics 100
3.1.3 Van der Pol; perturbation method; Duffing 112
3.2. Lyapunov–Schmidt method 126
3.2.1 Floquet theory 129
3.2.2 Critical time step 133
3.2.3 Mathieu–Hill system 136
xi

xii Contents
3.3. Basin of attractions and control of chaos 142
3.3.1 Lyapunov methods 142
3.3.2 Robust control; sliding mode; adaptive control; FIV model 147
Chapter 4. Flow-Induced Vibrations 163
4.1. Attenuator and flutter suppression 163
4.1.1 Helmholtz resonator; attenuator design 164
4.1.2 Reynolds equations; spatial and temporal discretizations;
suppression results 169
4.2. Stability issues of axial flow models 183
4.2.1 Internal flow; concentric flow 183
4.2.2 Buckling and flutter; parametric instability and Bolotin method 194
4.2.3 Collapsible tube; tube law; transverse and axial oscillations 214
4.3. Dynamics of thin structures 229
4.3.1 Stationary and moving panel; MITC elements 230
4.3.2 Bending and torsion; aeroelasticity models 244
Chapter 5. Boundary Integral Approaches 267
5.1. Underlying physics and formulations 267
5.1.1 Potential flow 267
5.1.2 Acoustic medium 273
5.1.3 Stokes flow 274
5.2. Green’s functions and boundary integrals 275
5.2.1 Reciprocal relations 275
5.2.2 Boundary element approaches 277
5.3. FSI systems with potential flows 279
5.3.1 Interaction with rigid body; added mass; lift surface 279
5.3.2 Radiation and scattering 299
5.3.3 Rigid body surface waves interactions 305
Chapter 6. Computational Linear Models 313
6.1. Potential and displacement-based formulations 313
6.1.1 φ-u and p-φ-u formulations 314
6.1.2 u/p and u-p-Λ formulations 321
6.1.3 Solvability and stability; FSI interfaces; zero modes 328
6.2. Solution procedures and convergence issues 344
6.2.1 Mode superposition; direct integration methods 347
6.2.2 Acoustoelastic/slosh FSI systems 356
6.2.3 Inf-Sup conditions of mixed formulations 386

Contents xiii
Chapter 7. Computational Nonlinear Models 407
7.1. Upwind and stabilization 407
7.1.1 One-dimensional model 408
7.1.2 Multi-dimensional model 411
7.2. Nonlinear finite element formulations for FSI systems 427
7.2.1 Mixed formulations for fluid and solid domains 427
7.2.2 Direct method with Jacobian matrix 436
7.2.3 Matrix-free Newton–Krylov; multigrid 441
7.2.4 Laminar and turbulent flows; porous interface; benchmark tests 455
7.3. Immersed methods 487
7.3.1 Immersed boundary method; nodal forces; incompressible
continuum 489
7.3.2 Particulate flow and blood modeling; mapping and kernel 499
7.3.3 Fictitious domain method; immersed continuum method 527
7.3.4 Implicit/compressible solver; FSI systems with immersed
solids 537
References 547
Subject Index 567