Axisa F., Antunes J. Modelling of Mechanical Systems: Fluid-Structure Interaction. Volume 3

Butterworth-Heinemann, Elsevier, 2007. 795 p. — ISBN13: 978-0-750-66847-7, ISBN10: 0-7506-6847-4.The study of fluid-structure interactions in this third volume covers the coupled dynamics of solids and fluids, restricted to the case of oscillatory motions about a state of static equilibrium. Fluid motion induced by a vibrating structure results from various distinct coupling mechanisms operating together but with a relative importance which varies enormously from one case to the other. Physical and mathematical aspects of modelling these mechanisms are described in depth and illustrated by numerous worked out exercises.Introduction to fluid-structure coupling
A short outline of fluid-structure coupled systems
Basic mechanism of fluid-structure dynamical coupling
A few elementary experiments
Dynamic equations of fluid-structure coupled systems
Elastic vibrations of solid structures
Dynamic equations of Newtonian fluids
Eulerian acceleration and material derivative
Mass-conservation equation
Momentum equation
Pressure and fluid elasticity
Fluid elasticity and equation of state of a gas
Cavitation of a liquid
Viscous stresses
Navier-Stokes equations
Linear approximation of the fluid equations
Linearized fluid equations about a quiescent state
Linear Navier-Stokes equations
The linear Euler equations
The sound wave equation in terms of a single field
Linearized boundary conditions
Fluid-structure coupling term at a wetted wall
Free surface of a liquid in a gravity field
Surface tension at the interface between two fluids
Physical quantities and oscillations of the fluid
Mean value of fluid density
Gravity field
Surface tension
Fluid elasticity
Fluid viscosityInertial couplingDiscrete systems
The fluid column model
Single degree of freedom systems
Piston-fluid system: tube of uniform cross-section
Piston-fluid system as a dynamically coupled system
Piston-fluid system: tube of variable cross-section
Hole and inertial impedance
Response to a seismic excitation
Nonlinear inertia in piping systems
Systems with spherical symmetry
Breathing mode of a spherical shell immersed in a liquid
Early stage of a submarine explosion
Piston-fluid system with two degrees of freedom
Natural modes of vibration
Lagrange’s equations
Newtonian treatment of the problem
Continuous systems
Modal added mass matrix
Strip model of elongated fluid-structure systems
Cylindrical shells of revolution
Cylindrical shell immersed in an infinite extent of liquid
Inertial coupling of two coaxial circular cylindrical shells
Thin fluid layer approximation
Concentric cylindrical shells of revolution
Extension to other geometries
Mode shapes modified by fluid inertia
Rigid rod partly immersed in a liquid
Coaxial cylindrical shells of revolution
Water tank with flexible lateral walls
D problems
Plate immersed in a liquid layer of finite depth
Circular cylindrical shell of low aspect ratio
Vertical oscillation of an immersed spherical object
The immersed sphere used as an inverted pendulumSurface wavesGravity waves
Harmonic waves in a rectilinear canal
Group velocity and propagation of wave energy
Shallow water waves ( ) 1 kH
Application of the shallow wave theory to tsunamis
Seismic tsunami waves
Meteorological tsunamis
Nonlinear limitation of resonant waves
Deep water waves ( ) 1 kH
Space and time profiles of progressive waves
Wake of a moving boat and Kelvin wedge
Water waves at intermediate depths: solitary waves
Wave impacting a rigid wall
Surface tension
Capillary waves, or ripples
Surface tension and cavitation
Static equilibrium of a micro-bubble, or cavitation nucleus
The collapse of cavitation bubbles
Oscillations and activation of the cavitation nuclei
Rayleigh-Plesset equation
Sloshing modes
Discrete systems
U tube
Interconnected tanks
Continuous systems
Rectangular tank
Circular tank
Fluid-structure interaction
Coupling between sloshing and structural modes
Floating structuresBuoyancy of a boat
Stability of the static equilibrium
Natural frequencies of the rigid body modes
Example 1: heave mode of a floating circular cylindrical buoy
Example 2: rectangular cross-section
Rolling induced by the swell
Antiresonant absorber for rollingPlane acoustical waves in pipe systemsAcoustics and sound perception
Acoustics in the context of fluid-structure interaction
Linear and conservative acoustical wave equation
Free sound waves in pipe systems: plane and harmonic waves
Acoustic impedances and standing sound waves
Plane wave approximation in pipes
Plane wave equations in pipes
Travelling waves in a uniform tube and tube impedance
Reflected and transmitted waves at a change of impedance
Reflected and transmitted waves through three media
Boundary conditions and terminal impedances
Radiation damping and complex impedance
Acoustical modes in a uniform tube
Application to wind musical instruments
Horns: Webster and Schrödinger equations
Bessel horns
Transfer matrix method (TMM)
Transfer matrix of a uniform tube element
Assembling of two tube elements
Two connected tubes of distinct cross-sectional areas
Two connected tubes filled with distinct fluids
Helmholtz resonators
Higher plane wave modes of an enclosure tube assembly
Enclosure-tube assembly: case of a very short tube
Forced waves
Concentrated acoustical sources
Volume velocity (monopole) source
Pressure (dipole) source
Transfer functions for a uniform tube
Transfer matrix method
Modal expansion method
Acoustical isolation of a piping system
Cavity inserted in series with the main circuit
Cavity connected in derivation to the main circuit
Computational procedures suited to TMM softwares
Formulation of the forced acoustical system
Matrix equation of a tube element
Impedances and external sources
Single branched circuits
Multi-branched circuits
Application to the acoustical isolation of a forced flow loop
Speed of sound
Speed of sound and fluid compressibility
Isothermal versus adiabatic speed of sound in gases
Speed of sound in a gas liquid mixture (bubbly liquid)
Quasi-static homogeneous model
Dispersive model accounting for the bubble vibrations
Speed of sound of a fluid contained within elastic walls3D Sound waves
3D Standing sound waves (acoustic modes)
Modal equations and general properties of acoustic modes
Interface separating two media and boundary conditions
Wave equation expressed in terms of displacement field
Wave equation expressed in terms of pressure
Analytical examples of acoustical modes
Rectangular enclosure
Circular cylindrical enclosure
Spherical enclosure
Guided wave modes and plane wave approximationRectangular waveguides
Guided mode waves
Physical interpretation
Cylindrical waveguides
Forced waves
Forced wave equations
Forced waves in rectangular enclosures
Green function
Response to a velocity source distributed over a surface
Response to a concentrated pressure, or dipole source
Modal expansion method for coupled enclosures
Forced waves in waveguides
Local and far acoustical fields
Impedance surface and mode coupling
Forced waves in open space: Green’s functions
3D unbounded medium
3D medium bounded by a fixed plane, image source method
3D medium bounded by a pressure nodal plane, dipole sources
Distributed monopole sources and 2D cylindrical waves
Distributed monopole sources and plane waves
Distributed monopole sources and first Rayleigh integral
Pressure field in the axial direction by a baffled circular piston
Directivity of sound radiated by a baffled circular piston
Dipole radiation by the unbaffled circular piston integral equation (KH)
Weighted integral formulations
The Kirchhoff-Helmholtz integral theorem
Particularization of K.H. integral to plane waves
Application to plane acoustic waves triggered by a transient
K.H. integral for 3D external and internal problems
Application: pressure field induced by the unbaffled circular pistonVibroacoustic coupling
Local equilibrium equations
Mixed and non symmetrical formulation
Symmetrical formulation in terms of displacements
Mixed and symmetrical formulation
Piston-fluid column system
Modal problem
Analytical solution
Modal expansion method: displacement as the fluid variable
Modal expansion method: pressure as the fluid variable
Pressure and displacement potential as two fluid variables
Analytical solutions of forced problems
Piston coupled to a fluid column and forced harmonically
Response to a transient force exerted on the piston
Tube excited by a transient pressure source
Expansion methods to solve forced problems
Displacement field as the fluid variable
Response to a seismic excitation
Vibroacoustic coupling in tube and ducts circuits
Simplifications inherent in the tubular geometry
Tubular vibroacoustic coupling model
Incompressible transverse coupling terms
Vibroacoustic coupling at a change in the cross-section
Vibroacoustic coupling at bends
Vibroacoustic coupling at closed ends and tube junctions
Equation of motion of a pipe filled with a fluid
Application to a few problems
Vibroacoustic modes of cylindrical vessels
Longitudinal vibroacoustic modes of a straight vessel
Numerical aspects related to the modal projection method
Vibroacoustic modes of an inflated toroidal shell
Thermal expansion lyre filled with incompressible fluid
Thermal expansion lyre filled with compressible fluid
Simplified model of a drum using modal expansions
Vibroacoustic consequences of cavitation
One dimensional model of cavitation
Analytical example
Finite element methodVariational formulation of the vibroacoustic equations
Formulation in terms of fluid displacement
Mixed ( ) , S X p formulation
Mixed ( ) ,,sX p Π formulation
Discretization in finite elements
Finite element equations in the ( ) ,s fX X variables
Finite element equations in the ( ) ,sX p variables
Finite element equations in the ( ) ,,sX p Π variables
Example: 1D acoustic finite elementEnergy dissipation by the fluid
Preliminary survey on linear modelisation of dissipation
Diversity and importance of the dissipative processes
The viscous damping model
Damped harmonic oscillator
Multiple degrees of freedom systems
Damped acoustical modes in a tube
Transfer matrix method
Forced damped waves
Spectral domain
Time domain: dissipative terminal impedance
Time domain: dissipative fluid
Radiation damping
Radiation of acoustic waves
Sound intensity and power levels
Piston-fluid column system: motion of the piston
Piston-fluid column system: acoustic waves
Piston-fluid system: terminal impedance for an open tube
Spherical shell pulsating in an infinite medium
Kirchhoff-Helmholtz integral applied to the spherical radiator
Rigid sphere oscillating rectilinearly
Radiation of circular cylindrical shells
Sound transmission through interfaces
Transmission loss at the interface separating two fluids
Transmission through a flexible wall: infinite and finite wall models
Vibroaoustic travelling waves in an infinite membrane
Sound transmission through an infinite membrane, or plate
Transmission through a finite plate
Radiation of water waves
Energy considerations
Boundary value problem
Dissipation induced by viscosity of the fluid
Viscous shear waves
Fluid-structure coupling, incompressible case
Piston-fluid system
Flexible plates coupled by a liquid layer
Rigid plate coupled to a thin liquid layer
Cylindrical annular gap
Application to fluid induced damping of multisupported tubes
Dissipation in acoustic waves
Viscous dissipation
Plane unconfined waves
Importance of fluid confinement in viscous dissipation
Plane waves confined in a circular cylindrical tube
Miscellaneous dissipative mechanisms in acoustic waves
Heat conduction and thermoacoustic coupled waves
Relaxation mechanisms
Appendix A A few elements of thermodynamics
Thermodynamic refresher
Law of energy conservation
Compressibility and thermal expansion coefficients
Second law: entropy
Maxwell relations
Thermodynamic relations particularized to perfect gases
Heat transfer and energy losses
Appendix A Mechanical properties of common materials
Phase diagram
Gas properties
Liquid properties
Solid properties
Appendix A The Green identity
Appendix A Bessel functions
Definition
Bessel functions of the first kind
Bessel functions of the second kind
Recurrence relations
Remarkable integrals
Lommel integrals
Hankel functions
Asymptotic forms for large values of the argument
Modified Bessel functions of the first and second kinds
Appendix A Spherical functions
Legendre functions and polynomials
Recurrence and orthogonality relations for Legendre polynomials
Spherical Bessel functions
Recurrence relations for spherical Bessel functions
Spherical Hankel functions
Appendix A Specific impedances of several substances