J. Planchard

Eigenfrequencies of a tube bundle placed in a confined fluid

Abstract The problem of natural vibration of a group of immersed tubes is investigated, and it is shown that the presence of the fluid has the effect of spreading the mechanical resonance frequencies of the tubes over intervals of varying width, and of introducing eigenfrequencies between these ranges. The case of compressible or uncompressible fluid is examined. Several computation methods are described.

Added mass and damping in fluid-structure interaction

Abstract This paper is concerned with the added mass matrix for a mechanical structure vibrating in an incompressible liquid. It is shown in particular that this matrix does not depend on viscosity and, from this fact, can be calculated as if the fluid is perfect. The viscous effect on the mechanical system can then be represented by a damping term of type time-convolution. The presence of a flowing fluid around the structure leads to additional damping terms proportional to the fluid density.

Existence and location of eigenvalues for fluid-solid structures

In this paper, we study some existence and location theorems for the eigenvalues of a mathematical model that describes the eigenfrequencies and eigenmotions of a tube bundle immersed in an inviscid compressible fluid. The model is a non-standard eigenvalue problem because it involves a non-local boundary condition, and the eigenvalues of the problem appear in two places: namely, in the equations and in this non-local boundary condition. The problem is treated by using methods from functional analysis. First, the existence of eigenvalues is stated by proving that the original eigenvalue problem is equivalent to that of determining the characteristic values of a linear, compact, selfadjoint operator. Next, the location theorems are established by obtaining explicit bounds for the eigenvalues. While some of these bounds are derived by applying general location theorems of the theory of compact selfadjoint operators, others are deduced by homotopy and continuity arguments.

Natural frequencies of tube bundle in an uncompressible fluid

Abstract A method of eigenfrequency computation for a cluster of tubes immersed in an uncompressible liquid is described. Several numerical examples are presented, showing that the presence of fluid involves the spreading of the natural frequencies of the tubes in vacuum. A lower bound of these eigenfrequencies can be obtained for a spatially periodic bundle.

A quadratic eigenvalue problem involving Stokes equations

Abstract An existence theorem for the eigenvalues of a spectral problem is studied in this paper. The physical situation behind this mathematical problem is the determination of the eigenfrequencies and eigenmotions of a fluid-solid structure. The liquid part in this structure is represented by a viscous incompressible fluid, while the solid part is a set of parallel rigid tubes. The spectral problem governing this system is a quadratic eigenvalue problem which involves Stokes equations with a non-local boundary condition. The strategy for tackling the question of existence of eigenvalues consists of proving that the original problem is equivalent to that of determining the characteristic values of a linear (non-selfadjoint) compact operator. Sharp estimates for the eigenvalues give precise information about the region of ω where the eigenvalues are located. In particular, we prove that this problem admits a countable set of eigenvalues in which only a finite number of them have a non-zero imaginary part.

Computation of the acoustic eigenfrequencies of cavities containing a tube bundle

Abstract The acoustic eigenfrequencies of a cavity containing a periodic tube array are determined by computing the effective sound velocity in planes normal to the tubes by means of homogenization techniques. Also the case of an elastic bundle is examined.

A method of finding the eigenvalues and eigenfunctions of self-adjoint elliptic operators

Abstract A method for computing the eigenvalues and eigenvectors of the laplacian operator on a bounded domain is presented. This is achieved by solving Helmholtzs equation and constructing a unitary operator defined on the boundary, depending on the excitatory frequency but independent of the boundary conditions. This method can be extended to free elastic vibration problems.

Asymptotic Analysis Relating Spectral Models in Fluid--Solid Vibrations

An asymptotic study of two spectral models which appear in fluid--solid vibrations is presented in this paper. These two models are derived under the assumption that the fluid is slightly compressible or viscous, respectively. In the first case, min-max estimations and a limit process in the variational formulation of the corresponding model are used to show that the spectrum of the compressible case tends to be a continuous set as the fluid becomes incompressible. In the second case, we use a suitable family of unbounded non-self-adjoint operators to prove that the spectrum of the viscous model tends to be continuous as the fluid becomes inviscid. At the limit, in both cases, the spectrum of a perfect incompressible fluid model is found. We also prove that the set of generalized eigenfunctions associated with the viscous model is dense for the L2-norm in the space of divergence-free vector functions. Finally, a numerical example to illustrate the convergence of the viscous model is presented.

Homogenization and Bloch wave method for fluid tube bundle interaction

The aim of this paper is to investigate the problem of the vibrations of large arrays of elastic rods immersed in a perfect incompressible fluid. The case of an infinite spatially periodic bundle is firstly considered leading to use the Bloch wave method in order to describe the resonance spectrum of the coupled system. When the bundle is contained in a bounded domain, the homogenization technique combined with the Bloch wave method allows to obtain the eigenspectrum which is formed of two eigenfrequencies (of infinite multiplicity), and of a continuous spectrum.

Un problème de fréquences propres en couplage fluide-structure

The aim of this paper is the determination of the eigenfrequencies of elastic tube-bundles immersed in a perfect fluid. Two computational methods in the case of tube bundles with periodic structure are described. These methods are based on homogenization techniques and pseudo-periodic functions.