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Forced Oscillations of a Spring-Mounted Body by a Viscous Liquid: Rotational Case

Denis Bonheure, Giovanni Paolo Galdi, Clara Patriarca

TL;DR

The paper addresses forced oscillations of a spring-mounted rotating body immersed in a viscous incompressible fluid, driven by a time-periodic far-field flow. It develops a body-fixed reformulation of the coupled Navier–Stokes and rigid-body system, and proves the existence of a $T$-periodic weak solution for any driving period with uniform bounds on linear and angular displacements, independent of resonance proximity. The authors extend previous work by incorporating rotation with an undamped restoring torque and by establishing pointwise amplitude bounds, using mollified bounded-domain problems and an invading-domain approach to pass to the full space. The results demonstrate that fluid viscosity provides sufficient dissipation to prevent resonance in this class of fluid–structure interaction problems, with implications for designing rotating structures subject to periodic flows.

Abstract

We study the periodic motions of the coupled system $\mathscr S$, consisting of an incompressible Navier-Stokes fluid interacting with a structure formed by a rigid body subject to {\em undamped} elastic restoring forces and torque around its rotation axis. The motion of $\mathscr S$ is driven by the uniform flow of the liquid, far away from the body, characterized by a time-periodic velocity field, $\mathbf{V}$, of frequency $f$. We show that the corresponding set of governing equations always possesses a time-periodic weak solution of the same frequency $f$, whatever $f>0$, the magnitude of $\mathbf{V}$ and the values of physical parameters. Moreover, we show that the amplitude of linear and rotational displacement is always pointwise in time uniformly bounded by one and the same constant depending on the data, regardless of whether $f$ is or is not close to a natural frequency of the structure. Thus, our result rules out the occurrence of resonant phenomena.

Forced Oscillations of a Spring-Mounted Body by a Viscous Liquid: Rotational Case

TL;DR

The paper addresses forced oscillations of a spring-mounted rotating body immersed in a viscous incompressible fluid, driven by a time-periodic far-field flow. It develops a body-fixed reformulation of the coupled Navier–Stokes and rigid-body system, and proves the existence of a -periodic weak solution for any driving period with uniform bounds on linear and angular displacements, independent of resonance proximity. The authors extend previous work by incorporating rotation with an undamped restoring torque and by establishing pointwise amplitude bounds, using mollified bounded-domain problems and an invading-domain approach to pass to the full space. The results demonstrate that fluid viscosity provides sufficient dissipation to prevent resonance in this class of fluid–structure interaction problems, with implications for designing rotating structures subject to periodic flows.

Abstract

We study the periodic motions of the coupled system , consisting of an incompressible Navier-Stokes fluid interacting with a structure formed by a rigid body subject to {\em undamped} elastic restoring forces and torque around its rotation axis. The motion of is driven by the uniform flow of the liquid, far away from the body, characterized by a time-periodic velocity field, , of frequency . We show that the corresponding set of governing equations always possesses a time-periodic weak solution of the same frequency , whatever , the magnitude of and the values of physical parameters. Moreover, we show that the amplitude of linear and rotational displacement is always pointwise in time uniformly bounded by one and the same constant depending on the data, regardless of whether is or is not close to a natural frequency of the structure. Thus, our result rules out the occurrence of resonant phenomena.

Paper Structure

This paper contains 10 sections, 4 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main results
  3. Body-fixed frame formulation
  4. $T$-periodic weak solutions and statement of the main result
  5. The initial-boundary value problem in bounded domains
  6. Local dissipation of the total energy
  7. Proof of Theorem \ref{['main:th']}
  8. Step $1$: Existence of $T$-periodic weak solutions in $\Omega_R$
  9. Step $2$: Invading domains procedure and proof of Theorem \ref{['main:th']}
  10. The Case of a symmetric body

Key Result

Theorem 2.1

Let $V\in H^1(0,T;\mathbb{R})$ be $T$-periodic for some $T>0$. Then, there is at least one corresponding $T$-periodic weak solution to (eq:01-translation+rotation_or). Furthermore, there exists $C>0$ depending only on the physical and geometrical parameters of the problem, such that and where

Figures (1)

  • Figure 1: Left: a generic representation of the system body-liquid. Right: a 2D-schematic view of the cross-section of the deck of a bridge subjected to a periodic wind-flow.

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Definition 5.1
  • ...and 2 more