Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem

TL;DR

This paper rigorously establishes conditions under which a fluid-structure system exhibits a Hopf bifurcation, leading to oscillations driven by fluid flow, without restrictions on oscillation frequency, and highlights the potential for large amplitude oscillations near natural frequencies.

Contribution

It provides the first rigorous analysis of Hopf bifurcation in a fluid-structure interaction problem with detailed spectral conditions.

Findings

Existence of a bifurcating time-periodic branch above a flow velocity threshold.

No restriction on the bifurcation frequency, allowing resonance with natural frequencies.

Large oscillation amplitudes possible near natural frequencies when fluid density is small.

Abstract

We furnish necessary and sufficient conditions for the occurrence of a Hopf bifurcation in a particularly significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $\lambda>0$. In particular, if the relevant linearized operator meets suitable spectral properties, there exists a threshold $\lambda_o>0$ above which a bifurcating time-periodic branch stems out of the branch of steady-state solutions. The most remarkable feature of our result is that no restriction is imposed on the frequency $\omega$ of the bifurcating solution, which may thus coincide with one of the natural structural frequencies $\omega_{\sf n}$ of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when $\omega\to\omega_{\sf n}$, the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first {\it rigorous} investigation of the existence of a Hopf bifurcation in a fluid-structure interaction problem.