No abstract

We computationally investigate coupling of a nonlinear rotational dissipative element to a sprung circular cylinder allowed to undergo transverse vortex-induced vibration (VIV) in an incompressible flow. The dissipative element is a ‘nonlinear energy sink’ (NES), consisting of a mass rotating at fixed radius about the cylinder axis and a linear viscous damper that dissipates energy from the motion of the rotating mass. We consider the Reynolds number range $20\leqslant Re\leqslant 120$, with $Re$ based on cylinder diameter and free-stream velocity, and the cylinder restricted to rectilinear motion transverse to the mean flow. Interaction of this NES with the flow is mediated by the cylinder, whose rectilinear motion is mechanically linked to rotational motion of the NES mass through nonlinear inertial coupling. The rotational NES provides significant ‘passive’ suppression of VIV. Beyond suppression however, the rotational NES gives rise to a range of qualitatively new behaviours not found in transverse VIV of a sprung cylinder without an NES, or one with a ‘rectilinear NES’, considered previously. Specifically, the NES can either stabilize or destabilize the steady, symmetric, motionless-cylinder solution and can induce conditions under which suppression of VIV (and concomitant reduction in lift and drag) is accompanied by a greatly elongated region of attached vorticity in the wake, as well as conditions in which the cylinder motion and flow are temporally chaotic at relatively low $Re$.

The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original formulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit timescale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram-Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation.

We introduce a class of acquisition functions for sample selection that lead to faster convergence in applications related to Bayesian experimental design and uncertainty quantification. The approach follows the paradigm of active learning, whereby existing samples of a black-box function are utilized to optimize the next most informative sample. The proposed method aims to take advantage of the fact that some input directions of the black-box function have a larger impact on the output than others, which is important especially for systems exhibiting rare and extreme events. The acquisition functions introduced in this work leverage the properties of the likelihood ratio, a quantity that acts as a probabilistic sampling weight and guides the active-learning algorithm toward regions of the input space that are deemed most relevant. We demonstrate the proposed approach in the uncertainty quantification of a hydrological system as well as the probabilistic quantification of rare events in dynamical systems and the identification of their precursors in up to 30 dimensions.

We investigate the dynamics of a two-dimensional circular cylinder mounted on a linear spring, restricted to move in the cross-flow direction and undergoing vortex-induced vibration, incorporating a strongly nonlinear (i.e., non-linearizable) internal element consisting of a mass that is free to rotate about the cylinder axis and whose angular motion is restrained by a linear viscous damper. The conjunction of the essentially nonlinear inertial coupling with the dissipative element makes the internal attachment behave as a nonlinear energy sink that is able to extract and dissipate energy from the motion of the cylinder and (indirectly) the surrounding fluid. At the intermediate Reynolds number Re = 100, we find that the cylinder with rotator undergoes repetitive cycles of slowly decaying oscillations interrupted by chaotic bursts; during the slowly decaying portion of each cycle, the dynamics of the cylinder is regular and can lead to significant vortex street elongation with partial stabilization of the wake. We construct a reduced-order model of the fluid-structure interaction dynamics based on the data obtained by direct numerical simulation, and employ analytical techniques such as complexification/averaging and the multiple-scales method to show that the strongly modulated response is the manifestation of a resonance capture into a slow invariant manifold (SIM) that leads to targeted energy transfer from the cylinder to the rotator. Capture into the SIM corresponds to transient cylinder stabilization, whereas escape from the SIM leads to chaotic bursts. Hence, the action of the nonlinear rotator on the resonance dynamics of the fluid-structure interaction is clarified.