Derivation of a slow phase model of vortex-induced vibrations for smooth and turbulent oncoming flows

Abstract

This paper analyzes the influence of turbulence on a wake-oscillator model. Turbulence is introduced by randomizing the model proposed by Facchinetti et al. under the quasisteady assumption. A multiple scale analysis of the deterministic model shows that the response is governed by a dimensionless group D, expressed as a function of the amplitudes of the forcing terms in the two governing equations, the total (aerodynamic plus structural) damping and the parameter ε of the fluid Van der Pol oscillator. The influence of turbulence is interpreted as a stochastic noise of small intensity and with a slower timescale than the (fast) oscillations, which is typical of wind engineering applications. A slow phase model of the problem is then derived by assuming that the small turbulence drives the system only slightly away from its limit cycle in smooth flow conditions. Standard modeling techniques borrowed from other fields of physics, in particular the observation of phase shifts and their accumulation, are used to highlight conditions under which the turbulence of the oncoming flow might reduce the amplitudes of vibrations of the body. The slow phase model is derived in smooth flow conditions, then extended to turbulent flow. It recalls that the phase plays a central role in synchronization problems, and that the response amplitude should only be considered as a sub-product of the slow phase. The slow phase model is expressed by means of a first order nonlinear differential equation for the phase and a memoryless transformation for the response amplitudes. Its solution is explicit and simple in some limiting cases. In particular, for small turbulence intensity, the response is shown to be insensitive to turbulence when its frequency content is not low enough. This major dependence upon the frequency content of the turbulence explains that the reduction of VIV due to turbulence cannot be explained by the turbulence intensity only, as usually considered today. The required relative smallnesses of the turbulence and its frequency content naturally appear in the derivation, which is led in a dimensionless manner. Finally, the present study constitutes an analysis of a phenomenological model which could be used in a much wider concept than of the elastically-mounted circular cylinder.

Parameter identification of wake-oscillator from wind tunnel data

“…For this reason, we suppose ξ = εξ 0 . Another choice could be ξ = ε 2 ξ 0 , as proposed by Denoël (2020) who showed that it will lead to the same results after mathematical developments as soon as ξ is as large as ε.…”

“…Ω ∼ 1. In forced vibration, Ω is driven by ε whilst in free vibration (present case), it is more appropriate to rescale the mistuning as Ω = 1 + ξ δ, see (Denoël, 2020), it is justified by results below. Because ξ is a small parameter, δ is the mistuning parameter of order 1.…”

“…The dimensionless group D 1 leads to asymmetric VIV curves. It is equal to zero in the Facchinetti's model (Denoël, 2020). The mistuning δ is the driving parameter in Eq.…”

“…When D 0 = D 1 = 0, there is no coupling term and it corresponds to type A (forced models). The cases D 0 = 0 or D 1 = 0 are shown to make the link with other kinds of VIV models that use only one coupling term in the fluid equation (only the acceleration (D 1 = 0) for Facchinetti model (Facchinetti et al, 2004;Denoël, 2020) or the velocity (D 0 = 0) for Hartlen-Currie model (Hartlen and Currie, 1970)). Tamura's wake-oscillator model is thus a generalized model that combines the features of Facchinetti and Hartlen-Currie models.…”

Generalized lift force model under vortex shedding

“…( 6) is thus not sufficient and a randomization of this generalized model is therefore necessary. Such a randomization has been suggested in Denoël (2020) in the context of stochastic Vortex Induced Vibration (VIV) by adding a term (1 + I u ) in front of U ∞ , to study the influence of turbulence intensity and length scale on VIV using a randomized version of Facchinetti's model (another model based on Van der Pol equation). Nielsen and Krenk (1997) also studied a stochastic version of their energy balanced model for VIV by replacing the fluid velocity with (1 + R(t))U ∞ , with the turbulence represented by a non-dimensional stochastic process R(t).…”