Abstract
A nonlinear mathematical model is developed in the time domain to simulate the behaviour of two identical flexibly mounted cylinders in tandem while undergoing vortex-induced vibration (VIV). Subsequently, the model is validated and modified against experimental results. Placing an array of bluff bodies in proximity frequently happens in different engineering fields. Chimney stacks, power transmission lines and oil production risers are few engineering structures that may be impacted by VIV. The coinciding of the vibration frequency with the structure natural frequency could have destructive consequences. The main objective of this study is to provide a symplectic and reliable model capable of capturing the wake interference phenomenon. This study shows the influence of the leading cylinder on the trailing body and attempts to capture the change in added mass and damping coefficients due to the upstream wake. The model is using two coupled equations to simulate the structural response and hydrodynamic force in each of cross-flow and stream-wise directions. Thus, four equations describe the fluid–structure interaction of each cylinder. A Duffing equation describes the structural motion, and the van der Pol wake oscillator defines the hydrodynamic force. The system of equations is solved analytically. Two modification terms are added to the excitation side of the Duffing equation to adjust the hydrodynamic force and incorporate the effect of upstream wake on the trailing cylinder. Both terms are functions of upstream shedding frequency (Strouhal number). Additionally, the added mass modification coefficient is a function of structural acceleration and the damping modification coefficient is a function of velocity. The modification coefficients values are determined by curve fitting to the difference between upstream and downstream wake forces, obtained from experiments. The damping modification coefficient is determined by optimizing the model against the same set of experiments. Values of the coefficients at seven different spacings are used to define a universal function of spacing for each modification coefficient so that they can be obtained for any given distance between two cylinders. The model is capable of capturing lock-in range and maximum amplitude.
Highlights
Based on their observation, when the angle between two cylinders is ψ < 30◦, the flow pattern could be divided into three groups: (i) at small pitch ratio and small angle of incident, the upstream shear layers reattached on the trailing cylinder; (ii) as ψ grew larger, the reattachment could not be maintained so the shear layer was deflected into and rolled up in the gap between two cylinders which induced separation on the trailing cylinder; and (iii) while ψ was still small, if the gap grew larger, the deflected shear layers in the gap could form a fully developed Kármán vortex street which was referred to as vortex impingement flow pattern
The model was developed based on a coupled system of a van der Pol wake oscillator and a Duffing equation
Duffing equation was considered to describe the structural response of the cylinders in both directions, and it is capable of capturing the structural nonlinearity of the system
Summary
There are mathematical models simulating VIV, but they are limited to an isolated cylinder and do not capture wake interference These models typically utilize two differential equations to describe the structural response and hydrodynamic forces and couple them together to represent the fluid–structure interaction [13]. Facchinetti et al [14] conducted an extensive study on simulation of the fluid force on a rigid isolated cylinder by a van der Pol wake oscillator and simulated structure response with a mass–damping system The proposed stall parameter was defined as a function of shedding frequency and cylinder velocity that provided negative values for large structural motion and could couple the wake force to the structural motion equation These attempts are limited to an isolated cylinder. An overall model is proposed which is able to capture the onset of lock-in, maximum amplitude and lock-in range width
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