The effect of two degrees of freedom on vortex-induced vibration at low mass and damping

Abstract

Although there are a great many papers dedicated to the problem of a cylinder vibrating transverse to a fluid flow ($Y$-motion), there are almost no papers on the more practical case of vortex-induced vibration in two degrees of freedom ($X,Y$ motion) where the mass and natural frequencies are precisely the same in both $X$- and $Y$-directions. We have designed the present pendulum apparatus to achieve both of these criteria. Even down to the low mass ratios, where $m^*\,{=}\,6$, it is remarkable that the freedom to oscillate in-line with the flow affects the transverse vibration surprisingly little. The same response branches, peak amplitudes, and vortex wake modes are found for both $Y$-only and $X,Y$ motion. There is, however, a dramatic change in the fluid–structure interactions when mass ratios are reduced below $m^*\,{=}\,6$. A new amplitude response branch with significant streamwise motion appears, in what we call the ‘super-upper’ branch, yielding massive amplitudes of 3 diameters peak-to-peak ($A^*_Y \,{\sim}\, 1.5$). We discover a corresponding periodic vortex wake mode, comprising a triplet of vortices being formed in each half-cycle, in what we define as a ‘2T’ mode. We qualitatively interpret the principal vortex dynamics and vortex forces which yield a positive rate of energy transfer ($\dot{e}_V$) causing the body vibration, using the following simple equation: \[\dot{e}_V = 2 {\Gamma}^* U^*_V \skew3\dot{Y}\] where $\Gamma^*$ is vortex strength, $U^*_V$ is the speed downstream of the dominant near-wake vorticity, and $\skew3\dot{Y}$ is the transverse velocity of the body. This simple approach suggests that the massive amplitude of vibration for the 2T mode is principally attributed to the energy transfer from the ‘third’ vortex of each triplet, which is not present in the lower-amplitude 2P mode. We also find two low-speed streamwise vibration modes, which is not unexpected, since they correspond to the first and second excitation modes of vibration for flexible cantilevers. By considering equations of motion for the two degrees of freedom, we find a critical mass, $m^*_{\hbox{\scriptsize\it crit}} \,{=}\, 0.52$, similar to recent $Y$-only studies, below which the large-amplitude vibrations persist to infinite flow velocity. We show that the critical mass $m^*_{\hbox{\scriptsize\it crit}}$ is the same for the $X$- and $Y$-directions, which ensures that the shapes of $X, Y$ trajectories can retain their form as the velocity becomes large. The extensive studies of vortex-induced vibration for $Y$-only body motions, built up over the last 35 years, remain of strong relevance to the case of two degrees of freedom, for $m^* \,{>}\, 6$. It is only for ‘small’ mass ratios, $m^* \,{<}\, 6$, that one observes a rather dramatic departure from previous results, which would suggest a possible modification to offshore design codes.

Time domain simulation of vortex-induced vibrations in stationary and oscillating flows

“…As the equations of motion governing viscous flow are difficult to solve, experiments have been the most important source of new insight. Examples of typical experiments are free vibration of elastically mounted rigid cylinders (Feng, 1968;Vikestad, 1998;Govardhan and Williamson, 2000;Jauvtis and Williamson, 2004) and cylinders undergoing forced motions (Sarpkaya, 1978;Moe and Wu, 1990;Morse and Williamson, 2009;Aglen and Larsen, 2011;Yin and Larsen, 2012). Experiments with long flexible structures have also been performed, both under controlled laboratory conditions (Chaplin et al, 2005;Trim et al, 2005;HueraHuarte et al, 2014) and in more realistic field environments (Huse et al, 1998;Vandiver et al, 2006).…”

Two-dimensional vortex-induced vibration suppression through the cylinder transverse linear/nonlinear velocity feedback

“…10 Influence of m * on controlled amplitudes for cylinder with ξ = 0.00361 and β K = γ G = 0.1 for NC, respectively. Experimental θ xy data (squares) of Jauvtis and Williamson [23] and numerical (θ xy , θ qy ) results (solid lines) in the absence of control are also overlapped in Fig. 7a-d.…”

“…Accordingly, it is deduced that β K and γ G should be of the order of unity. Accordingly, the low-mass-damping 2-DOF cylinder (m * = 2.6, ξ = 0.00361) tested by Jauvtis and Williamson [23] is considered. Comparisons of analytical (circles), numerical (triangles) and experimental (squares) results are displayed in Fig.…”

“…Several experimental [11,23,38] and CFD [2,30] 2-DOF VIV studies have enabled the recently modified wake oscillator models to capture the experimentally observed hydro-elastic phenomena governing coupled cross-flow/in-line VIV alongside the lift/drag hydrodynamic features [41]. By using a CFD approach, Hasheminejad et al [19] recently proposed an adaptive fuzzy sliding mode controller for suppressing the 2-DOF VIV for a low-mass cylinder in a laminar flow with Re = 90.…”

Estudo experimental das vibrações induzidas pela emissão de vórtices em cilindros flexíveis inclinados em relação à correnteza.

“…Além dos artigos citados acima é importante salientar os grandes avanços e fundamentais contribuições dos estudos de: Lyons & Patel (1986), Meneghini (1993), Jauvtis & Williamson (2004.…”