Stability analysis of the wake-induced vibration of tandem circular and square cylinders

Abstract

We present a stability analysis to investigate the underlying fluid–structure modes during transverse wake-induced vibration (WIV) of an elastically mounted downstream cylinder in a tandem arrangement at low Reynolds number. The upstream cylinder with an equal diameter is kept stationary, whereas the downstream cylinder in the tandem arrangement is free to vibrate in the transverse direction. The WIV involves complex interaction dynamics of the upstream wake with the freely vibrating downstream cylinder, which leads to a relatively large transverse force and vibration in the post-lock-in region. We consider a data-driven model reduction approach to construct an eigenvalue representation of WIV system and perform the stability analysis to examine the underlying process of WIV for the tandem circular and square cylinders. The model reduction of the fluid system is constructed by the eigensystem realization algorithm (ERA) and coupled with a transversely vibrating bluff body in a state-space format. Unlike the wake-oscillator model, the ERA-based ROM does not rely on any empirical formulation and captures naturally the essential linear fluid dynamics through the solution of the Navier–Stokes equations. Results show that the WIV region and the onset reduced velocity can be predicted accurately by tracing the eigenvalue trajectories of the ERA-based low-dimensional model for a range of reduced velocities. The stability analysis reveals that there exists a persistently unstable eigenvalue branch that sustains WIV, which also implies that the highly nonlinear behavior of WIV has its linear origin. The sharp corner of the square cylinder is found to have the stabilizing effects, namely (i) the reduced velocity for the WIV onset is larger than its circular cylinder counterpart, (ii) the transverse response amplitude and the lift force are consistently lesser in both lock-in and post-lock-in regimes for the square cylinder configuration. This work has a potential impact on the development of control strategies for reducing undesired vibrations and loads in flexible structures undergoing wake interference effects.

Appendix A: Fluid-Structure Energy Transfer

Following our previous work on VIV [34], the displacement and lift coefficient can be obtained for the linear FSI system as:

$$\begin{aligned} \left. \begin{array}{ll} \displaystyle Y={\hat{Y}} e^{\lambda _r t}\cos (\lambda _i t) \\ \displaystyle C_l=\hat{C_l} e^{\lambda _r t}\cos (\lambda _i t + \phi ) \end{array}\right\} , \end{aligned}$$

(A.1)

where $\lambda =\lambda _r+i\lambda _i$ is eigenvalue with real $\lambda _r$ and imaginary $\lambda _i$ components, ${\hat{Y}}$ and $\hat{C_l}$ denote the magnitudes of eigenmodes. The phase angle difference is derived by plugging Eq. (A.1) into Eq. (12):

$$\begin{aligned} \sin {\phi } = \frac{2\lambda _r\lambda _i}{\sqrt{(\lambda _r^2 + (2\pi F_s)^2 + \lambda _i^2)^2 - (4\pi \lambda _i F_s)^2 }}. \nonumber \\ \end{aligned}$$

(A.2)

Refer to [34] for further details. Next, the energy transfer per cycle is evaluated as:

$$\begin{aligned} \displaystyle E(t)= & {} \int _{t}^{t+\frac{2 \pi }{\lambda _i}} {\dot{Y}} \hat{C_l} \mathrm{d}t \nonumber \\ \displaystyle= & {} \frac{1}{2}(\lambda _i \sin (\phi ) \nonumber \\&+\, \lambda _r \cos (\phi ) + \lambda _r \cos (2 \lambda _i t + \phi )) \int _{t}^{t+\frac{2 \pi }{\lambda _i}} e^{2 \lambda _r t} dt\nonumber \\ \end{aligned}$$

(A.3)

Using Eq. (A.3), the energy transfer coefficient $E_c$ can be defined by excluding the exponential growth/decay rate $\lambda _r$

$$\begin{aligned} E_c = \pi {\hat{Y}} \hat{C_l} \sin (\phi ) \end{aligned}$$

(A.4)