Numerical simulation of vortex-induced vibration of a circular cylinder with low mass-damping in a turbulent flow
Introduction
Vortex shedding behind bluff bodies arises in many fields of engineering, such as heat exchanger tubes, marine cables, flexible risers in petroleum production and other marine applications, bridges, and chimneys stacks. These examples are only a few of a large number of problems where vortex-induced vibrations are important. The practical significance of vortex-induced vibrations has led to a large number of fundamental studies [see Sarpkaya (1979); Griffin and Ramberg (1982); Bearman (1984); Parkinson (1989); Blevins (1977)]. The case of an elastically mounted cylinder vibrating as a result of fluid forcing is one of the most basic and revealing cases in the general subject of vortex-induced bluff–body fluid–structure interactions. Consequently, determination of the unsteady forces on the cylinder is of central importance to the dynamics of such interactions. Despite the extensive force measurements for a cylinder undergoing transverse-forced vibration, there have appeared no direct lift-force measurements in the literature for an elastically mounted arrangement. Sarpkaya (1995) recently presented a set of drag measurements for a cylinder which can oscillate both in-line and transverse to the flow. We should also note that Hover et al. (1997) have developed a novel feedback control system to study free motions. Khalak and Williamson (1997) have presented new force measurements of lift and drag for a hydroelastic cylinder of very low mass and damping. Consequently, for numerics, comparisons of forces with experimental data are difficult.
Vortex-induced vibration is generally associated with the so-called “lock-in” phenomenon where the motion of the structure is believed to dominate the shedding process, thus synchronizing the shedding frequency. Lock-in is characterized by a shifting of the vortex shedding frequency to the system natural frequency . Lock-in can also refer to the coalescence of the shedding, the cylinder oscillation and natural frequency . Numerous studies in vortex-induced vibration literature support the existence of lock-in [e.g., Griffin et al., (1973); Anagnostopoulos and Bearman (1992); Goswami et al., (1993); Brika and Laneville (1993); Blackburn and Henderson (1996); Fujarra et al., (1998)]. However, Gharib (1999) has noted the absence of lock-in behavior from almost all experimental studies for small mass ratios [=(oscillating mass)/(displaced fluid mass)]. In almost all the literature, the problem of vortex-induced vibration of a cylinder with a large mass ratio has been well studied. However, there remain some rather basic questions concerning vibration phenomena under the conditions of very low mass and damping, for which there are few laboratory investigations. As one reduces the mass ratio to 1% of the value used in the classical study of Feng (1968), it is of significant and fundamental interest to know what is the dominant response frequency during excitation, what is the range of normalized velocity for significant oscillations or lock-in, and what is the amplitude of response as a function of normalized velocity?
Recently, Govardhan and Williamson (2000) published a review for these phenomena. Their experiments show that there exists two distinct types of response for the transverse oscillations of an elastically mounted rigid cylinder. At low mass-damping, they find three amplitude response branches, denoted “the initial branch”,“ the upper branch” and “the lower branch”. At high mass-damping, corresponding to the classical experiments of Feng (1968), only two branches exist; the upper branch is absent. At low mass-damping, the transitions between the modes of response are discontinuous. The mode change between initial and upper response branches involves a hysteresis. This contrasts with the intermittent switching of modes for the transition between upper–lower branches. Much recent two-dimensional numerical studies with low mass-damping or even with zero damping (Blackburn and Karniadakis 1993) and (Newman and Karniadakis 1997)) give amplitude results very similar (A/D=0.6) but smaller than the expected values. For these simulations, the Reynolds number typically used is Re=100−200. In the simulations of Saltara et al. (1998) and Evangelinos (1999), the Reynolds number is higher, Re=1000, and the amplitude reaches values of A/D=0.7. But all these simulations are carried out for low Reynolds numbers. This is contrasted with typical values of Re=103−105 for the experimental data. It seems that only the lower branch of amplitudes is picked up by the numerical simulations and the upper branch is absent. Recently, Blackburn et al. (2001) presented computations which predict the upper branch. However, these authors do not predict the amplitude of the experimental upper branch.
The main objective of this paper is to investigate numerically the dynamics and fluid forcing on an elastically mounted rigid cylinder with low mass-damping, constrained to oscillate transversely to a free stream. The numerical results are compared with the experimental data obtained by Khalak and Williamson (1996).
Section snippets
Fluid modeling
The aim is a computational study to predict the two-dimensional fluid motion induced by the oscillation of a circular cylinder. Equations are presented throughout in nondimensional form. The velocity scale is the reference velocity, U∞, the length scale is the cylinder diameter, D, and time is nondimensionalized by the aerodynamic scale D/U∞.
The unsteady incompressible Reynolds-averaged Navier–Stokes equations can be written in the following strong-conservation form, in the inertial system,
Discrete equations
The collocated cell-centered grid lay-out is used. So, the Cartesian velocity components and the pressure share the same location at the center of the control volume (Fig. 1).
In the following, Uk(NN) will be the unknown kth Cartesian velocity component at point NN and the flux at cell interface pN is identified as (JUi)(pN). The discrete divergence of the flux φ over the control volume is simplyso that the discrete continuity equation results from .
Initial conditions
To start a computation at a normalized velocity and a Reynolds number Re given, we need to initialize the flow. Thus, three initial conditions are used.
- (i)
Firstly, the condition denoted from rest. In this case, for each Reynolds number considered, the solution of vortex shedding behind a fixed cylinder is attempted. When the lift force becomes periodic, the elastically mounted cylinder is allowed to oscillate.
- (ii)
The second condition is called increasing velocity that is to say with the cylinder
Spatial grid independence
In order to establish a grid-independent solution, computations have been performed for several meshes with 202×150, 252×187, 302×225, 352×262, 402×300 grids for a reduced velocity and a Reynolds number Re=3757. The r.m.s displacement, yrms and the maximum displacement, ymax, are presented in Table 1. The difference in displacement between the coarse grid and the finest grid is 2% for the r.m.s. values and less than 3% for the maximum value. Hence, all computational results are obtained
Preamble
Before presenting results, some numerical parameters need to be specified. The equations are solved on an O-type structured grid. A mesh with 202 points in the angular direction and 150 points in the radial direction is used. The first points of the mesh in fluid are located at y=0.001D away from the wall what represents a maximum distance y+⩽0.6 for the range of Reynolds numbers considered. The outer flow boundary is located at 25 diameter lengths away from the cylinder. A nondimensional time
Conclusions
This numerical study of vortex-induced vibrations of a circular cylinder with low mass-damping shows the steady response of the cylinder is a hysteresis phenomenon. Three initial conditions have been used: from rest, increasing velocity and decreasing velocity. According these initial conditions, the response of the cylinder differs. With the from rest and decreasing velocity conditions, the simulations predict only the lower branch. On the other hand, with the increasing velocity condition,
Acknowledgements
The authors gratefully acknowledge the Scientific Committee of IDRIS (projects 00.0129 and 01.0129) for the attribution of CPU time on the Nec SX5 of IDRIS. The authors are also grateful to Charles H.K. Williamson for placing his experimental data at our disposal and for helpful discussions.
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