CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 4, Fall 1996

A FINITE DIFFERENCE CALCULATION FOR A TRANSVERSE SUPERIMPOSED OSCILLATION

SERPIL KOCABIYIK AND PHU NGUYEN

ABSTRACT. A spectral-finite difference method is wed for analysis of unsteady, laminar flow past an impulsively started oscillating and translating circular cylinder. The flow is incompressible and twedimensional and the cylinder oscillations are harmonic. These oscillations are only allowed when the oscillatory-tetranslationalvelocity ratio is 0.5. The analysis is developed in terms of the scalar vorticity and stream function. The Navier-Stokes equations in a noninertial frame attached to the cylinder are solved in a rectangular grid, based on a modified polar system. The parameters involved are the velocity ratio, Reynolds number and Strouhal number. The Reynolds number ranges between 200 and lo3 and Strouhal number ranges between s/4 and r / 2 . The effects of Reynolds number and of the Strouhal number on the laminar asymmetric wake evolution are studied. The surface pressure, drag, lift and surface vorticity are also extracted from numerical results.

1. Introduction. One of the most common causes of flow-induced vibration of high aspect ratio bluff bodies is the regular shedding vortices. Oscillations most frequently occur in a direction transverse to that of the mainstream and often accompanied by large changes in the structure of the shed vortices. The physics of vortex street formation and the near-wake flow have been the focal point for many past studies as reported in a recent review article by Griffin and Hall [9]. One reason for this interest has been the importance of knowing how the mean and fluctuating fluid forces are generated on the body due to vortex shedding. Numerous experimental and theoretical studies have been performed using the circular cylinder, and this shape is chosen for the present investigation. Physical insight into the vortex shedding process by means of an accurate mathematical formulation and by means of streamline plotting for transverse rectilinear motion is one of the goals of the present paper. The results of this research are of both theoretical and practical importance since it adds to the knowledge of

Accepted for publication in ~ecember 1996. Copyright 01996 Rocky Mountain Mathematics Consortium

382 S. KOCABIYIK AND P. NGUYEN

vortex formation and hydrodynamic forces acting on the cylinder and because this problem can be related to several engineering applications.

In the present paper we consider two-dimensional flow caused by an infinitely long circular cylinder set in motion impulsively which trans- lates with uniform velocity and also undergoes a harmonic transverse oscillation. A frame of reference which translates and oscillates with the cylinder is employed where the flow variables chosen to describe the motion of the viscous incompressible fluid are taken to be the stream function and the vorticity. The unsteady two-dimensional Navier- Stokes equations are solved using a spectral finite-difference method, but with the boundary vorticity calculated using global integral con- ditions rather than local fini te-difference approximations. Thus, the present method differs from the previous similar studies: [ll, 5, 131. All of these simulations were carried out using the primitive variables except for Lecointe and Piquet [13] in which vortex shedding from both the suddenly started and oscillating cylinders was investigated using the AD1 method. In the present study we investigate a higher Reynolds number range than those previously considered. Detailed features of the present numerical method and systematic validations have been outlined in Nguyen and Kocabiyik [14] and Kocabiyik and Nguyen [12]; therefore, Section 2 only briefly summarizes the equations and the numerics. The study of the unsteady wake behind a circular cylinder is then considered for transverse superimposed motion.

In the case of flow past a stationary cylinder, vortices are shed at a constant nondimensional natural shedding frequency, for a flow with a particular Reynolds number. The vortices induce a periodic transverse or lift force at shedding frequency and a periodic drag force at twice the shedding frequency. Within a range of cylinder oscillation frequency, vortex shedding is controlled by the oscillation of the cylinder, and a considerable increase in the fluid forces is observed. This is known as the "lock-in," "wake capture" or "synchronization" phenomenon. During the transverse oscillation, lock-in occurs when the cylinder oscillation frequency approaches the natural shedding frequency causing a considerable increase in the drag force with the vortices being shed at the same frequency as that of the cylinder. The lock-in range is somewhat dependent on Reynolds number and cylinder amplitude, and the maximum range is f40 percent of the midpoint frequency (see [lo]). Outside lock-in, two types of forces