A numerical survey of wake modes and energy transfers for an oscillating cylinder at Re=200

A lack of information regarding the connections between energy transfer and wake mode of oscillating bodies in low Reynolds number flow prompted a two-dimensional numerical investigation at Re = 200. The region in which vortex shedding synchronized with the cylinder oscillation frequency was defined and the energy transfers calculated. The mode of Kármán vortex shedding, in which two vortices of opposite sign were shed each cycle (2S), displayed a gradual change from positive to negative energy transfer with increasing amplitude of motion. This change in energy transfer was directly related to the relative velocity of the fluid, which altered the position of the stagnation point and upper and lower shear layers on the body. As amplitude was increased the 2S shedding mode evolved into an asymmetric mode of shedding in which a pair and a single vortex (P+S) were shed each motion cycle. Energy transfers were calculated in the region of primary lock-in. The P+S mode occurred only in the region of negative energy transfer. Introduction A numerical investigation was conducted into the wake states and energy transfers experienced by a cylinder, undergoing forced oscillations transverse to the flow at Re=200. By investigating the parameter space defined by reduced velocity, Vr, and amplitude of the motion, A, the resulting lift trace was recorded and flow visualisations obtained. Changes in the form or phase of the lift trace, relative to the cylinder displacement result in a change in the direction or magnitude of the energy transfer of the system (Carberry et al., 2001; Blackburn and Henderson, 1999). The forced oscillations experienced by the cylinder supress the three-dimensionality of the flow and oscillations near the natural shedding frequency, fn, may extend the laminar flow range from Re = 150 up to approximately 350 (Koopmann, 1967; Griffin, 1971). Subsequently, all simulations were restricted to two-dimensions. Particular attention was given to the region in which the vortex shedding frequency synchronised with the imposed cylinder motion. Carberry et al. (2001) found experimentally that changes in the lift force, and hence the energy transfer, of a cylinder undergoing forced oscillation were intrinsically linked to the characteristics of the near wake. When traversing the parameter space defined by frequency and amplitude of the body motion, a discontinuity in the phase between the lift force and cylinder displacement was observed. This discontinuity was noted by Bishop and Hassan (1964) and has since been observed at frequencies both above and below the natural shedding frequency at a range of Reynolds numbers (Lu and Dalton, 1996; Hover et al., 1998). Difficulties in detecting the phase jump at low Reynolds number led Blackburn and Henderson (1999) to conclude that for flows with Re < 400, there exists a viscous dissipation inhibiting the switch and only for Reynolds number greater than this can the phenomenon be accurately observed. Currently, a lack of information exists concerning the link between energy transfer and wake modes throughout the wider parameter space at lower Reynolds numbers. The current investigation attempts to fill this gap. Additionally, the calculation of energy transfer throughout the parameter space may be useful in predicting flow regimes likely to result in vortex-induced vibration (Hover et al.,1998). Methodology The two-dimensional numerical code utilized a spectral-element scheme with three-step time-splitting to solve the incompressible Navier-Stokes equations with an additional forcing term. Eighth-order Lagrangian interpolating polynomials were used to approximate the solution variables within the macro-elements of the mesh. Both the freestream flow speed, U, and diameter, D, were normalised. For a more detailed description of the numerical scheme, see Thompson et al. (1996). The mesh (shown in figure 1) consisted of quadrilateral elements with the domain extending 23D downstream and 15D upstream and to each of the transverse boundaries. The mesh was attached to the non-inertial frame of reference of the cylinder and the free-stream flow and cylinder motion were started impulsively from rest. A no-slip condition was enforced at the cylinder body.