Stability and Bifurcation Analysis of a Forced Cylinder Wake

We present a study on the dynamics of a cylinder wake subjected to forced excitation. Williams et al. (1992) discovered that the spatial symmetry of the excitation plays a crucial role in determining the resulting wake dynamics. Reflection-symmetric forcing was found to affect the Karman wake much more strongly compared to 2 ( , ) Z κ π asymmetric forcing. For low forcing amplitudes, the existence of a nonlinear mode interaction mechanism was postulated to explain the observed “beating” phenomenon observed in the wake. Previous work by the authors (Mureithi et al. 2002, 2003) presented general forms of the modal interaction amplitude equations governing the dynamics of the periodically forced wake. In the present work, numerical CFD computations of the forced cylinder wake are presented. It is shown that the experimentally observed wake bifurcations can be reproduced by numerical simulations with reasonable accuracy. The CFD computations show that the forced wake first looses reflection symmetry followed by a bifurcation associated with vortex merging as the forcing amplitude is increased. A bifurcation analysis of a simplified amplitude equation shows that these two transitions are due to a pitchfork bifurcation and a perioddoubling bifurcation of mixed mode solutions. INTRODUCTION The Karman wake behind a circular cylinder is a global instability mode manifested following spatio-temporal (absolute) instability over a ‘large’ region of the cylinder near wake. Two such global modes are possible. The Karman wake corresponds to the so called ‘sinuous’ mode. A symmetrical ‘varicose’ mode may also be manifested in the form pairs of vortices symmetrically shed behind the cylinder. Although the symmetrical varicose mode is intrinsically unstable, it may be stabilized by cylinder motion. Thus interaction between symmetrical shedding and structural motion has been shown to lead to self-excited vibration of the structure (King 1977, Naudascher 1987). The problem of forced cylinder motion in flow has been considered, for instance, by Naudascher (1986), Williamson & Roshko (1988), Ongoren & Rockwell (1988) and Williams et al. (1992). Lateral (cross-flow) and inflow forced oscillation of a cylinder was performed over a range of frequencies and amplitudes. In this parameter space, complex sheddings modes synchronized to the cylinder oscillations were discovered. Synchronized patterns (modes) labeled according to the number of vortices shed per cycle of cylinder oscillation, included 2S, 2P, P+S and 2P+2S; S and P indicate single vortex and pairs of vortices, respectively. Wake (spatio-temporal) symmetry was found to depend strongly on natural shedding to forcing frequency ratio. For low amplitude forcing, modes involving coalescence of large numbers of vortices were found. In Williams et al.’s experiments apparent period-doubling was found to occur during symmetric forcing for intermediate forcing amplitudes. ‘Simple’ symmetry rules were proposed to describe the fundamental wake structure differences for symmetric and antisymmetric forcing. Recent work by Mureithi et al. (2002-2004) has quantitatively established the important role that symmetry plays in the dynamics of the forced cylinder wake. In particular 1 Copyright © 2005 by ASME amplitude equations governing the nonlinear dynamics of mode interactions have been derived based on symmetry equivariant theory. The present paper reports recent developments in this ongoing work. While period-doubling was observed in experimental tests reported in Mureithi et al. (2002), the complexity of the turbulent flow made it difficult to discern the dynamics underlying the observed transitions in the wake flow. To overcome this difficulty, the cylinder wake flow is simulated numerically eliminating the effects of extraneous experimental noise. Flow turbulence, known to play at best a secondary role in the wake bifurcation dynamics, is ignored. CFD COMPUTATIONS Numerical simulations have been performed using the CFD code Fluent. A rectangular domain around a circular cylinder, having diameter D, was generated using the Fluent preprocessor Gambit. Computation domain boundaries were set at 15D upstream, 40D downstream and 20D on the lateral sides. Special care was required to avoid the influence of the boundaries. To solve the Navier-Stokes equations, Fluent uses a standard path in finite volume. A second order upwind scheme was used to obtain the face values needed for the integral computations. A second order temporal integration was performed during the simulations. Modelling of the cylinder motion was done using time dependent inertia forces and boundary conditions to simulate the problem in the nonGalilean cylinder reference frame. Time dependent conditions were input to Fluent by defining a UDF (User Defined Function). Simulations were all carried out for a flow at Re=1000. At this Reynolds number the cylinder wake is three dimensional and turbulent. However, as also justified in Blackburn et al. (1999), the two dimensional harmonic forcing of the cylinder will produce flows more two dimensional than the fixed cylinder wake. Furthermore the stochastic threedimensional perturbations associated with flow turbuluence are not believed to be of primary importance vis-à-vis the global wake bifurcation dynamics considered in this study. As an example, the primary Hopf bifurcation responsible for the Karman wake shedding is independent of the detailed flow turbulence structure being governed by the form of the velocity profile in the cylinder wake. Test case The case of the fixed cylinder was used to validate the numerical model and the accuracy of the simulation results. A series of structured meshes, each time more refined, were tested up to spatial convergence. The final mesh chosen has the following characteristics: 260000 elements (corresponding to 125000 nodes and 135000 cells). Special care was required in meshing the cylinder boundary layer, which is excited by the moving cylinder. The time step was chosen to capture the dynamics of the wake. It was equal to one hundredth of the vortex-shedding period. The accuracy of our results has been verified by comparing our drag and lift coefficient (respectively Cd, Cl) and Strouhal number (St) values to some experimental and numerical data, as shown in Table 1. The computed drag coefficient has a mean value of 1.27. The rms lift coefficient is 0.85, while the Strouhal number is approximately 0.24. Re = 1000 St Cd Cl Our results 0.237 1.27 0.85 Hua et al. (2005) computation 0.239 1.49±0.2 0.98 Roshko (1954) experiments 0.21 1.2 Table 1. Summary of present results compared to other computational and experimental data. The Strouhal number is high compared to experimental results but the value is reasonable when compared to other two dimensional CFD simulations. The drag coefficient is larger than that found experimentally by Roshko (1954). This can be explained by the absence of turbulence modelling in our simulations as justified earlier. The lift coefficient results are close to those of Hua et al. (2005). Figure 1 shows the Von Karman wake downstream of the cylinder. The change in the wake structure beyond ten diameters downstream is in agreement with experimental observations (see Van dyke (1982)) and the computation results of Hua et al. (2005). The foregoing verification is for the fixed cylinder case. The validation of the moving cylinder simulations may be found in the results and discussion sections of the paper. Figure 1. Vorticity field (s) past a fixed cylinder