Floquet and Force Analysis of Fluid Structures Generated by Oscillating Cylinders

Significant Achievements, Anticipated Outcomes and Future Work

Investigations into the flow resulting from oscillating a circular cylinder in an initially quiescent flow have been conducted. A number of flow states have been identified and the transitions between these states have been studied using a combination of Direct-Numerical simulation (DNS), Floquet analysis and particle tracking using DNS.

Our studies of the two-dimensional symmetry breaking transitions in the time-periodic flow generated by harmonic cylinder motion are summarised in Figure 1. This figure, which shows results obtained using Floquet analysis, shows the curve of marginal stability for the two modes of two-dimensional symmetry breaking that have been observed.

The transitions to a synchronous mode (I), with real multipliers, and a quasi-periodic mode (II), with complex multipliers, have been located. Experimental results are shown with a dashed line. An illustration of a quasi-periodic mode can be seen in Figure 2 which shows a quasi-periodic flow with particles in the flow at a Stokes number (frequency of oscillation) of 40 and KC= 4.7 (amplitude). The particle tracking matches closely with experimental results and shows the periodic symmetries present in the flow. The large scale vortices, shown at distance from the cylinder (located in the centre of the image), are formed quasi-periodically.

Our three-dimensional investigations are summarised in Figure 3 which shows the curves of marginal stability in three-dimensional space. Our results from floquet analysis, indicated by the solid and hollow dots, match well with experimental results, shown as lines through the solid dots. Two sets of floquet analyses are shown here with the solid dot set representing a series of simulations where the symmetry of the base flow was unrestricted (as it would be in experiments). The hollow dot series shows a set of simulations where symmetry was enforced about the axis of cylinder motion. Figure 4 shows the spanwise wavenumbers at the onset of the three-dimensional modes. Experimental values from a paper by Tatsuno & Bearman (1990) are indicated with a cross (x).

Computational Techniques Used

We use a parallel DNS code to simulate time-dependent incompressible flows. Essentially this amounts to time-integration of Navier-Stokes problems; a time-splitting scheme is used to reduce the complexity of each integration sub-step combined with a spectral element/Fourier spatial discretisation. The spectral element method is a high-order finite element technique that combines the geometric flexibility of finite elements with the high accuracy of spectral methods. The method was pioneered in the mid 1980s by Anthony Patera and students at MIT.

Floquet analysis is achieved by developing two-dimensional time-periodic flows using the spectral element method described previously. This "base" flow is then perturbed with random noise and used in a specially modified version of the spectral element solver which permits us to monitor the growth rates for individual spanwise wavenumbers. The predictions for growth are then tested using the full version of the solver.

Publications, Awards and External Funding

External Funding and Awards

None.

Publications

J. Elston, J. Sheridan and H. M. Blackburn, Two-Dimensional Floquet Stability Analysis of the Flow Produced by an Oscillating Circular Cylinder in Quiescent Fluid. European Journal of Mechanics B/Fluids 23, pp99-106, 2004.

J. Sheridam, J. Elston and H.M.Blackburn, Flow over an oscillating cylinder, International Union of Theoretical and Applied Mechanics Conference on Fluid-Structure Interaction (Keynote address) Rutgers University, USA, June, 2003.