Ronald D. Henderson

Three-dimensional Floquet stability analysis of the wake of a circular cylinder

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

Details of the drag curve near the onset of vortex shedding

A closer look at the drag curve for a circular cylinder near the onset of vortex shedding reveals that there is a sharp transition in the forces acting on a body moving through a fluid when it produces an unsteady wake. In this Letter results from high‐resolution computer simulations are presented to quantify the change in viscous drag, pressure drag, and base pressure coefficients.

Three-dimensional instability in flow over a backward-facing step

Results are reported from a three-dimensional computational stability analysis of flow over a backward-facing step with an expansion ratio (outlet to inlet height) of 2 at Reynolds numbers between 450 and 1050. The analysis shows that the first absolute linear instability of the steady two-dimensional flow is a steady three-dimensional bifurcation at a critical Reynolds number of 748. The critical eigenmode is localized to the primary separation bubble and has a flat roll structure with a spanwise wavelength of 6.9 step heights. The system is further shown to be absolutely stable to two-dimensional perturbations up to a Reynolds number of 1500. Stability spectra and visualizations of the global modes of the system are presented for representative Reynolds numbers.

A study of two-dimensional flow past an oscillating cylinder

In this paper we describe a detailed study of the wake structures and flow dynamics associated with simulated two-dimensional flows past a circular cylinder that is either stationary or in simple harmonic cross-flow oscillation. Results are examined for Re = 500 and a fixed motion amplitude of y(max)/D = 0.25. The study concentrates on a domain of oscillation frequencies near the natural shedding frequency of the fixed cylinder. In addition to the change in phase of vortex shedding with respect to cylinder motion observed in previous experimental studies, we note a central band of frequencies for which the wake exhibits long-time-scale relaxation oscillator behaviour. Time-periodic states with asymmetric wake structures and non-zero mean lift were also observed for oscillation frequencies near the lower edge of the relaxation oscillator band. In this regime we compute a number of bifurcations between different wake configurations and show that the flow state is not a unique function of the oscillation frequency. Results are interpreted using an analysis of vorticity generation and transport in the base region of the cylinder. We suggest that the dynamics of the change in phase of shedding arise from a competition between two different mechanisms of vorticity production.

Nonlinear dynamics and pattern formation in turbulent wake transition

Results are reported on direct numerical simulations of transition from two-dimensional to three-dimensional states due to secondary instability in the wake of a circular cylinder. These calculations quantify the nonlinear response of the system to three-dimensional perturbations near threshold for the two separate linear instabilities of the wake: mode A and mode B. The objectives are to classify the nonlinear form of the bifurcation to mode A and mode B and to identify the conditions under which the wake evolves to periodic, quasi-periodic, or chaotic states with respect to changes in spanwise dimension and Reynolds number. The onset of mode A is shown to occur through a subcritical bifurcation that causes a reduction in the primary oscillation frequency of the wake at saturation. In contrast, the onset of mode B occurs through a supercritical bifurcation with no frequency shift near threshold. Simulations of the three-dimensional wake for fixed Reynolds number and increasing spanwise dimension show that large systems evolve to a state of spatiotemporal chaos, and suggest that three-dimensionality in the wake leads to irregular states and fast transition to turbulence at Reynolds numbers just beyond the onset of the secondary instability. A key feature of these ‘turbulent’ states is the competition between self-excited, three-dimensional instability modes (global modes) in the mode A wavenumber band. These instability modes produce irregular spatiotemporal patterns and large-scale ‘spot-like’ disturbances in the wake during the breakdown of the regular mode A pattern. Simulations at higher Reynolds number show that long-wavelength interactions modulate fluctuating forces and cause variations in phase along the span of the cylinder that reduce the fluctuating amplitude of lift and drag. Results of both two-dimensional and three-dimensional simulations are presented for a range of Reynolds number from about 10 up to 1000.

Secondary instability in the wake of a circular cylinder

Secondary instability of flow past a circular cylinder is examined using highly accurate numerical methods. The critical Reynolds number for this instability is found to be Rec=188.5. The secondary instability leads to three‐dimensionality with a spanwise wavelength at onset of 4 cylinder diameters. Three‐dimensional simulations show that this bifurcation is weakly subcritical.

Adaptive Spectral Element Methods for Turbulence and Transition

These notes present an introduction to the spectral element method with applications to fluid dynamics. The method is introduced for one-dimensional problems, followed by the discretization of the advection and diffusion operators in multi-dimensions, and efficient ways of dealing with these operators numerically. We also discuss the mortar element method, a technique for incorporating local mesh refinement using nonconforming elements; this is the foundation for adaptive methods. An adaptive strategy based on analyzing the local polynomial spectrum is presented and shown to give accurate solutions even for problems with weak singularities. Finally we describe techniques for integrating the incompressible Navier-Stokes equations, including methods for performing computational linear and nonlinear stability analysis of non-parallel and time-periodic flows.

Spectral Elements: Adaptivity and Applications in Fluid Dynamics

At the heart of all numerical methods for solving mathematical problems lies the requirement that numerical errors be computed, estimated, or at least bounded in some way. Except for the simplest cases, i.e. linear problems or benchmark cases with known solutions, this is done in a refinement study by systematically varying some parameter related to the accuracy of the numerical method. For example, in structured algorithms like regular finite differences and spectral methods the accuracy of the solution is governed by a single parameter: the mesh spacing, h, or the order of the approximating functions, N. Refinement is easy: just change the single parameter. For the sake of analogy, we compare this to finding the minimum of a function in 1-dimension that only decreases in one direction.