H. M. Blackburn

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.

Topology of fine-scale motions in turbulent channel flow

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.

Convective instability and transient growth in flow over a backward-facing step

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes

Previous studies dealing with Floquet secondary stability analysis of the wakes of circular and square cross-section cylinders have shown that there are two synchronous instability modes, with long (mode A) and short (mode B) spanwise wavelengths. At intermediate wavelengths another mode arises, which reaches criticality at Reynolds numbers higher than modes A or B. Here we concentrate on these intermediate-wave number modes for the wakes of circular and square cylinders. It is found that in both cases these modes possess complex-conjugate pair Floquet multipliers, and can be combined to produce either standing or traveling waves. Both these states are quasiperiodic.

The influence of pipe length on turbulence statistics computed from direct numerical simulation data

In this paper, direct numerical simulation of fully developed turbulent pipe flow is carried out at Reτ≈170 and 500 to investigate the effect of the streamwise periodic length on the convergence of turbulence statistics. Mean flow, turbulence intensities, correlations, and energy spectra were computed. The findings show that in the near-wall region (below the buffer region, r+≤30), the required pipe length for all turbulence statistics to converge needs to be at least a viscous length of O(6300) wall units and should not be scaled with the pipe radius (δ). It was also found for convergence of turbulence statistics at the outer region that the pipe length has to be scaled with pipe radius and a proposed pipe length of 8πδ seems sufficient for the Reynolds numbers considered in this study.

Symmetry breaking of two-dimensional time-periodic wakes

A number of two-dimensional time-periodic flows, for example the Karman street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincare map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-period- flip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

Lock-in behavior in simulated vortex-induced vibration

Abstract Results are presented for a two-dimensional numerical simulation of vortex-induced vibration of a circular cylinder at a Reynolds number of 250. A spectral element spatial discretization and a stiffly stable time integration scheme were employed to solve the Navier-Stokes equations in an accelerating frame of reference attached to the cylinder. The response envelope of the vibrating cylinder was similar to those previously obtained in experiments, and lock-in (coalescence of cross-flow oscillation and vortex-shedding frequencies at a frequency close to the cylinder natural frequency) was observed. Chaotic cylinder responses were observed over a range of cylinder natural frequencies. Simulations of flows past a cylinder in forced cross-flow oscillation are also discussed, and a result showing an asymmetric P + S vortex shedding mode is presented.

Spectral element filtering techniques for large eddy simulation with dynamic estimation

Spectral element methods have previously been successfully applied to direct numerical simulation of turbulent flows with moderate geometrical complexity and low to moderate Reynolds numbers. A natural extension of application is to large eddy simulation of turbulent flows, although there has been little published work in this area. One of the obstacles to such application is the ability to deal successfully with turbulence modelling in the presence of solid walls in arbitrary locations. An appropriate tool with which to tackle the problem is dynamic estimation of turbulence model parameters, but while this has been successfully applied to simulation of turbulent wall-bounded flows, typically in the context of spectral and finite volume methods, there have been no published applications with spectral element methods. Here, we describe approaches based on element-level spectral filtering, couple these with the dynamic procedure, and apply the techniques to large eddy simulation of a prototype wall-bounded turbulent flow, the plane channel, using a mixing length-based eddy viscosity subgrid-scale model. The methods outlined here may be carried over without modification to more complex geometries.

Convective instability and transient growth in steady and pulsatile stenotic flows

We show that suitable initial disturbances to steady or long-period pulsatile flows in a straight tube with an axisymmetric 75%-occlusion stenosis can produce very large transient energy growths. The global optimal disturbances to an initially axisymmetric state found by linear analyses are three-dimensional wave packets that produce localized sinuous convective instability in extended shear layers. In pulsatile flow, initial conditions that trigger the largest disturbances are either initiated at, or advect to, the separating shear layer at the stenosis in phase with peak systolic flow. Movies are available with the online version of the paper.