Thomas Leweke

Wake transition of a rolling sphere

Graphical Abstract

A merging criterion for two-dimensional co-rotating vortices

We propose a quantitative criterion for the merging of a pair of equal two-dimensional co-rotating vortices. A cross-validation between experimental and theoretical analyses is performed. Experimental vortices are generated by the roll-up of a vortex sheet originating from the identical and impulsive rotation of two plates. The phenomenon is then followed up in time until a rapid pairing transition occurs for which critical parameters are measured. In the theoretical approach, the nonlinear Euler solution representing a pair of equal vortices is computed for various nonuniform vorticity distributions. The stability analysis of such a configuration then provides critical values for the onset of merging. From this data set, a criterion depending on global impulse quantities is extracted for different shapes of the vorticity distribution. This theoretical statement agrees well with our experimentally based criterion.

Three-dimensional instabilities in wake transition

Abstract It is now well-known that the wake transition regime for a circular cylinder involves two modes of small-scale three-dimensional instability, modes “A” and “B”, occurring in different Reynolds number ranges. These modes are quite distinct in spanwise lengthscale and in symmetry, and they are found to scale on different physical features of the flow. Mode A has a large spanwise wavelength of around 3–4 cylinder diameters, and scales on the larger physical structure in the flow, namely the core of the primary Karman vortices. The feedback from one vortex to the next gives an out-of-phase streamwise vortex pattern for this mode. In contrast, the mode B instability has a distinctly smaller spanwise wevelength (1 diameter) which scales on the smaller physical structure in the flow, namely the braid shear layer. The symmetry of mode B is determined by the reverse flow behind the bluff cylinder, leading to a system of streamwise vortices which are in phase between successive half cycles. The symmetries of both modes are the same as the ones found in the vortex system evolving from perturbed plane wakes studied by Meiburg and Lasheras (1988) and Lasheras and Meiburg (1990). Furthermore, the question of the physical origin of these three-dimensional instabilities is addressed. We present evidence that they are linked to general instability mechanisms found in two-dimensional linear flows. In particular, mode A seems to be a result of an elliptic instability of the near-wake vortex cores; predictions based on elliptic instability theory concerning the initial perturbation shape and the spanwise wevelength are in good agreement with experimental observations. For the mode B instability, it is suggested that it is a manifestation of a hyperbolic instability of the stagnation point flow found in the braid shear layer linking the primary vortices.

Three-dimensional instability during vortex merging

The interaction of two parallel vortices of equal circulation is observed experimentally. For low Reynolds numbers (Re), the vortices remain two dimensional and merge into a single one, when their time-dependent core size exceeds approximately 30% of the vortex separation distance. At higher Re, a three-dimensional (3-D) instability is discovered, showing the characteristics of an elliptic instability of the vortex cores. The instability rapidly generates small-scale turbulent motion, which initiates merging for smaller core sizes and produces a bigger final vortex than for laminar 2-D flow.

Elliptic instability of a co-rotating vortex pair

In this paper, we report experimental results concerning a three-dimensional short-wave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le Dizes & Laporte (2002) concerning elliptic instability of co-rotating vortices show very good agreement. At later stages of the flow, when the perturbation amplitude becomes sufficiently large, the two vortices merge into a single structure. This happens for smaller cores sizes than in the case of two-dimensional merging. The three-dimensional merging leads to a final vortex characterized by turbulent small-scale motion, whose size appears to be larger than it would have been without instability. The vorticity profile of the final vortex is non-Gaussian after both two-dimensional and three-dimensional merging. The profile contains more vorticity outside the inner core than a Gaussian vortex, resulting from the ejection of vorticity filaments during the merging stage.

Sphere-wall collisions: vortex dynamics and stability

For moderate Reynolds numbers, a sphere colliding with a wall in the normal direction will lead to a trailing recirculating wake, threading over the sphere after impact and developing into a complex vortex-ring system as it interacts with vorticity generated at the wall. The primary vortex ring, consisting of the vorticity from the wake of the sphere prior to impact, persists and convects, relatively slowly, outwards away from the sphere owing to the motion induced from its image. The outward motion is arrested only a short distance from the axis because of the strong interaction with the secondary vorticity. In this paper, the structure and evolution of this combined vortex system, consisting of a strong compact primary vortex ring surrounded by and interacting with the secondary vorticity, is quantified through a combined experimental and numerical study. The Reynolds-number range investigated is 100 < Re < 2000. At Reynolds numbers higher than about 1000, a non-axisymmetric instability develops, leading to rapid distortion of the ring system. The growth of the instability does not continue indefinitely, because of the dissipative nature of the flow system; it appears to reach a peak when the wake vorticity first forms a clean primary vortex ring. A comparison of the wavelength, growth rate and perturbation fields predicted from both linear stability theory and direct simulations, together with theoretical predictions, indicates that the dominant physical mechanism for the observed non-axisymmetric instability is centrifugal in nature. The maximum growth occurs at the edge of the primary vortex core, where the vorticity changes sign. Notably, this is a physical mechanism different from that proposed previously to explain the development of the three-dimensional flow of an isolated vortex ring striking a wall.

Steady inlet flow in stenotic geometries: convective and absolute instabilities

Steady inlet flow through a circular tube with an axisymmetric blockage of varying size is studied both numerically and experimentally. The geometry consists of a long, straight tube and a blockage, semicircular in cross-section, serving as a simplified model of an arterial stenosis. The stenosis is characterized by a single parameter, the aim being to highlight fundamental behaviours of constricted flows, in terms of the total blockage. The Reynolds number is varied between 50 and 2500 and the stenosis degree by area between 0.20 and 0.95. Numerically, a spectral-element code is used to obtain the axisymmetric base flow fields, while experimentally, results are obtained for a similar set of geometries, using water as the working fluid. At low Reynolds numbers, the flow is steady and characterized by a jet flow emanating from the contraction, surrounded by an axisymmetric recirculation zone. The effect of a variation in blockage size on the onset and mode of instability is investigated. Linear stability analysis is performed on the simulated axisymmetric base flows, in addition to an analysis of the instability, seemingly convective in nature, observed in the experimental flows. This transition at higher Reynolds numbers to a time-dependent state, characterized by unsteadiness downstream of the blockage, is studied in conjunction with an investigation of the response of steady lower Reynolds number flows to periodic forcing.

Flow around an impulsively arrested circular cylinder

The vortex dynamics of the flow around a suddenly arrested translating circular cylinder is investigated by direct numerical simulation and water tank experiments. In the numerical study, a method of visualization of streaklines in simulated-particle tracking computations is proposed, which is based on a variable-variance two-dimensional Gaussian-weighted summation of particles in the vicinity of each interpolation point, and for which a close similarity with physical dye visualizations is found. This technique is used to identify the trajectory of both the wake vortices, as well as the secondary vortices induced as the original wake convects over the arrested cylinder. Observations show that, in a fashion similar to the flow past an arresting sphere, each wake vortex induces a counter-rotating vortex pair, which subsequently self-propels over a range of sometimes surprising trajectories as the Reynolds number and cylinder translation distance are varied. At low Reynolds numbers and short translation dist...

The wake behind a cylinder rolling on a wall at varying rotation rates

A study investigating the flow around a cylinder rolling or sliding on a wall has been undertaken in two and three dimensions. The cylinder motion is specified from a set of five discrete rotation rates, ranging from prograde through to retrograde rolling. A Reynolds number range of 20–500 is considered. The effects of the nearby wall and the imposed body motion on the wake structure and dominant wake transitions have been determined. Prograde rolling is shown to destabilize the wake flow, while retrograde rotation delays the onset of unsteady flow to Reynolds numbers well above those observed for a cylinder in an unbounded flow. Two-dimensional simulations show the presence of two recirculation zones in the steady wake, the lengths of which increase approximately linearly with the Reynolds number. Values of the lift and drag coefficient are also reported for the steady flow regime. Results from a linear stability analysis show that the wake initially undergoes a regular bifurcation from a steady two-dimensional flow to a steady three-dimensional wake for all rotation rates. The critical Reynolds number Re c of transition and the spanwise wavelength of the dominant mode are shown to be highly dependent on, but smoothly varying with, the rotation rate of the cylinder. Varying the rotation from prograde to retrograde rolling acts to increase the value of Re c and decrease the preferred wavelength. The structure of the fully evolved wake mode is then established through three-dimensional simulations. In fact it is found that at Reynolds numbers only marginally (~5%) above critical, the three-dimensional simulations indicate that the saturated state becomes time dependent, although at least initially, this does not result in a significant change to the mode structure. It is only at higher Reynolds numbers that the wake undergoes a transition to vortex shedding. An analysis of the three-dimensional transition indicates that it is unlikely to be due to a centrifugal instability despite the superficial similarity to the flow over a backward-facing step, for which the transition mechanism has been speculated to be centrifugal. However, the attached elongated recirculation region and distribution of the spanwise perturbation vorticity field, and the similarity of these features with those of the flow through a partially blocked channel, suggest the possibility that the mechanism is elliptic in nature. Some analysis which supports this conjecture is undertaken.

The evolution of a subharmonic mode in a vortex street

The development of a subharmonic three-dimensional instability mode in a vortex street is investigated both numerically and experimentally. The flow past a ring is considered as a test case, as a previous stability analysis has predicted that for a range of aspect ratios, the first-occurring instability of the vortex street is subharmonic. For the flow past a circular cylinder, the development of three-dimensional flow in the vortex street is known to lead to turbulent flow through the development of spatio-temporal chaos, whereas subharmonic instabilities have been shown to cause a route to chaos through the development of a period-doubling cascade. The three-dimensional vortex street in the flow past a ring is analysed to determine if a subharmonic instability can alter the route to turbulence for a vortex street. A linear stability analysis and non-axisymmetric computations are employed to compute the flow past a ring with an aspect ratio