Anders Peter Andersen

Unsteady aerodynamics of fluttering and tumbling plates

We investigate the aerodynamics of freely falling plates in a quasi-two-dimensional flow at Reynolds number of

Vortex wakes of a flapping foil

10^{3}

Analysis of transitions between fluttering, tumbling and steady descent of falling cards

, which is typical for a leaf or business card falling in air. We quantify the trajectories experimentally using high-speed digital video at sufficient resolution to determine the instantaneous plate accelerations and thus to deduce the instantaneous fluid forces. We compare the measurements with direct numerical solutions of the two-dimensional Navier–Stokes equation. Using inviscid theory as a guide, we decompose the fluid forces into contributions due to acceleration, translation, and rotation of the plate. For both fluttering and tumbling we find that the fluid circulation is dominated by a rotational term proportional to the angular velocity of the plate, as opposed to the translational velocity for a glider with fixed angle of attack. We find that the torque on a freely falling plate is small, i.e. the torque is one to two orders of magnitude smaller than the torque on a glider with fixed angle of attack. Based on these results we revise the existing ODE models of freely falling plates. We get access to different kinds of dynamics by exploring the phase diagram spanned by the Reynolds number, the dimensionless moment of inertia, and the thickness-to-width ratio. In agreement with previous experiments, we find fluttering, tumbling, and apparently chaotic motion. We further investigate the dependence on initial conditions and find brief transients followed by periodic fluttering described by simple harmonics and tumbling with a pronounced period-two structure. Near the cusp-like turning points, the plates elevate, a feature which would be absent if the lift depended on the translational velocity alone.

Unsteady motion: escape jumps in planktonic copepods, their kinematics and energetics.

We present an experimental study of a symmetric foil performing pitching oscillations in a vertically flowing soap film. By varying the frequency and amplitude of the oscillation we visualize a variety of wakes with up to 16 vortices per oscillation period, including von Karman vortex street, inverted von Karman vortex street, 2P wake, 2P + 2S wake and novel wakes ranging from 4P to 8P. We map out the wake types in a phase diagram spanned by the width-based Strouhal number and the dimensionless amplitude. We follow the time evolution of the vortex formation near the round leading edge and the shedding process at the sharp trailing edge in detail. This allows us to identify the origins of the vortices in the 2P wake, to understand that two distinct 2P regions are present in the phase diagram due to the timing of the vortex shedding at the leading edge and the trailing edge and to propose a simple model for the vorticity generation. We use the model to describe the transition from 2P wake to 2S wake with increasing oscillation frequency and the transition from the von Karman wake, typically associated with drag, to the inverted von Karman wake, typically associated with thrust generation.

Mechanisms and feasibility of prey capture in ambush-feeding zooplankton

We study the dynamics of a rigid card falling in air using direct numerical simulations of the two-dimensional Navier–Stokes equation and a fluid force model based on ordinary differential equations derived from recent experiments and simulations. The system depends on three non-dimensional parameters, i.e. the thickness-to-width ratio, the dimensionless moment of inertia, and the Reynolds number. By increasing the thickness-to-width ratio in the direct numerical simulations and thereby the non-dimensional moment of inertia we observe a transition from periodic fluttering to periodic tumbling with a wide transition region in which the cards flutter periodically but tumble once between consecutive turning points. In the transition region the period of oscillation diverges and the cards fall vertically for distances of up to 50 times the card width. We analyse the transition between fluttering and tumbling in the ODE model and find a heteroclinic bifurcation which leads to a logarithmic divergence of the period of oscillation at the bifurcation point. We further discuss the bifurcation scenario of the ODE model in relation to our direct numerical simulations and the phase diagrams measured by willmarth, Hawk & Harvey (1964) and belmonte, Eisenberg & Moses (1998).

The bathtub vortex in a rotating container

We describe the kinematics of escape jumps in three species of 0.3–3.0 mm-sized planktonic copepods. We find similar kinematics between species with periodically alternating power strokes and passive coasting and a resulting highly fluctuating escape velocity. By direct numerical simulations, we estimate the force and power output needed to accelerate and overcome drag. Both are very high compared with those of other organisms, as are the escape velocities in comparison to startle velocities of other aquatic animals. Thus, the maximum weight-specific force, which for muscle motors of other animals has been found to be near constant at 57 N (kg muscle)−1, is more than an order of magnitude higher for the escaping copepods. We argue that this is feasible because most copepods have different systems for steady propulsion (feeding appendages) and intensive escapes (swimming legs), with the muscular arrangement of the latter probably adapted for high force production during short-lasting bursts. The resulting escape velocities scale with body length to power 0.65, different from the size-scaling of both similar sized and larger animals moving at constant velocity, but similar to that found for startle velocities in other aquatic organisms. The relative duration of the pauses between power strokes was observed to increase with organism size. We demonstrate that this is an inherent property of swimming by alternating power strokes and pauses. We finally show that the Strouhal number is in the range of peak propulsion efficiency, again suggesting that copepods are optimally designed for rapid escape jumps.

Swimming behavior and prey retention of the polychaete larvae Polydora ciliata (Johnston)

Many marine zooplankters, particularly among copepods, are “ambush feeders” that passively wait for their prey and capture them by fast surprise attacks. This strategy must be very demanding in terms of muscle power and sensing capabilities, but the detailed mechanisms of the attacks are unknown. Using high-speed video we describe how copepods perform spectacular attacks by precision maneuvering during a rapid jump. We show that the flow created by the attacking copepod is so small that the prey is not pushed away, and that the attacks are feasible because of their high velocity (≈100 mm·s−1) and short duration (few ms), which leaves the prey no time for escape. Simulations and analytical estimates show that the viscous boundary layer that develops around the attacking copepod is thin at the time of prey capture and that the flow around the prey is small and remains potential flow. Although ambush feeding is highly successful as a feeding strategy in the plankton, we argue that power requirements for acceleration and the hydrodynamic constraints restrict the strategy to larger (> 0.25 mm), muscular forms with well-developed prey perception capabilities. The smallest of the examined species is close to this size limit and, in contrast to the larger species, uses its largest possible jump velocity for such attacks. The special requirements to ambush feeders with such attacks may explain why this strategy has evolved to perfection only a few times among planktonic suspension feeders (few copepod families and chaetognaths).

Structure of a steady drain-hole vortex in a viscous fluid

We study the time-independent free-surface flow which forms when a fluid drains out of a container, a so-called bathtub vortex. We focus on the bathtub vortex in a rotating container and describe the free-surface shape and the complex flow structure using photographs of the free surface, flow visualizations, and velocity measurements. We find that the velocity field in the bulk of the fluid agrees with predictions from linear Ekman theory for the boundary layer at the bottom, and we discuss the limitations of linear Ekman theory for the source–sink flow in the experiment. We introduce a radial expansion approximation of the central vortex core and reduce the model to a single first-order equation. We solve the equation numerically and find that the axial velocity depends linearly on height whereas the azimuthal velocity is almost independent of height. We discuss the model of the bathtub vortex introduced by Lundgren ( J. Fluid Mech. vol. 155, 1985, p. 381) and compare it with our experiment. We find that the measured velocities and surface profiles are described well by the model when Ekman upflow and surface tension effects are included.

Hydraulic jumps in a channel

SUMMARY The behavior of the ubiquitous estuarine planktotrophic spionid polychaete larvae Polydora ciliata was studied. We describe ontogenetic changes in morphology, swimming speed and feeding rates and have developed a simple swimming model using low Reynolds number hydrodynamics. In the model we assumed that the ciliary swimming apparatus is primarily composed of the prototroch and secondarily by the telotroch. The model predicted swimming speeds and feeding rates that corresponded well with the measured speeds and rates. Applying empirical data to the model, we were able to explain the profound decrease in specific feeding rates and the observed increase in the difference between upward and downward swimming speeds with larval size. We estimated a critical larval length above which the buoyancy-corrected weight of the larva exceeds the propulsion force generated by the ciliary swimming apparatus and thus forces the larva to the bottom. This modeled critical larval length corresponded to approximately 1 mm, at which, according to the literature, competence for metamorphosis and no more length increase is observed. These findings may have general implications for all planktivorous polychaete larvae that feed without trailing threads. We observed bell shaped particle retention spectra with a minimum prey size of approximately 4 μm equivalent spherical diameter, and we found that an ontogenetic increase in maximum prey size add to a reduction in intra-specific food competition in the various larval stages. In a grazing experiment using natural seawater, ciliates were cleared approximately 50% more efficiently than similar sized dinoflagellates. The prey sizes retainable for P. ciliata larvae covers the microplankton fraction and includes non-motile as well as motile prey items, which is why the larvae are trophically positioned among the copepods and dinoflagellates. Not only do larval morphology and behavior govern larval feeding, prey behavior also influences the feeding efficiency of Polydora ciliata.

An averaging method for nonlinear laminar Ekman layers

We use direct numerical simulations to study a steady bathtub vortex in a cylindrical tank with a central drain-hole, a flat stress-free surface and velocity prescribed at the inlet. We find that the qualitative structure of the meridional flow does not depend on the radial Reynolds number, whereas we observe a weak overall rotation at a low radial Reynolds number and a concentrated vortex above the drain-hole at a high radial Reynolds number. We introduce a simple analytically integrable model that shows the same qualitative dependence on the radial Reynolds number as the simulations and compares favourably with the results for the radial velocity and the azimuthal velocity at the surface. Finally, we describe the height dependence of the radius of the vortex core and the maximum of the azimuthal velocity at a high radial Reynolds number, and we show that the data on the radius of the vortex core and the maximum of the azimuthal velocity as functions of height collapse on single curves by appropriate scaling.