Silas Alben

Drag reduction through self-similar bending of a flexible body

The classical theory of high-speed flow predicts that a moving rigid object experiences a drag proportional to the square of its speed. However, this reasoning does not apply if the object in the flow is flexible, because its shape then becomes a function of its speed—for example, the rolling up of broad tree leaves in a stiff wind. The reconfiguration of bodies by fluid forces is common in nature, and can result in a substantial drag reduction that is beneficial for many organisms. Experimental studies of such flow–structure interactions generally lack a theoretical interpretation that unifies the body and flow mechanics. Here we use a flexible fibre immersed in a flowing soap film to measure the drag reduction that arises from bending of the fibre by the flow. Using a model that couples hydrodynamics to bending, we predict a reduced drag growth compared to the classical theory. The fibre undergoes a bending transition, producing shapes that are self-similar; for such configurations, the drag scales with the length of self-similarity, rather than the fibre profile width. These predictions are supported by our experimental data.

Foldable structures and the natural design of pollen grains

Upon release from the anther, pollen grains of angiosperm flowers are exposed to a dry environment and dehydrate. To survive this process, pollen grains possess a variety of physiological and structural adaptations. Perhaps the most striking of these adaptations is the ability of the pollen wall to fold onto itself to prevent further desiccation. Roger P. Wodehouse coined the term harmomegathy for this folding process in recognition of the critical role it plays in the survival of the pollen grain. There is still, however, no quantitative theory that explains how the structure of the pollen wall contributes to harmomegathy. Here we demonstrate that simple geometrical and mechanical principles explain how wall structure guides pollen grains toward distinct folding pathways. We found that the presence of axially elongated apertures of high compliance is critical for achieving a predictable and reversible folding pattern. Moreover, the intricate sculpturing of the wall assists pollen closure by preventing mirror buckling of the surface. These results constitute quantitative structure-function relationships for pollen harmomegathy and provide a framework to elucidate the functional significance of the very diverse pollen morphologies observed in angiosperms.

The mechanics of active fin-shape control in ray-finned fishes

We study the mechanical properties of fin rays, which are a fundamental component of fish fin structure. We derive a linear elasticity model that predicts the shape of fin rays, given the input muscle actuation and external loading. We then test the model using experiments that measure (i) the ray deflection for a given actuation at the muscular interface, and (ii) the force–displacement response under conditions of actuation and non-actuation. The model agrees well with the experiment; both show a concentration of curvature at the ray base or at the point of an externally applied force, and a variation in ray stiffness over more than an order of magnitude depending on actuation at the bases of the fin rays.

Dynamics of freely swimming flexible foils

We use modeling and simulations guided by initial experiments to study thin foils which are oscillated at the leading edge and are free to move unidirectionally under the resulting fluid forces. We find resonant-like peaks in the swimming speed as a function of foil length and rigidity. We find good agreement between the inviscid model and the experiment in the foil motions (particularly the wavelengths of their shapes), the dependences of their swimming speeds on foil length and rigidity, and the corresponding flows. The model predicts that the foil speed is proportional to foil length to the −1/3 power and foil rigidity to the 2/15 power. These scalings give a good collapse of the experimental data.

Optimal flexibility of a flapping appendage in an inviscid fluid

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50 % (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ∼k −5 , corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderateto-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

How flexibility induces streamlining in a two-dimensional flow

Recent work in bio-fluid dynamics has studied the relation of fluid drag to flow speed for flexible organic structures, such as tree leaves, seaweed, and coral beds, and found a reduction in drag growth due to body reconfiguration with increasing flow speed. Our theoretical and experimental work isolates the role of elastic bending in this process. Using a flexible glass fiber wetted into a vertical soap-film flow, we identify a transition in flow speed beyond which fluid forces dominate the elastic response, and yield large deformations of the fiber that greatly reduce drag. We construct free-streamline models that couple fluid and elastic forces and solve them in an efficient numerical scheme. Shape self-similarity emerges, with a scaling set by the balance of forces in a small “tip region” about the flow’s stagnation point. The result is a transition from the classical U2 drag scaling of rigid bodies to a new U4/3 drag law. We derive an asymptotic expansion for the fiber shape and flow, based on the le...

Simulating the dynamics of flexible bodies and vortex sheets

We present a numerical method for the dynamics of a flexible body in an inviscid flow with a free vortex sheet. The formulation is implicit with respect to body variables and explicit with respect to the free vortex sheet. We apply the method to a flexible foil driven periodically in a steady stream. We give numerical evidence that the method is stable and accurate for a relatively small computational cost. A continuous form of the vortex sheet regularization permits continuity of the flow across the bodys trailing edge. Nonlinear behavior arises gradually with respect to driving amplitude, and is attributed to the rolling-up of the vortex sheet. Flow quantities move across the body in traveling waves, and show large gradients at the body edges. We find that in the small-amplitude regime, the phase difference between heaving and pitching which maximizes trailing edge deflection also maximizes power output; the phase difference which minimizes trailing edge deflection maximizes efficiency.

Wake-mediated synchronization and drafting in coupled flags

A recent experiment has shown ‘inverted drafting’ in flags: the drag force on one flag is increased by excitation from the wake of another. Here we use vortex sheet simulations to show that inverted drafting occurs when the flag wakes add coherently to form strong vortices. By contrast, normal drafting occurs for higher frequency oscillations, when the vortex wake becomes more complex and mixed on the scale of the flag. The types of drafting and dynamics (synchronization and erratic flapping) depend on the separation distance between the flags. For both tandem and side-by-side flags in synchronized flapping, the phase difference depends nearly monotonically on separation distance.

Passive Robotic Models of Propulsion by the Bodies and Caudal Fins of Fish

Considerable progress in understanding the dynamics of fish locomotion has been made through studies of live fishes and by analyzing locomotor kinematics, muscle activity, and fluid dynamics. Studies of live fishes are limited, however, in their ability to control for parameters such as length, flexural stiffness, and kinematics. Keeping one of these factors constant while altering others in a repeatable manner is typically not possible, and it is difficult to make critical measurements such as locomotor forces and torques on live, freely-swimming fishes. In this article, we discuss the use of simple robotic models of flexing fish bodies during self-propulsion. Flexible plastic foils were actuated at the leading edge in a heave and/or pitch motion using a robotic flapping controller that allowed moving foils to swim at their self-propelled speed. We report unexpected non-linear effects of changing the length and stiffness of the foil, and analyze the effect of changing the shape of the trailing edge on self-propelled swimming speed and kinematics. We also quantify the structure of the wake behind swimming foils with volumetric particle image velocimetry, and describe the effect of flexible heterocercal and homocercal tail shapes on flow patterns in the wake. One key advantage of the considerable degree of control afforded by robotic devices and the use of simplified geometries is the facilitation of mathematical analyses and computational models, as illustrated by the application of an inviscid computational model to propulsion by a flapping foil. This model, coupled with experimental data, demonstrates an interesting resonance phenomenon in which swimming speed varies with foil length in an oscillatory manner. Small changes in length can have dramatic effects on swimming speed, and this relationship changes with flexural stiffness of the swimming foil.

Passive and active bodies in vortex-street wakes

We model the swimming of a finite body in a vortex street using vortex sheets distributed along the body and in a wake emanating from its trailing edge. We determine the magnitudes and distributions of vorticity and pressure loading on the body as functions of the strengths and spacings of the vortices. We then consider the motion of a flexible body clamped at its leading edge in the vortex street as a model for a flag in a vortex street and find alternating bands of thrust and drag for varying wavenumber. We consider a flexible body driven at its leading edge as a model for tail-fin swimming and determine optimal motions with respect to the phase between the bodys trailing edge and the vortex street. For short bodies maximizing thrust or efficiency, we find maximum deflections shifted in phase by 90° from oncoming vortices. For long bodies, leading-edge driving should reach maximum amplitude when the vortices are phase-shifted from the trailing edge by 45° (to maximize thrust) and by 135° (to maximize efficiency). Optimal phases for intermediate lengths show smooth transitions between these values. The optimal motion of a body driven along its entire length is similar to that of the model tail fin driven only at its leading edge, but with an additional outward curvature near the leading edge. The similarity between optimal motions forced at the leading edge and all along the body supports the high performance attributed to fin-based motions.