Silas Alben Assistant Professor School of Mathematics Georgia Institute of Technology 686 Cherry St. Atlanta, GA  30332-0160 Email: last name at math.gatech.edu Office: Skiles 238, tel. 404-894-3312 Lab: Skiles 238A

Teaching: Fall 2009: Math 2403: Differential Equations                   Math/CSE 6643: Numerical Linear Algebra Spring 2009: Math 6644: Iterative Methods for Systems of Equations Fall 2008: Math 6643: Numerical Linear Algebra Spring 2008: Math 6646: Numerical ODEs Fall 2007: Math 2406: Abstract Vector Spaces Research: My research addresses problems from biology (especially biomechanics) and engineering which can be studied with the tools of applied mathematics and continuum mechanics. My work consists of modeling, theoretical analysis and development of numerical methods, with the general goal of obtaining new physical insight into these problems.

Wake-mediated synchronization and drafting in coupled flags
S. Alben, Journal of Fluid Mechanics (2009).
Movie of tandem flags, synchronized ; Movie of tandem flags, unsynchronized ; Movie of side-by-side flags

Recent experiments have 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.

Passive and active bodies in vortex-street wakes
S. Alben, Journal of Fluid Mechanics (2009).

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 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 wave number. 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 body’s trailing edge and the vortex street.

On the swimming of a flexible body in a vortex street
S. Alben, Journal of Fluid Mechanics, 635, 27-45 (2009).

We formulate a new theoretical model for the swimming of a flexible body in a vortex street. We consider the class of periodic travelling-wave body motions, in the limit of small amplitude. We determine the body wave which provides maximum output power for fixed amplitude and the body wave which maximizes efficiency for a given output power. We compare our results with previous experiments and simulations and give physical interpretations for agreements and disagreements in terms of the phase between the body wave and vortex street.

Collapse and folding of pressurized rings in two dimensions
E. Katifori, S. Alben, D.R. Nelson, Physical Review E, 79, 056604 (2009).

Hydrostatically pressurized circular rings confined to two dimensions (or cylinders constrained to have only z-independent deformations) undergo Euler-type buckling when the outside pressure exceeds a critical value. We perform a stability analysis of rings with arclength-dependent bending moduli and determine how weakened bending modulus segments affect the buckling critical pressure. Rings with a fourfold symmetric modulation are particularly susceptible to collapse. In addition we study the initial postbuckling stages of the pressurized rings to determine possible ring folding patterns.

Simulating the dynamics of flexible bodies and vortex sheets
S. Alben, Journal of Computational Physics, 228, 2587–2603 (2009) .

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 body’s 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.

A cascade of length scales in elastic rings under confinement
K. Spears and S. Alben, Chaos, 18, 041109 (2008).

Elastic objects under confinement are common in mechanics and biology. Examples include mitochondria and chromosomes, for which conformation and function are strongly determined by confining forces. When elastic objects grow in a confined space, minimization of elastic energy creates a complex spatial configuration and force network. To simulate a two-dimensional ring growing within a rigid circular boundary with a fixed radius, we take a long strip of elastic material (mylar) of fixed length, join the ends to form a closed loop, and then shrink the confining ring boundary. We find a distribution of curvatures which is inversely proportional to the local number of overlapping layers. We give a simple quantitative argument for this relationship.

Packings of a charged line on a sphere
S. Alben, Physical Review E, 78, 066603 (2008).

We find equilibrium configurations of open and closed lines of charge on a sphere, and track them with respect to varying sphere radius. Closed lines transition from a circle to a spiral-like shape through two low-wave-number bifurcations—“baseball seam” and “twist”—which minimize Coulomb energy. The spiral shape is the unique stable equilibrium of the closed line. Other unstable equilibria arise through tip-splitting events. An open line transitions smoothly from an arc of a great circle to a spiral as the sphere radius decreases. Under repulsive potentials with faster-than-Coulomb power-law decay, the spiral is tighter in initial stages of sphere shrinkage, but at later stages of shrinkage the equilibria for all repulsive potentials converge on a spiral with uniform spacing between turns. Multiple stable equilibria of the open line are observed.

 
Optimal flexibility of a flapping appendage in an inviscid fluid
S. Alben, Journal of Fluid Mechanics, 614, 355 - 380 (2008).

We study propulsive forces generated by a flexible body with a vortex-sheet wake pitched periodically at the leading edge. We find that the thrust power generated by the body has a series of damped 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.  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 a power-law distribution of resonant rigidities. We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

The flapping-flag instability as a nonlinear eigenvalue problem
S. Alben, Physics of Fluids, 20, 104106 (2008).

We reconsider the classical problem of the instability of a flapping flag in an inviscid background flow with a vortex sheet wake, and reformulate it as a nonlinear eigenvalue problem. We solve the problem numerically for the 20 lowest wave number modes in the parameter space of flag mass and flag rigidity. We study the connection between the modes and the growth rates in the eigenvalue and initial value problems. Using an infinite flag model we compute the parameters of the most unstable flag and show that a classical mechanism for the instability correlating pressure lows to flag amplitude peaks does not hold.

Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos
S. Alben and M.J. Shelley, Physical Review Letters, 100, 074301 (2008).

We investigate the "flapping flag" instability through a model for an inextensible flexible sheet in an inviscid 2D flow with a free vortex sheet. We solve the fully-nonlinear dynamics numerically and find a transition from a power spectrum dominated by discrete frequencies to an apparently continuous spectrum of frequencies. We compute the linear stability domain which agrees with previous approximate models in scaling but differs by large multiplicative factors. We also find hysteresis, in agreement with previous experiments. Erratum: correction of parameters and Fig. 2 Movies of Flapping Flags (.avi files): First Periodic State                   Second Periodic State             Third Periodic State Chaotic State

An implicit method for coupled flow-body dynamics
S. Alben, Journal of Computational Physics, 227, 4912--4933 (2008).

We propose an efficient method for computing coupled flow-body dynamics. The time-stepping is implicit, and uses an iterative method (preconditioned GMRES) to solve the flow-body equations. The preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices, and requires a small number of iterations per time step. We use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions: flapping and rising.

How bumps on whale flippers delay stall: an aerodynamic model
E.A. van Nierop, S. Alben, and M.P. Brenner, Physical Review Letters, 100, 054502 (2008).

Wind tunnel experiments have shown that bumps on the leading edge of model humpback whale flippers cause them to "stall" (i.e., lose lift dramatically) more gradually and at a higher angle of attack. Here we develop an aerodynamic model which explains the observed increase in stall angle. The model predicts that as the amplitude of the bumps is increased, the lift curve flattens out, leading to potentially desirable control properties. See also: "Whale-Inspired Windmills," MIT Technology Review Mar. 6, 2008 "Fluid dynamics: Lifting a whale," Nature, Research Highlights Feb. 21, 2008

The mechanics of active fin-shape control in ray-finned fishes.
S. Alben, P.G. Madden and G.V. Lauder, Journal of the Royal Society Interface, 4, 243--256 (2007).

We have studied the mechanical properties of fin rays, which are a fundamental component of fish fin structure. We have derived a linear elasticity model which predicts the shape of fin rays given the input muscle actuation and external loading. The model agrees well with experiments: 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.

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The self-assembly of flat sheets into closed surfaces
S. Alben and M.P. Brenner, Physical Review E, 75, 056113 (2007).

A recent experiment (Boncheva et al. PNAS 102, 3924-3929 (2005)) introduced the possibility of initiating the self-assembly of a 3D structure from a flat elastic sheet. The ultimate utility of this method for assembly depends on whether it leads to incorrect, metastable structures. Here we examine how the number of metastable states depends on the sheet shape and thickness. Using simulations and theory we have identified out-of-plane buckling as the key event leading to metastability. The buckling strain that arises from joining edges of a planar sheet can be estimated using the theory of dislocations in elastic media. The number of metastable states increases rapidly with increasing variability in the boundary curvature and decreasing sheet thickness. See also: Self-assembly could simplify nanotech construction, New Scientist, June 7, 2007

Coherent locomotion as an attracting state for a free flapping body
Proceedings of the National Academy of Sciences of the U.S.A., 2005, 102 (32), 11163-11166;  S. Alben and M.J. Shelley.

We study numerically a fluid flow problem at the transition between low- and high-Reynolds-number locomotion, motivated by an experiment at the Courant Institute Applied Math Lab. In our study, a 2-D rigid body is flapped in the vertical direction and is free to move horizontally. Above a critical flapping frequency, the wing becomes unstable to horizontal motion. For certain ranges of wing shape and mass, this instability saturates to unidirectional flapping flight. We have found that the typical event which triggers "take-off" is a fortuitous collision of the body with vortices shed on previous flapping strokes. Dynamics of a free flapping body Take-off (.mov)    Take-off (.avi) Back-and-forth (.mov)     Back-and-forth (.avi) Chaotic (.mov)    Chaotic (.avi) Thin body (.mov)     Thin body (.avi)

How flexibility induces streamlining in a two-dimensional flow Physics of Fluids 16 (5): 1694-1713 (2004); S. Alben, M. Shelley, and J. Zhang Drag reduction through self-similar bending of a flexible body Nature 420, 479-481 (2002); S. Alben, M. Shelley, and J. Zhang See also: Nature's Secret to Building for Strength: Flexibility, New York Times, Dec. 17, 2002

Nature abounds with organisms utilizing body flexibility in order to survive in flowing fluids.  An experiment in the Applied Mathematics Lab at Courant studied aspects of this using a length of fiber optic glass -- a flexible body -- immersed in the the quasi- two-dimensional flow of a running soap film.  As the flow speed increases the shape of the flexible body bends and becomes more and more streamlined -- the two left panels -- and consequently the fluid drag on the body grows much more slowly than if it were rigid.  The rightmost figure shows the numerical solution of our model of a flexible body deformed by an surrounding  flow and wake.  This theory shows an emerging self-similarity in shape arising from a balance of fluid and elastic forces at the tip.  This self-similarity yields a new, reduced drag law where drags grows as the 4/3 power, rather than the square, of the flow velocity.

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