Silas Alben

Silas Alben

Assistant Professor

School of Mathematics
Georgia Institute of Technology
686 Cherry St.
Atlanta, GA 30332-0160

Email: last name at math dot gatech dot edu
Office: Skiles 238, tel. 404-894-3312
Lab: Skiles 238A

Current course, Fall 2008
Math 6643: Numerical Linear Algebra

Spring 2008: Math 6646: Numerical Ordinary Differential Equations
Fall 2007: Math 2406: Abstract Vector Spaces

Research:

My recent research can be summarized as problems arising in biology (especially biomechanics) and materials science 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.

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.

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, to appear (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 ver-
sion 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. In both cases a linear instability to horizontal motion sets in above a critical Reynolds num-
ber, leading to a stable oscillatory state. The pressure forces play a destabilizing role against the stabilizing viscous forces,
with oscillatory time scales set by either external flapping or the intrinsic flow–body coupling. The latter lowers the insta-
bility threshold in Reynolds number.

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. We find that stall delay is insensitive to the wavelength of the bumps, in
accordance with experimental observations.

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.
Journal of the Royal Society Interface, published online 11/29/06; S. Alben, P.G. Madden and G.V. Lauder.

Ray-finned fishes are a group of over 28,000 species, comprising more than half of all vertebrates, that have
diversified into a wide variety of aquatic habitats and are known for their diversity of locomotory styles. One
of the key characteristics of ray-finned fishes is the presence of fins that extend into the water and act as control surfaces during locomotion.

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.

(a), Photograph of a bluegill sunfish. (b), Schematic showing the location of the fish fins. The pectoral fin is highlighted with the blue box. (c), Photograph of a cleared and stained pectoral fin. The short segments can be seen along the length of the rays. (d), Schematic showing dorsal views of a fin ray with two hemitrichs. Muscles (not shown) exert forces on the tendons attached to the head of each hemitrich.

(a), A comparison between the model (dashed line) and the experimental trajectory (photograph), when a point force is applied (aluminium triangle in the photograph) at 2/3 the distance from the base to the tip of the ray. (b), Experimental measurements of the force versus base shift, for a point force which holds a point 2/3 along the ray fixed as in panel a. (c), The data in (b) replotted on a log-log scale (green triangles), together with the corresponding data for the two models: a uniform-shear-modulus material (blue
circles) showing a linear growth of force with shift, and collagen springs (red crosses) showing a cubic growth of force with shift.

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

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

Representative cases for the assembly of flat sheets into curved surfaces. a, Discretization of the shape used by Boncheva et al. for the "flower" cut. b, A case where a random initial perturbation leads to the same sign of curvature at all cusps of the flower cut after buckling. c, The assembled flower cut. d, An initial buckling of the flower cut when five of the six cusps buckle upward (inset, black circles), and one buckles downward (inset, green circle). Such wrong-signed initial curvature persists in the final state of the sphere. e, Simulation of the final equilibrium of the flower shape with alternating positive and negative charges at the tips (with a discretization which is coarser than that in 'a'); misfolding occurs due to an instability among multiple interacting charges. f, Discretization of the shape used by Boncheva et al. for the "orange peel" cut. g, An intermediate stage in the assembly of the orange peel cut, in which the initial buckling has the same sign at both cusps. h, The final assembled orange peel cut, showing a concentration of stretching strain near the boundary. The stretching strain decays rapidly from a maximum of 20% at the boundary to less than 2% within 90% of the sheet. i, A simulation of the orange peel shape, where the charges are sufficiently strong to join the edges before the shape equilibrates elastically from previous zipping steps. The zipping deviates sufficiently from the case in panel `g' that incorrect curvature occurs. j, Discretization of the shape used by Boncheva et al. for the "equator cut." k, A simulation of the equator cut shape during zipping, showing an example of the generic phenomenon in which neighboring cusps have opposite-signed curvature, thus preventing the successful zipping of the equator cut into a sphere..

A diagram showing how dislocations arise in the joining of curved edges. When two discretized edges in 'a' are joined, a disclination appears, leading to the deformed mesh shown in panel `b.' In the continuuum limit, the disclination density equals the sum of the curvatures of the joined points, shown in panel `c.'

Ongoing work: fluid-body interactions

The coupled motion of fluids and solids in contact arises frequently in the biological world, and leads to interesting and unexpected phenomena in theoretical mechanics. These phenomena can often be understood by identifying new length and time scales intrinsic to the fully-coupled system

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 a recent experiment in the Courant Institute Wetlab. 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

The body and fluid dynamics (shown as a contour plot of the vorticity field) during the stages of takeoff for a flapping body in a sample case. For the first upstroke (0 < t < 0.5), a positive horizontal velocity perturbation is prescribed, increasing to 0.2% of the peak vertical velocity; subsequently the horizontal velocity is set by horizontal fluid forces. (A) Startup motion and flow are nearly vertically symmetric, so horizontal forces on the body are small. (B) Nearly four periods after the body is freed, the flow has become very asymmetric, while the body velocity is still nearly vertical because of body inertia. (C) The body collides with a previously shed vortex at t = 6.2, which decreases suction at the right edge by creating a vortex dipole. (D) The horizontal component of the pressure (blue) and viscous (red) stresses acting on the surface of the body, at the instant shown in C. The black line is the sum of the stresses. The arc length around the body surface (moving counterclockwise) is denoted by s, with the right edge at s = 0 and left edge at s = 2.14. The peak near s = 0 has a much smaller magnitude than the trough at s = 2.14, coincident with the formation of a vortex dipole at s = 0. (E) The horizontal pressure (blue line) and viscous (red line) forces acting on the body as a function of time. Dashed lines mark the instants of B, C, and F. Also shown is horizontal velocity of the body, u_b(t) (black line), which shows quasi-steady locomotion for t = 8. The green line shows the negative (for separation from other graphs) of cycle-averaged input power P(t). (F) The subsequent vortex street and center-of-mass trajectory (dotted line) in the quasi-steady state of unidirectional (leftward) flight. Shed vortices form a staggered array with net fluid momentum directed away from the body.

Figure 1:
(Download .avi file)
Quicktime (.mov) format--lower resolution

Figure 3a:
(Download .avi file)
(Quicktime (.mov) format--lower resolution)

The trajectory of a body with the same mass as above (body density is 65 times fluid density),

with the same start-up condition and Re_fr , but which is more slender (aspect ratio 10:1). We find
that thinner bodies move more directly and smoothly into steady locomotion. In this particular case
the wake consists of paired vortices at an oblique angle.

Figure 3c:
(Download .avi file)
(Quicktime (.mov) format--lower resolution )

The trajectory of a body with the same aspect ratio as in Fig. 1, but lighter (body density = fluid density), at the same Re_fr and with the same start-up condition. The motion of this light body is highly sensitive to the instantaneous fluid forces on it, and thus it interacts strongly with the vortices that it has shed. The lack of substantial inertia also means that the body cannot sustain a velocity against a period of drag, and the body-fluid system finds it difficult to access a locomoting state.

Example of Stable Nonlocomoting State
(Download .avi file)
(Quicktime (.mov) format--lower resolution )

The dynamics of a neutrally-buoyant body subject to the same start-up condition as in the above cases. The aspect ratio is 3.3:1, and Re_fr = 7.3, which is less than the transition Re_fr = 8 at which the body becomes unstable.

Example of Periodic Nonlocomoting State
(Download .avi file)
(Quicktime (.mov) format--lower resolution )

The dynamics of a neutrally-buoyant body subject to the same start-up condition as in the above cases. The aspect ratio is 3.3:1, and Re_fr = 11, which is above the transition Re_fr = 8 at which the body becomes unstable.
However, the ratio of body density to fluid density and the value of Re_frare not sufficient for forward locomotion.

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. A recent 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.