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Silas AlbenAssistant Professor School
of
Mathematics Office: Skiles 238, tel. 404-894-3312 Lab: Skiles 238A |
Current course, Spring 2008
Math 6646: Numerical Ordinary Differential Equations
Fall 2007: Math 2406: Abstract Vector Spaces
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Research:
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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. |
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.
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(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. |
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(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 |
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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.. |
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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.' |
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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. |
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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. |
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Figure 1: (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 Refr , 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 Refr 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 Refr = 7.3, which is less than the transition Refr = 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 Refr = 11, which is above the transition Refr = 8 at which the body becomes unstable. However, the ratio of body density to fluid density and the value of Refrare 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 |
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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. |