Series
EFMWR Seminar Series

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Event Series
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Associated Organization(s)
Associated Organization(s)

Publication Search Results

Now showing 1 - 10 of 10
  • Item
    Invariant Objects in Volume Preserving Maps and Flows
    ( 2014-11-21) de la Llave, Rafael
    We consider smooth volume preserving maps. Two important phenomena are transport and mixing. We present several geometric obstructions that prevent transport and mixing and numerical methods to compute them. These are quasiperiodic orbits of the maps and their treatment requires KAM (Kolmogorov-Arnold-Moser) techniques for an analytic treatment. The result we present has an a-posteriori format (an approximate solution with good condition numbers implies a true solution) and it also leads to very efficient algorithms (low storage requirements and low operation count). These algorithms have been implemented and run (by J. Meiss and A. Fox) and they formulated to conjectures about breakdown. A novelty of the method is that the topology also plays a role. Depending on the global topology the tori may be obstructions to mixing but not to transport or be obstructions to transport and mixing. This is joint work with T. Blass and with A. Fox.
  • Item
    Flow, Mass Transport and Surface Waves in Coastal Water with Vegetation
    ( 2014-11-07) Mei, Chiang C.
    Mangroves along the sea shores are known to provide partial protection against tsunamis. A theory is given for the effects of emergent coastal forests on the propagation of surface waves of small amplitudes. The forest is idealized by a periodic array of vertical cylinders. Simple models are employed to represent interstitial turbulence generated by flow through the cylinders. Multi-scale (homogenization) analysis is carried out to save computational labor. Analytical and numerical solutions for wave attenuation on the macro-scale for different bathymetries are presented. Numerical results are compared with a laboratory experiment motivated by a proposed measure for coastal protection against tsunamis. Emergent vegetation in lakes and coastal waters affects the flow and transport of nutrients ands pollutants. The multiple-scale theory is extended to the nonlinear flow and dispersion in a current through a periodic array of vertical cylinders standing on a horizontal bed. Using the drag coefficient measured for an array in open channels, eddy viscosity in the interstitial flow on the micro-scale is calculated for a wide range of Reynolds numbers. Nonlinear correction of the classic Darcy's law is deduced on the macroscale. The interstitial velocity is used to derive the macro-scale convection diffusion equation for the solute concentration. Taylor dispersivity and the total effective diffusivity tensors are deduced for a wide range of flow rates and solid fractions.
  • Item
    The Search for Exact Coherent Structures in a Quasi-Two-Dimensional Flow
    ( 2014-10-17) Tithof, Jeffrey
    Recent theoretical advances suggest that turbulence can be characterized using unstable solutions of the Navier-Stokes equations having regular temporal behavior, called Exact Coherent Structures (ECS). Due to their experimental accessibility and theoretical tractability, two-dimensional flows provide an ideal setting for the exploration of turbulence from a dynamical systems perspective. In our talk, we present a combined numerical and experimental study of electromagnetically driven flows in a shallow layer of electrolyte. On the numerical front we present our research concerning the search for ECS in a two-dimensional Kolmogorov-like flow. We discuss the change in the dynamics of the flow as the Reynolds number is varied. For a weakly turbulent flow, we show that the turbulent trajectory explores a region of state space which contains a number of ECS. We then discuss the occurrence of states similar to these numerically computed ECS in an experimental quasi-two-dimensional Kolmogorov-like flow.
  • Item
    Noise is your friend, or: How well can we resolve state space?
    ( 2014-09-05) Cvitanović, Predrag
    All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. What is the best resolution possible for a given physical system? It turns out that for nonlinear dynamical systems the noise itself is highly nonlinear, with the effective noise different for different regions of system's state space. The best obtainable resolution thus depends on the observed state, the interplay of local stretching/contraction with the smearing due to noise, as well as the memory of its previous states. We show how that is computed, orbit by orbit. But noise also associates to each orbit a finite state space volume, thus helping us by both smoothing out what is deterministically a fractal strange attractor, and restricting the computation to a set of unstable periodic orbits of finite period. By computing the local eigenfunctions of the Fokker-Planck evolution operator, forward operator along stable linearized directions and the adjoint operator along the unstable directions, we determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive noise. The space of all chaotic spatiotemporal states is infinite, but noise kindly coarse-grains it into a finite set of resolvable states.