Chaotic dynamics of two-dimensional flows around a cylinder

We study flow around a cylinder from a dynamics perspective, using drag and lift as indicators. We observe that the mean drag coefficient bifurcates from the steady case when the Karman vortex street emerges. We also find a jump in the dimension of the drag/lift attractor just above Reynolds number 100. We compare the simulated drag values with experimental data obtained over the last hundred years. Our simulations suggest that a vibrational resonance in the cylinder would be unlikely for Reynolds numbers greater than 1000, where the drag/lift behavior is fully chaotic.Comment: 27 pages, including appendi

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as $\alpha \rightarrow 0$ (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as $\alpha \rightarrow 0$ (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero