Simon Tavener

Bifurcation of Low Reynolds Number Flows in Symmetric Channels

The flowfields in two-dimensional channels with discontinuous expansions are studied numerically to understand how the channel expansion ratio influences the symmetric and nonsymmetric solutions that are known to occur. For improved confidence and understanding, two distinct numerical techniques are used. The general flowfield characteristics in both symmetric and asymmetric regimes are ascertained by a time-marching finite difference procedure. The flowfields and the bifurcation structure of the steady solutions of the Navier-Stokes equations are determined independently using the finite element technique. The two procedures are then compared both as to their predicted critical Reynolds numbers and the resulting flowfield characteristics. Following this, both numerical procedures are compared with experiments.

Bifurcation for flow past a cylinder between parallel planes

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented. A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem. A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion. Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

DYNAMICS OF PRION DISEASE TRANSMISSION IN MULE DEER

Chronic wasting disease (CWD), a contagious prion disease of the deer family, has the potential to severely harm deer populations and disrupt ecosystems where deer occur in abundance. Consequently, understanding the dynamics of this emerging infectious disease, and particularly the dynamics of its transmission, has emerged as an important challenge for contemporary ecologists and wildlife managers. Although CWD is contagious among deer, the relative importance of pathways for its transmission remains unclear. We developed seven competing models, and then used data from two CWD outbreaks in captive mule deer and model selection to compare them. We found that models portraying indirect transmission through the environment had 3.8 times more support in the data than models representing transmission by direct contact between infected and susceptible deer. Model-averaged estimates of the basic reproductive number (R0) were 1.3 or greater, indicating likely local persistence of CWD in natural populations under conditions resembling those we studied. Our findings demonstrate the apparent importance of indirect, environmental transmission in CWD and the challenges this presents for controlling the disease.

The numerical analysis of bifurcation problems with application to fluid mechanics

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

An A Posteriori-A Priori Analysis of Multiscale Operator Splitting

In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction-diffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori-a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting.

A Posteriori Analysis and Adaptive Error Control for Multiscale Operator Decomposition Solution of Elliptic Systems I: Triangular Systems

this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, residuals, and the generalized Greens function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.

Simulation of boundary condition influence in a second-order ferroelectric phase transition

Using a two-dimensional Ginzburg–Landau model and the finite-element computational method, we have calculated stable domain configurations resulting from a second-order ferroelectric phase transition for a finite-sized system. The boundary conditions applied here correspond to fully charge compensated situations, either by surface electrodes or by the injection of charges (or defects) near the sample surface. The domain wall thickness of a finite system without surface electrodes was found to become thinner as it approaches sample surfaces. This is distinctively different from that of an infinite system for which a planar wall assumption can be used. The orientation of the macroscopic polarization of a finite system without surface electrodes was found to be determined by its aspect ratio. A size effect was observed when all the dimensions were reduced simultaneously. The relaxation process in the formation of domains and the switching process have also been simulated for charge neutral boundary condition...

On the creation of stagnation points near straight and sloped walls

We report the results of an experimental and numerical study of the creation of stagnation points in a rotating cylinder of fluid where one endwall is rotated. Good agreement is found with previous results where the stagnation points are formed on the free core of the primary columnar vortex. The effect of adding a small cylinder along the center of the flow is then investigated and the phenomena are found to be robust despite the qualitative change in the boundary conditions. Finally, we show that sloping the inner cylinder has a dramatic effect on the recirculation such that it can either be intensified or suppressed.

A Posteriori Analysis and Improved Accuracy for an Operator Decomposition Solution of a Conjugate Heat Transfer Problem

We consider the accuracy of an operator decomposition finite element method for a conjugate heat transfer problem consisting of two materials coupled through a common boundary. We derive accurate a posteriori error estimates that account for the transfer of error between components of the operator decomposition method as well as the differences between the adjoints of the full problem and the discrete iterative system. We use these estimates to guide adaptive mesh refinement. In addition, we address a loss of order of convergence that results from the decomposition and show that the order of convergence is limited by the accuracy of the transferred gradient information. We employ a boundary flux recovery method to regain the expected order of accuracy in an efficient manner.

Numerical and Asymptotic Methods for Certain Viscous Free-Surface Flows

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.