John M. Stockie

Analysis and computation of immersed boundaries, with application to pulp fibres

Immersed fibres are a very useful tool for modeling moving, elastic interfaces that interact with a surrounding fluid. The Immersed Boundary Method is a computational tool based on the immersed fibre model which has been used successfully to study a wide range of applications including blood flow in the heart and arteries and motion of suspended particles. This work centres around a linear analysis of an isolated fibre in two dimensions, which pinpoints a discrete set of solution modes associated solely with the fibre. We investigate the stability and stiffness characteristics of the fibre modes and then relate the results to the severe time step restrictions experienced in explicit and semi-implicit immersed boundary computations. A subset of the modes corresponding to tangential oscillations of the fibre are the main source of stiffness, which intensifies when the fibre force is increased or fluid viscosity is decreased--this explains why computations are limited to unrealistically small Reynolds numbers. We also investigate the effects of smoothing the fibre over a given thickness, which corresponds to the delta function approximation that is central to the discrete scheme. The results can be applied to explore the accuracy of various interpolation methods in an idealised setting. The analysis is next extended to predict time step restrictions and convergence rates for various explicit and semi-implicit discretisations. The results are verified in numerical experiments. Finally, we introduce a novel application of the Immersed Boundary Method to the motion of pulp fibres in a two-dimensional shear flow. The method is shown to reproduce both theoretical results and experimentally observed behaviour over a wide range of parameter values.

Stability analysis for the immersed fiber problem

A linear stability analysis is performed on a two-dimensional version of the “immersed fiber problem,” formulated by C. Peskin to model the flow of fluid in the presence of a mesh of moving, elastic fibers. The purpose of the analysis is to isolate the modes in the solution which are associated with the fiber and thereby determine the effect of the presence of a fiber on the fluid. The results are used not only to make conclusions about the stability of the problem but also to suggest guidelines for developing numerical methods for flows with immersed fibers.

An inverse Gaussian plume approach for estimating atmospheric pollutant emissions from multiple point sources

Abstract A method is developed for estimating the emission rates of contaminants into the atmosphere from multiple point sources using measurements of particulate material deposited at ground level. The approach is based on a Gaussian plume type solution for the advection–diffusion equation with ground-level deposition and given emission sources. This solution to the forward problem is incorporated into an inverse algorithm for estimating the emission rates by means of a linear least squares approach. The results are validated using measured deposition and meteorological data from a large lead–zinc smelting operation in Trail, British Columbia. The algorithm is demonstrated to be robust and capable of generating reasonably accurate estimates of total contaminant emissions over the relatively short distances of interest in this study.

Adiabatic Relaxation of Convective-Diffusive Gas Transport in a Porous Fuel Cell Electrode

The gas diffusion layer in the electrode of a proton exchange membrane fuel cell is a highly porous material which acts to distribute reactant gases uniformly to the active catalyst sites. We develop a mathematical model for flow of a multicomponent mixture of ideal gases in a highly porous electrode. The model is comprised of a porous medium equation for the evolution of the gas mixture and a singularly perturbed convection-diffusion equation for the interspecies mass transfer within the mixture. The equations are coupled through nonlinear boundary conditions which describe consumption of reactants and generation of end products at the catalyst layer. Through a two-time-scale analysis, we derive a single reduced equation which captures the slow, diffusively driven, adiabatic relaxation to the steady state at each electrode. The asymptotic results are compared with one- and two-dimensional computations of the full system.

A moving mesh method with variable mesh relaxation time

We propose a moving mesh adaptive approach for solving time-dependent partial differential equations. The motion of spatial grid points is governed by a moving mesh PDE (MMPDE) in which a mesh relaxation time @t is employed as a regularization parameter. Previously reported results on MMPDEs have invariably employed a constant value of the parameter @t. We extend this standard approach by incorporating a variable relaxation time that is calculated adaptively alongside the solution in order to regularize the mesh appropriately throughout a computation. We focus on singular, parabolic problems involving self-similar blow-up to demonstrate the advantages of using a variable relaxation time over a fixed one in terms of accuracy, stability and efficiency.

Numerical Simulations of Particle Sedimentation Using the Immersed Boundary Method

We study the settling of solid particles in a viscous incompressible fluid contained within a two-dimensional channel, where the mass density of the particles is greater than that of the fluid. The fluid-structure interaction problem is simulated numerically using the immersed boundary method, where the added mass is incorporated using a Boussinesq approximation. Simulations are performed with a single circular particle, and also with two particles in various initial configurations. The terminal particle settling velocity and drag coefficient correspond closely with other theoretical, experimental and numerical results, and the particle trajectories reproduce the expected behavior qualitatively. In particular, simulations of a pair of interacting particles similar drafting-kissing-tumbling dynamics to that observed in other experimental and numerical studies.

Bayesian estimation of airborne fugitive emissions using a Gaussian plume model

Abstract A new method is proposed for estimating the rate of fugitive emissions of particulate matter from multiple time-dependent sources via measurements of deposition and concentration. We cast this source inversion problem within the Bayesian framework, and use a forward model based on a Gaussian plume solution. We present three alternate models for constructing the prior distribution on the emission rates as functions of time. Next, we present an industrial case study in which our framework is applied to estimate the rate of fugitive emissions of lead particulates from a smelter in Trail, British Columbia, Canada. The Bayesian framework not only provides an approximate solution to the inverse problem, but also quantifies the uncertainty in the solution. Using this information we perform an uncertainty propagation study in order to assess the impact of the estimated sources on the area surrounding the industrial site.

An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver

We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures that has the computational complexity of a completely explicit method and also has excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible Navier-Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical simulations of several fluid-structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. We also compare the proposed algorithm with other second-order projection-based fluid solvers. Lastly, the execution time and scaling properties of the proposed algorithm are investigated and compared to alternate approaches.

A Mathematical Model of Sap Exudation in Maple Trees Governed by Ice Melting, Gas Dissolution, and Osmosis

We develop a mathematical model for sap exudation in a maple tree that is based on a purely physical mechanism for internal pressure generation in trees in the leafless state. There has been a long...

Heat and Mass Transfer Modeling of Dry Gases in the Cathode of PEM Fuel Cells

The transport of three gas species, O2, H2O and N2, through the cathode of a proton exchange membrane (PEM) fuel cell is studied numerically. The diffusion of the individual species is modeled via the Maxwell–Stefan equations, coupled with appropriate conservation equations. Two mechanisms are assumed for the internal energy sources in the system: a volumetric heat source due to the electrical current flowing through the cathode; and heat flow towards the cathode at the cathode-membrane interface due to the exothermic chemical reaction at this interface, in which water is generated. The governing equations of the unsteady fluid motion are written in fully conservative form, and consist of the following: (i) three equations for the mass conservation of the species; (ii) the momentum equation for the mixture, which is approximated using Darcys Law for flow in porous media; and (iii) an energy equation, written in a form that has enthalpy as the dependent variable.