Conditional moment closure for chemical reactions in laminar chaotic flows

We implement conditional moment closure (CMC) for simulation of chemical reactions in laminar chaotic flows. The CMC approach predicts the expected concentration of reactive species, conditional upon the concentration of a corresponding nonreactive scalar. Closure is obtained by neglecting the difference between the local concentration of the reactive scalar and its conditional average. We first use a Monte Carlo method to calculate the evolution of the moments of a conserved scalar; we then reconstruct the corresponding probability density function and dissipation rate. Finally, the concentrations of the reactive scalars are determined. The results are compared (and show excellent agreement) with full numerical simulations of the reaction processes in a chaotic laminar flow. This is a preprint of an article published in AlChE Journal copyright (2007) American Institute of Chemical Engineers: http://www3.interscience.wiley.com/

Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Error estimation and adaptive mesh refinement for aerodynamic flows

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated.

High-order hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows

This article is concerned with the construction of general isotropic and anisotropic adaptive strategies, as well as hp-mesh refinement techniques, in combination with dual-weighted-residual a posteriori error indicators for the discontinuous Galerkin finite element discretization of compressible fluid flow problems.

Reduced models of chemical reaction in chaotic flows

We describe and evaluate two reduced models for nonlinear chemical reactions in a chaotic laminar flow. Each model involves two separate steps to compute the chemical composition at a given location and time. The “manifold tracking model” first tracks backwards in time a segment of the stable manifold of the requisite point. This then provides a sample of the initial conditions appropriate for the second step, which requires solving one-dimensional problems for the reaction in Lagrangian coordinates. By contrast, the first step of the “branching trajectories model” simulates both the advection and diffusion of fluid particles that terminate at the appropriate point; the chemical reaction equations are then solved along each of the branched trajectories in a second step. Results from each model are compared with full numerical simulations of the reaction processes in a chaotic laminar flow.