Diffusion Effects on Slow Invariant Manifolds

Diffusion effects on the Slow Invariant Manifold (SIM) of a closed reactive system are examined with the goal of developing a model reduction technique which rigorously accounts for the coupling of reaction and diffusion processes. A robust method of constructing a one-dimensional SIM by calculating equilibria and then integrating heteroclinic orbits is extended to systems with diffusion across a small characteristic length. First, a spatially homogeneous system of NO production is discussed. Diffusion is then added as a correction to the spatially homogeneous system using a Galerkin method to project the infinite dimensional dynamical system onto a low dimensional approximate inertial manifold. A critical length is identified, above which a perturbed SIM is found by similar techniques of calculating equilibria and integrating heteroclinic orbits. Introduction Reactive flow problems are known to display multiscale phenomena that cause challenges in the numerical simulations of such problems. Verification of these simulations requires grid resolution that captures the full range of scales in both space and time. Large disparity in scales induces simulations that require significant computational effort. A disparity in temporal scales can be caused by the reaction mechanism alone, while the addition of diffusion couples the differing reaction time scales to a disparity in length scales. Recently, considerable effort has been expended in identification of model reduction techniques for reactive flows in order to reduce the computational cost, while maintaining as much consistency with the underlying reactive flow physics as possible. The reviews of Griffiths [1] or Lu and Law [2] are good references for these techniques in general. Most of the methods described therein address only reaction mechanisms. Some current research that extends these methods to systems with diffusion are Singh, et al. [3], Ren and Pope [4], Davis [5, 6], Bykov and Maas [7], Lam [8], Adrover, et al. [9], and Goussis, et al. [10] The study of Davis and Skodje [11] is particularly relevant. In their study, which was performed on spatially homogeneous reactive systems, the authors calculate a one-dimensional Slow Invariant Manifold (SIM) of the system by integrating a heteroclinic orbit between the system’s physical and non-physical equilibrium points. This technique has recently been refined by Al-Khateeb et al. [12] to examine larger systems. The SIM is a unique trajectory of the dynamical system that describes the long time dynamics Corresponding author: [email protected] Proceedings of the 2010 Technical Meeting of the Central States Section of the Combustion Institute of the system’s evolution efficiently, which makes the SIM one of the premiere reduction techniques available; however, the SIM has almost exclusively been used for spatially homogenous systems. Specific Objectives The objective of this paper is to examine how the addition of diffusion to a reactive system affects the SIM. Here, we focus on short length scales; in the limit of an infinitesimal length, diffusion will have a negligible effect, and the system will remain spatially homogeneous. We aim to identify a critical length scale at which diffusion has a significant effect on the spatially homogeneous system. We find that at this critical length a bifurcation occurs in the SIM. This critical length can serve as a rough division between what Goussis et al. [10] refer to as ‘local’ or ‘global’ analysis. The objective of the current paper will be to focus on the global analysis, where the time scales from diffusion modification are faster than the reaction time scales. Methodology By the addition of a linear Fick’s law diffusion model to a spatially homogeneous reaction mechanism, the governing equations change from ordinary differential equations (ODEs),