Rational reduction of reactive flow models and efficient computation of their solutions

This study will focus on the development of methods for rational reduction of model equations of reactive systems and their efficient numerical simulations. The method of Intrinsic Low-Dimensional Manifold (ILDM), which is used to obtain reduced model equations for spatially homogeneous reactive systems modeled by a system of stiff ordinary differential equation (ODEs), is described in detail. A less stiff reduced system of ODEs is obtained using the ILDM method by equilibrating the fast time scale chemical processes and resolving only the slow time scale chemical processes. The accuracy of the standard ILDM approximation is clarified, and it is shown that the ILDM is a good approximation of the Slow Invariant Manifold (SIM) for stiff ODEs and small manifold curvature. Efficient construction of multi-dimensional ILDMs in a polar parametric space is also presented. Subsequently, an operator splitting method is used to extend the use of the ILDM method to spatially inhomogeneous reactive systems or reactive flow systems modeled by a system of partial differential equations (PDEs). This procedure is implemented on a one-dimensional viscous detonation problem for a mixture of hydrogen-oxygen-argon in a shocktube. The operator splitting method allows each spatial point to be treated as a spatially homogeneous premixed reactor in the reaction step, so that the ILDM method can be implemented. The Wavelet Adaptive Multilevel Representation (WAMR) is used in the convection diffusion step for efficient resolution of the fine spatial structures in the flow. In this problem, the ILDM and WAMR methods, together, allow for a numerical simulation which is three times faster than the numerical simulation of the full model equations. The construction of slow manifolds for PDEs modeling the reactive flow systems is also addressed. An improved extension of the standard ILDM method to reactive flow systems is given. Reduced model equations are obtained by equilibrating the fast dynamics of a closely coupled reaction/convection/diffusion system and resolving only the slow dynamics of the same in order to reduce computational costs, while maintaining a desired level of accuracy. The improvement is realized through formulation of an elliptic system of partial differential equations which describe the infinite-dimensional Approximate Slow Invariant Manifold (ASIM) for the reactive flow system. This is demonstrated on a simple reaction diffusion system, where it is shown that the error incurred when using the ASIM is less than that incurred by use of Mass-Pope Projection (MPP) of the diffusion effects onto the ILDM. This comparison is further done for ozone decomposition in a premixed laminar flame where an error analysis shows a similar trend.