Hierarchical Structure of Slow Manifolds of Reacting Flows

Abstract

Abstract Nowadays the mathematical description of chemically reacting flows uses very often reaction mechanisms with far above hundred or even thousand chemical species (and, therefore, a large number of partial differential equations must be solved), which possibly react within more than a thousand of elementary reactions. These chemical kinetic processes cover time scales from nanoseconds to seconds. An analogous scaling problem arises for the length scales. Due to these scaling problems the detailed simulation of three-dimensional turbulent flows in practical systems is beyond the capacity of even today's super-computers. Using simplified sub-models is a way out of this problem. The question arising in mathematical modeling of reacting flows is then: How detailed, or down to which scale has each process to be resolved (chemical reaction, chemistry-turbulence-interaction, molecular transport processes) in order to allow a reliable description of the entire process. Both the chemical source term and the transport term have one important property, namely, they cause the existence of low-dimensional attractors in composition space. When these manifolds can be constructed (described) and parametrized by a small number of variables, it can be used to reformulate and reduce the mathematical description for modeling reacting flows. In this work the hierarchical nature of these low-dimensional manifolds of slow motions is discussed. It is demonstrated how this important feature of reacting flows is accounted for by the standard model reduction methods (like e.g. PEA and QSSA methods) as well as by recently developed concepts of model reduction. The use of the hierarchical nature for identification of the low-dimensional manifolds to devise hierarchical modeling concepts (e.g. for turbulent reacting flows) is additionally discussed.