On slow manifolds of chemically reactive systems (original) (raw)
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In calculations of chemically reactive flows, dimension reduction of reactive systems via the use of slow attracting manifolds is an effective approach to reducing the computational burden. In the reduced description, the reactive system is described in terms of a smaller ...
Amethod is described which rationally corrects the method of intrinsic low dimensional manifolds (ILDM) to account for the effects of small convection and diffusion. The ILDM method is well suited for spatially homogeneous problems and provides a systematic way to overcome the severe stiffness which is associated with full models of detailed kinetics and thus significantly improves computational efficiency. Significant errors can arise however when the ILDM method is applied to systems which have convection and diffusion. Motivated by techniques from center manifold theory, and using the ILDM as a reference manifold, we project our entire system of equations onto a new basis, which is segregated into fast and slow sets of equations. The fast scale equations are equilibrated, requiring the solution of an elliptic equation in space. The slow equations are allowed to temporally evolve. Improvements in predictions relative to those of the traditional IDLM method are shown for a simple m...
Calculation of Slow Invariant Manifolds for Reactive Systems
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009
One-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady, isothermal, isochoric, spatially homogeneous, closed reactive systems are calculated. The technique is based on global analysis of the composition space of the reactive system. The identification of all the system's finite and infinite critical points plays a major role in calculating the system's slow invariant manifold. The slow invariant manifolds are constructed by calculating heteroclinic orbits which connect appropriate critical points to the critical point which corresponds to the unique stable physical critical point of chemical equilibrium. The technique is applied to small and large detailed kinetics mechanisms for hydrogen combustion.
Parameterisations of slow invariant manifolds: application to a spray ignition and combustion model
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A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.
Method of invariant manifold for chemical kinetics
Chemical Engineering Science, 2003
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of the MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to e ectively reduce description in chemical kinetics. ?
Physics of Fluids
Manifold based simplified chemistry is an efficient reduction technique for the chemical kinetics, which aims to reduce the computational effort in numerical simulations. While the concept of reduced chemistry has been used for decades and various models have been developed up to now, their coupling with turbulent physical processes (e.g., mixing processes) has not been investigated extensively. This is attributed to the fact that the turbulent physical processes act as perturbation to the chemistry which pulls the thermo-kinetic states away from the manifold, and these states must relax back onto the manifold again. The present work gives insight into the coupling of reduced kinetic and the turbulent mixing processes. Accordingly, a strategy based on the Intrinsic Low-Dimensional Manifold concept is proposed. This coupling strategy is validated through the well-known Sandia Flame series. It is shown that the numerical results agree very well with those using detailed chemistry (no ...
2018
The paper has two goals: It presents basic ideas, notions, and methods for reduction of reaction kinetics models: quasi-steady-state, quasi-equilibrium, slow invariant manifolds, and limiting steps. It describes briefly the current state of the art and some latest achievements in the broad area of model reduction in chemical and biochemical kinetics, including new results in methods of invariant manifolds, computation singular perturbation, bottleneck methods, asymptotology, tropical equilibration, and reaction mechanism skeletonisation.