Abstract
The adoption of detailed mechanisms for chemical kinetics often poses two types of severe challenges: First, the number of degrees of freedom is large; and second, the dynamics is characterized by widely disparate time scales. As a result, reactive flow solvers with detailed chemistry often become intractable even for large clusters of CPUs, especially when dealing with direct numerical simulation (DNS) of turbulent combustion problems. This has motivated the development of several techniques for reducing the complexity of such kinetics models, where, eventually, only a few variables are considered in the development of the simplified model. Unfortunately, no generally applicable a priori recipe for selecting suitable parameterizations of the reduced model is available, and the choice of slow variables often relies upon intuition and experience. We present an automated approach to this task, consisting of three main steps. First, the low dimensional manifold of slow motions is (approximately) sampled by brief simulations of the detailed model, starting from a rich enough ensemble of admissible initial conditions. Second, a global parametrization of the manifold is obtained through the Diffusion Map (DMAP) approach, which has recently emerged as a powerful tool in data analysis/machine learning. Finally, a simplified model is constructed and solved on the fly in terms of the above reduced (slow) variables. Clearly, closing this latter model requires nontrivial interpolation calculations, enabling restriction (mapping from the full ambient space to the reduced one) and lifting (mapping from the reduced space to the ambient one). This is a key step in our approach, and a variety of interpolation schemes are reported and compared. The scope of the proposed procedure is presented and discussed by means of an illustrative combustion example.
Highlights
The solution of detailed models for chemical kinetics is often dramatically time consuming, owing to a large number of variables evolving in processes with a wide range of time and space scales.As a result, fluid dynamic flow solvers coupled with detailed chemistry still present a challenge, even for modern clusters of CPUs, especially when dealing with direct numerical simulation (DNS) of turbulent combustion systems
It is well known that, due to the presence of fast and slow dynamics, the above systems are characterized by low dimensional manifolds in the concentration space, where a typical solution trajectory is initially rapidly attracted towards the manifold, while afterwards, it proceeds to the thermodynamic equilibrium point, always remaining in close proximity to the manifold
It is worth stressing that the above description briefly outlines only one possible approach for coupling a model reduction method to a flow solver: the case where the low dimensional manifolds of the homogeneous problem are identified in advance in the entire phase-space
Summary
The solution of detailed models for chemical kinetics is often dramatically time consuming, owing to a large number of variables evolving in processes with a wide range of time and space scales. A large number of governing equations for chemical species (a few hundred for mechanisms of standard hydrocarbon fuels) are to be solved at (typically) millions of distinct discretization points in the computational domain This has motivated the development of a plethora of approaches aiming at reducing the computational complexity of such detailed combustion models, ideally by recasting them in terms of only a few new reduced variables. Upon identification of the slow manifold, we propose a generally applicable methodology for selecting a suitable parameterization; we investigate various interpolation/extrapolation schemes that need to be used in the solution of a reduced dynamical system expressed in terms of the variables learned This latter step, and its integration with the previous two in an overall computer-assisted approach, constitute the methodological novelty of the paper.
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