Zhuyin Ren

The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics

This work addresses the construction and use of low-dimensional invariant manifolds to simplify complex chemical kinetics. Typically, chemical kinetic systems have a wide range of time scales. As a consequence, reaction trajectories rapidly approach a hierarchy of attracting manifolds of decreasing dimension in the full composition space. In previous research, several different methods have been proposed to identify these low-dimensional attracting manifolds. Here we propose a new method based on an invariant constrained equilibrium edge (ICE) manifold. This manifold (of dimension nr) is generated by the reaction trajectories emanating from its (nr-1)-dimensional edge, on which the composition is in a constrained equilibrium state. A reasonable choice of the nr represented variables (e.g., nr major species) ensures that there exists a unique point on the ICE manifold corresponding to each realizable value of the represented variables. The process of identifying this point is referred to as species reconstruction. A second contribution of this work is a local method of species reconstruction, called ICE-PIC, which is based on the ICE manifold and uses preimage curves (PICs). The ICE-PIC method is local in the sense that species reconstruction can be performed without generating the whole of the manifold (or a significant portion thereof). The ICE-PIC method is the first approach that locally determines points on a low-dimensional invariant manifold, and its application to high-dimensional chemical systems is straightforward. The inputs to the method are the detailed kinetic mechanism and the chosen reduced representation (e.g., some major species). The ICE-PIC method is illustrated and demonstrated using an idealized H2O system with six chemical species. It is then tested and compared to three other dimension-reduction methods for the test case of a one-dimensional premixed laminar flame of stoichiometric hydrogen/air, which is described by a detailed mechanism containing nine species and 21 reactions. It is shown that the error incurred by the ICE-PIC method with four represented species is small across the whole flame, even in the low temperature region.

Second-order splitting schemes for a class of reactive systems

We consider the numerical time integration of a class of reaction-transport systems that are described by a set of ordinary differential equations for primary variables. In the governing equations, the terms involved may require the knowledge of secondary variables, which are functions of the primary variables. Specifically, we consider the case where, given the primary variables, the evaluation of the secondary variables is computationally expensive. To solve this class of reaction-transport equations, we develop and demonstrate several computationally efficient splitting schemes, wherein the portions of the governing equations containing chemical reaction terms are separated from those parts containing the transport terms. A computationally efficient solution to the transport sub-step is achieved through the use of linearization or predictor-corrector methods. The splitting schemes are applied to the reactive flow in a continuously stirred tank reactor (CSTR) with the Davis-Skodjie reaction model, to the CO+H2 oxidation in a CSTR with detailed chemical kinetics, and to a reaction-diffusion system with an extension of the Oregonator model of the Belousov-Zhabotinsky reaction. As demonstrated in the test problems, the proposed splitting schemes, which yield efficient solutions to the transport sub-step, achieve second-order accuracy in time.

Computationally efficient implementation of combustion chemistry in parallel PDF calculations

In parallel calculations of combustion processes with realistic chemistry, the serial in situ adaptive tabulation (ISAT) algorithm [S.B. Pope, Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation, Combustion Theory and Modelling, 1 (1997) 41-63; L. Lu, S.B. Pope, An improved algorithm for in situ adaptive tabulation, Journal of Computational Physics 228 (2009) 361-386] substantially speeds up the chemistry calculations on each processor. To improve the parallel efficiency of large ensembles of such calculations in parallel computations, in this work, the ISAT algorithm is extended to the multi-processor environment, with the aim of minimizing the wall clock time required for the whole ensemble. Parallel ISAT strategies are developed by combining the existing serial ISAT algorithm with different distribution strategies, namely purely local processing (PLP), uniformly random distribution (URAN), and preferential distribution (PREF). The distribution strategies enable the queued load redistribution of chemistry calculations among processors using message passing. They are implemented in the software x2f_mpi, which is a Fortran 95 library for facilitating many parallel evaluations of a general vector function. The relative performance of the parallel ISAT strategies is investigated in different computational regimes via the PDF calculations of multiple partially stirred reactors burning methane/air mixtures. The results show that the performance of ISAT with a fixed distribution strategy strongly depends on certain computational regimes, based on how much memory is available and how much overlap exists between tabulated information on different processors. No one fixed strategy consistently achieves good performance in all the regimes. Therefore, an adaptive distribution strategy, which blends PLP, URAN and PREF, is devised and implemented. It yields consistently good performance in all regimes. In the adaptive parallel ISAT strategy, the type and extent of redistribution is determined on the fly based on the prediction of future simulation time. Compared to the PLP/ISAT strategy where chemistry calculations are essentially serial, a speed-up factor of up to 30 is achieved. The study also demonstrates that the adaptive strategy has acceptable parallel scalability.

Transport-chemistry coupling in the reduced description of reactive flows

The reduced description of inhomogeneous reactive flows by chemistry-based low-dimensional manifolds is complicated by the transport processes present and the consequent transport-chemistry coupling. In this study, we focus on the use of intrinsic low-dimensional manifolds (ILDMs) to describe inhomogeneous reactive flows. In particular we investigate three different approaches which can be used with ILDMs to incorporate the transport-chemistry coupling in the reduced description, namely, the Maas–Pope approach, the ‘close-parallel’ approach, and the approximate slow invariant manifold (ASIM) approach. For the Maas–Pope approach, we validate its fundamental assumption: that there is a balance between the transport processes and chemical reactions in the fast subspace. We show that even though the Maas–Pope approach makes no attempt to represent the departure of composition from the ILDM, it does adequately incorporate the transport-chemistry coupling in the dynamics of the reduced system. For the ‘close-parallel’ approach, we demonstrate its use with the ILDM to incorporate the transport-chemistry coupling. This approach is based on the ‘close-parallel’ assumption that the compositions are on a low-dimensional manifold which is close to and parallel to the ILDM. We show that this assumption implies a balance between the transport processes and chemical reaction in the normal subspace of the ILDM. The application of the ASIM approach in general reactive flows is investigated. We clarify its underlying assumptions and applicability. Also in the regime where the fast chemical time scales are much smaller than the transport time scales, we reformulate the ASIM approach so that explicit governing PDEs are given for the reduced composition. For the reaction–diffusion systems considered, we show that all the three approaches predict the same dynamics of the reduced compositions, i.e. each results in the same evolution equations for the reduced composition variables (to leading order). We also show that all the three approaches are valid only when the fast chemical time scales are much smaller than the transport time scales. Moreover, a simplified ASIM approach is proposed.

The geometry of reaction trajectories and attracting manifolds in composition space

In numerical simulations of combustion processes, the use of dimension reduction to simplify the description of the chemical system has the advantage of reducing the computational cost, but it is important also to retain accuracy and adequate detail. Most existing dimension reduction methods assume the existence of low-dimensional attracting manifolds in the full composition space and try to approximate or directly identify the low-dimensional attracting manifolds. However, questions remain about the geometry of the reaction trajectories in the full composition space, the existence of the low-dimensional attracting manifolds in low-temperature regions, and the minimum dimension of the attracting manifold required for describing a particular chemical system. This paper tries to address some of these issues by studying the reaction trajectories starting from a wide range of different initial compositions for both H2/air and CH 4/air mixtures. Along each trajectory, we study the tangent bundle of the trajectory, the eigenvalues of the Jacobian matrices, and the singular values of the sensitivity matrices (i.e. sensitivity with respect to initial composition). It is shown that the dimension of the affine space containing a trajectory (or of the tangent bundle along a trajectory) is much smaller than the dimension of the full composition space. Even at low temperatures, the Jacobian matrices still have a significant number of large (in magnitude) negative eigenvalues, which implies the existence of fast time scales and low-dimensional attracting manifolds (even at low temperatures). The geometrical significance of sensitivity matrices is explored. Based on the sensitivity matrices, a new method is proposed to determine the minimum dimension of the attracting manifold required for describing a chemical system with prescribed accuracy, and to identify the ‘principal subspace’ which is an approximation to the tangent space of the attracting manifold.