Viatcheslav Bykov

The extension of the ILDM concept to reaction–diffusion manifolds

In the present work, the method of simplifying chemical kinetics based on Intrinsic Low-Dimensional Manifolds (ILDMs) is modified to deal with the coupling of reaction and diffusion processes. Several problems of the ILDM method are overcome by a relaxation to an invariant system manifold (Reaction–Diffusion Manifold – REDIM). This relaxation process is governed by a multidimensional parabolic partial differential equation system, where, as an initial solution, an extended ILDM is used. Furthermore, a method for the solution and tabulation of the manifold is proposed in terms of generalized coordinates, with a subsequent procedure for the integration of the reduced system on the found manifold. This modification of the ILDM significantly improves the performance of the concept and allows us to extend its area of applicability. Illustrative comparative calculations of detailed and reduced models of flat laminar flames verify the approach.

Simple global reduction technique based on decomposition approach

Large and complex (nonlinear) models of chemical kinetics are one of the major obstacles in simulations of reacting flows. In the present work a new approach for an automatic reduction of chemical kinetics models, the so-called Global Quasi-Linearization (GQL) method is presented. The method is similar to the ILDM and CSP approaches in the sense that it is based on a decomposition into fast/slow motions and on slow invariant manifolds, but has a global character which allows us to overcome difficulties with the application of slow invariant manifolds and significantly simplifies the construction procedure for approximation of the slow invariant system manifold. The method is implemented within the standard ILDM method and applied to a number of model examples and to a meaningful combustion chemistry model.

Singularly perturbed vector fields

A geometrically invariant concept of singularly perturbed systems of ordinary differential equations (singularly perturbed vector fields) is proposed in this paper. Singularly perturbed vector fields can be represented locally as singularly perturbed systems (for corresponding coordinate system choice. The paper focuses on possible ways of fast and slow directions/manifolds evaluations. A special algorithm for the evaluation is proposed. The algorithm is called as a global quasi-linearization procedure. A practical application of the proposed algorithm for numerical simulations is the main issue of the paper.

Investigation of the Hierarchical Structure of Kinetic Models in Ignition Problems

Abstract In this paper a novel method for global analysis of chemical kinetic models is discussed and applied to auto ignition of high hydrocarbons. It is mainly based on the concept of decomposition of motions and follows major steps of the ILDM method. However, a very important difference of the suggested method is its ability of an explicit and global representation of the system decomposition as a standard singular perturbed system (SPS). The fact that the ILDM provides local information about the decomposition makes the ILDM quite accurate in description of the slow system dynamics, but for fast motions, which become very important in the context of ignition/extinction problems, there are up to now no reliable methods based on ILDM. Therefore, the current work is devoted to developing such a method which can be used efficiently for global analysis of the reaction mechanisms with subsequent formulation of explicit reduced models for unsteady combustion regimes like ignition processes. The suggested method is illustrated by a simple Lindemann kinetic model and then applied successfully to the auto ignition of a homogeneous n-heptane/air system.

Study of Extinction Limits of Diluted Hydrogen-Air Counter-Flow Diffusion Flames with the Redim Method

In the present study the reaction-diffusion manifolds (REDIM) method for model reduction is applied to detect extinction strain rates of highly diluted hydrogen/nitrogen and air counter-flow diffusion non-premixed flames at different ambient pressures. The considered problem inhibits miscellaneous challenges for modeling of reactive flows and is therefore well suited for testing the REDIM reduction method. First of all, by assigning critical strain rates the model’s ability to capture transient system behavior is investigated. Moreover, critical system regimes of extinction are strongly influenced by the pressure dependence of chain branching and termination mechanisms. This leads to a nonmonotonic dependence between system pressure and extinction limits. The aim of the present work is to find out to what extent the REDIM method is capable to reproduce these complex chemical chain branching and termination channels. Furthermore, in highly diluted hydrogen counter-flow diffusion flames the fast molecular transport of radicals into the reaction zone is strongly affecting the flame stability. Consequently, detailed transport models must be considered in this critical case, which represents an additional task for the reduced model. In the present work, results of reduced simulations based on the REDIM method are compared with detailed calculations and experimental data to show the ability of the REDIM model reduction method to account for such critical regimes of nonstationary combustion processes and to reproduce the nonmonotonic function of critical strain rate dependent on the system pressure in the case of a detailed transport model.

Reaction-Diffusion Manifolds and Global Quasi-linearization: Two Complementary Methods for Mechanism Reduction

The paper outlines the current state in the model reduction of systems governing reacting flows by manifold methods. The main idea of such approaches is based on the fact that any reduced model defines a manifold of low dimen- sion imbedded in the system composition/state space. In this respect the decomposition into relatively fast and slow mo- tions due to multiple time scales present in the system is a crucial property of the reacting system. It allows the application of the geometrical framework of slow and fast invariant manifolds to model reduction. Recently developed approaches, namely, the so-called Reaction-Diffusion Manifolds (REDIMs) and Global-Quasi Linearization (GQL) are in the focus of this work. The methods extend and follow the well known ILDM method. The paper discusses both the theoretical basis of the approaches and detailed implementation schemes for studying, reducing and simulating the reacting flows systems. Simple yet containing all features of the reacting flows models of n-heptane/air and syngas/air systems are used to illus- trate and verify the methods.

The Effect of Evaporation Models on Urea Decomposition from Urea-Water-Solution Droplets in SCR Conditions

Selective catalytic reduction using urea-water-solution to reduce emissions of NOx from diesel engines is commonly used in the automotive industry. For a high efficiency of this process, a good understanding of the formation of ammonia from urea-water-solution droplets is required. There are two main variants for the description of urea decomposition into ammonia and isocyanic acid from droplets of urea-water-solution based on an evaporation model: direct decomposition at the interface and decomposition in the gas phase by a chemical reaction. These variants have been compared using detailed one-dimensional simulations with a detailed model for the gas-liquid interface. In addition, the influence of gas phase chemistry and varying ambient conditions on the decomposition of urea was determined. It is shown that water evaporation and urea decomposition cannot be completely separated. Direct decomposition overestimates the production of ammonia due to the varying gas phase properties of ammonia and isocyanic acid. Decomposition in the gas phase correctly calculates the mass of ammonia produced by a droplet but the gas phase reaction couples strongly with the evaporation process. Especially at lower ambient temperatures, the evaporation rate is increased and it is more sensitive to changes of the ambient conditions and initial droplet diameter. Of the known relevant gas phase chemistry, only the hydrolysis of isocyanic acid happens in a time-scale similar to that of the droplet variation at temperatures typical for selective catalytic reduction.

Joint Characteristic Timescales and Entropy Production Analyses for Model Reduction of Combustion Systems

The reduction of chemical kinetics describing combustion processes remains one of the major topics in the combustion theory and its applications. Problems concerning the estimation of reaction mechanisms real dimension remain unsolved, this being a critical point in the development of reduction models. In this study, we suggest a combination of local timescale and entropy production analyses to cope with this problem. In particular, the framework of skeletal mechanism is in the focus of the study as a practical and most straightforward implementation strategy for reduced mechanisms. Hydrogen and methane/dimethyl ether reaction mechanisms are considered for illustration and validation purposes. Two skeletal mechanism versions were obtained for methane/dimethyl ether combustion system by varying the tolerance used to identify important reactions in the characteristic timescale analysis of the system. Comparisons of ignition delay times and species profiles calculated with the detailed and the reduced models are presented. The results of the application show transparently the potential of the suggested approach to be automatically implemented for the reduction of large chemical kinetic models.

Hierarchical Structure of Slow Manifolds of Reacting Flows

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 todays 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.

Scaling Invariant Interpolation for Singularly Perturbed Vector Fields (SPVF)

The problem of modelling, numerical simulations and interpretation of the simulations results of complex systems arising in reacting flows requires more and more sophisticated methods of qualitative system analysis. Recently, the concept of invariant, slow/fast, attractive manifolds has proven to be an efficient tool for such an analysis. In particular, it allows us to study main properties of detailed models describing the reacting flow by considering appropriate low dimensional manifolds, which appear naturally in the system state/composition space as a manifestation of a restricted number of real degrees of freedom exhibited by the system.In order to answer the question of what are the minimal number of the real degrees of freedom (real system dimension) and to approximate low dimensional manifolds (i.e., reduced system’s phase spaces) the concept of Singularly Perturbed Vector Fields (SPVF) has been suggested lately [1]. In the current work a scales invariant version of the SPVF will be presented and discussed.