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.     

Problem adapted reduced models based on Reaction-Diffusion Manifolds (REDIMs)

In this work a novel modification of the REDIM method is presented. The method follows the main concept of decomposition of time scales. It is based on the assumption of existence of invariant slow manifolds in the thermo-chemical composition space (state space) of a reacting flow. A central point of the current modification is its capability to include both transport and thermo-chemical processes and their coupling into the definition of the reduced model. This feature makes the method more problem oriented, and more accurate in predicting the detailed system dynamics. The manifold of the reduced model is approximated by applying the so-called invariance condition together with repeated integrations of the reduced model in an iterative way. The latter is needed to improve the estimate of gradients of the reduced model parameters (coordinates which define the reduced manifold locally). To verify the approach one-dimensional stationary laminar methane/air and syngas/air flames are investigated. In particular, it is shown that the adaptive REDIM method recovers the full stationary system dynamics governed by detailed chemical kinetics and the molecular transport in the case of a one dimensional reduced model and, therefore, includes the so-called flamelet method as a limiting case.     

Comparative analysis of two asymptotic approaches based on integral manifolds

A comparative analysis of the two powerful asymptotic methods,ILDM and MIM (intrinsic low-dimensional manifolds; method ofinvariant manifold), is presented in the paper. The two methodsare based on the general theory of integral manifolds. The ILDMmethod is able to handle large systems of ODEs, whereas theMIM method treats systems with a limited number of unknown variables.The MIM method allows one to conduct analytical explorationof the original system and to obtain final expressions in compactform, whereas the ILDM method is a numerical approach that yieldsthe numerical form of the desired surface. The ILDM method workswell in a region where a rough splitting of the initial systemexists. Regions of the phase space where splitting does notexist are problematic for the ILDM method. In these regionsthe MIM method provides additional information regarding thedynamical behaviour of the system. A number of simple examplesare considered and analysed. It is shown that for the Semenovmodel (singularly perturbed system of ODEs) the ILDM methodgives a surface which appears close to the first order (withrespect to the corresponding small parameter) approximationof the stable (attracting) invariant manifolds. The complementaryproperties of the two asymptotic approaches suggests a feasiblecombination of the two methods, which is the subject of a futurework.     

Relaxation of concentration perturbation in chemical kinetic systems

In a linear approximation, the relaxation of a concentration perturbation can be described by a matrix exponential, which can be evaluated using Jordan decomposition. In time-scale analysis, this approach has advantages when the Jacobian has degenerate eigenvalues, which may occur when the mechanism contains identical rate constants, characteristic to tropospheric chemistry and low-temperature combustion.     

On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

^** Email: vbykov{at}cs.bgu.ac.il^*** Email: goldfarb{at}cs.bgu.ac.il^**** Email: vladimir{at}bgumail.bgu.ac.il^***** Email: umaas{at}itt.mach.uni-karlsruhe.de Using the method of integral (invariant) manifolds, the intrinsiclow-dimensional manifolds (ILDM) method is analysed. This isa method for identifying invariant manifolds of a system's slowdynamics and has proven to be an efficient tool in modellingof laminar and turbulent combustion. It allows treating multi-scalesystems by revealing their hidden hierarchy and decomposingthe system dynamics into fast and slow motions. The performedanalysis shows that the original ILDM technique can be interpretedas one of the many possible realizations of the general framework,which is based on a special transformation of the original coordinatesin the state space. A modification of the ILDM is proposed basedon a new definition of the transformation matrix. The proposednumerical procedure is demonstrated on linear examples and highlynon-linear test problems of mathematical theory of combustionand demonstrates in some cases better performance with respectto the existing one.     

On-demand generation of reduced mechanisms based on hierarchically extended intrinsic low-dimensional manifolds in generalized coordinates

A new implementation scheme for reduced mechanisms based on hierarchically generated and extended intrinsic low-dimensional manifolds (ILDMs) created “on-demand” is presented. The algorithm includes the use of ILDMs in generalized coordinates and a new hierarchical concept for the extension of the ILDMs into the domain of slow chemistry. Problems of pre-calculated ILDM tables are overcome by generating ILDM cells on-demand during the flame calculation, yielding an increased efficiency of the table generation and implementation. In view of a future generation of ILDMs with adaptive dimension based on a local online error control, the presented algorithm includes the possibility to increase the ILDM dimension hierarchically after the stationary solution (solution after 10⁴ s) of the first flame calculation with an n_c-dimensional ILDM is reached and to re-calculate the result of this first flame calculation using higher-dimensional manifolds with a subsequent error test. The paper presents the generation of hierarchically extended ILDMs in generalized coordinates as well as the on-demand implementation scheme. A sample free flame calculation for the syngas-air system validates the algorithms.