Intrinsic Low-Dimensional Manifolds

Maas U and Pope S B 1992 Simplifying chemical kinetics intrinsic low-dimensional manifolds in composition space Comb. Flame 88 239... 

Maas, U., Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics, Comput. Visualization Sci. 1 (1998) 69-82. 

An example of a smart tabulation method is the intrinsic, low-dimensional manifold (ILDM) approach (Maas and Pope 1992). This method attempts to reduce the number of dimensions that must be tabulated by projecting the composition vectors onto the nonlinear manifold defined by the slowest chemical time scales.162 In combusting systems far from extinction, the number of slow chemical time scales is typically very small (i.e, one to three). Thus the resulting non-linear slow manifold ILDM will be low-dimensional (see Fig. 6.7), and can be accurately tabulated. However, because the ILDM is non-linear, it is usually difficult to find and to parameterize for a detailed kinetic scheme (especially if the number of slow dimensions is greater than three ). In addition, the shape, location in composition space, and dimension of the ILDM will depend on the inlet flow conditions (i.e., temperature, pressure, species concentrations, etc.). Since the time and computational effort required to construct an ILDM is relatively large, the ILDM approach has yet to find widespread use in transported PDF simulations outside combustion. 

Maas, U. and S. B. Pope (1992). Simplifying chemical kinetics Intrinsic low-dimensional manifolds in composition space. Combustion and Flame 88, 239-264. 

There are a number of alternatives and variations to the reduced mechanism method. The intrinsic low dimensional manifold (ILDM) approach [253] and similar methods [399] seek to decouple the fastest time scales in the chemistry. There is a wide range of time scales for chemical reactions in most high-temperature processes, from 10-9 second to seconds. Fast reactions, or reactions with small time scales, quickly bring composition points down to attracting manifolds in the composition space. Then composition points move along on manifolds. In the ILDM approach it is assumed that any movement of the... 

U. Maas and S.B. Pope. Implementation of Simplified Chemical Kinetics Based on Intrinsic Low-Dimensional Manifolds. Proc. Combust. Inst., 24 103-112,1992. 

A linear algebraic system of rate equations for the fast species results, which can be solved a priori. Hence a strongly reduced (in the number of species to be treated) system is obtained. This concept originates from astrophysical applications and from Laser physics. It is in some instances also referred to as collisional-radiative approximation , for the fast species, lumped species concept , bundle-n method or intrinsic low dimensional manifold (ILDM) method in the literature. We refer to [9,12,13] for further references on this. 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

There are possible alternative ways for the construction of algebraic models. If the look-up tables of the intrinsic low-dimensional manifold method are fitted by polynomials [234], the result is an algebraic model similar to a repro-model describing only slow variables. Polynomials can be fitted to the integrated solutions of few-step global mechanisms [233]. Such integrated solutions are found in the look-up tables used in the Monte-Carlo method for the simulation of turbulent flames. 

U.Maas, S.B.Pope Simplifying Chemical Kinetics Intrinsic Low-Dimensional Manifolds in Composition Space. Combustion and Flame, vol 88, pp. 239-264, (1992)... 

Maas, U., Pope, S.B., Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds. Preprint 92-06, Interdisziplinares Zentrum fur wissenschaftliches Rechnen, Universitat Heidelberg, (1992)... 

U. Maas and S.B. Pope (1992) Simplifying Chemical Kinetics Intrinsic Low-Dimensional Manifolds in Composition Space. Combust. Flame 88, 239-264 S.H. Lam (1993) Using CSP to Understand Complex Chemical Kinetics. Combust. Sci. and Tech. 89, 375-404 J. Wagenhuber, J. Chem. Phys., to be submitted. 

The method of intrinsic low-dimensional manifolds (ILDM) developed by Maas and Pope simplifies the detailed chemistry automatically [2]. The method is based on the dynamical systems approach and identifies and decouples automatically the fastest processes of the reactive system. Assuming constant pressure and enthalpy the chemical reaction corresponds to a movement along trajectories in the state space set up by the n, chemical species. It is observed that after a very short time the dynamics of chemistry is restricted to subspaces (so-called low dimensional manifolds) of the state space. 

Furthermore, mathematical procedures can be applied to the detailed mechanism or the skeletal mechanism which reduces the mechanism even more. These mathematical procedures do not exclude species, but rather the species concentrations are calculated by the use of simpler and less time-consuming algebraic equations or they are tabulated as functions of a few preselected progress variables. The part of the mechanism that is left for detailed calculations is substantially smaller than the original mechanism. These methods often make use of the wide range of time scales and are thus called time scale separation methods. The most common methods are those of (i) Intrinsic Low Dimensional Manifolds (ILDM), (ii) Computational Singular Perturbation CSF), and (iii) level of importance (LOl) analysis, in which one employs the Quasy Steady State Assumption (QSSA) or a partial equilibrium approximation (e.g. rate-controlled constraints equilibria, RCCE) to treat the steady state or equilibrated species. 

Three time-scale separation methods will be described below. These are the Intrinsic Low Dimensional Manifold method (ILDM), Computational Singular Perturbation (CSP), and the lifetime analysis based on tbe Level Of Imjaortance (LOI). 

Intrinsic Low Dimensional Manifolds 3.3.1 The standard approach to ILDM... 

Maas, U. Pope, S.B., (1994). Laminar Flame Calculations using Simplified Chemical Kinetics Bases on Intrinsic Low-Dimensional Manifolds, Proceedings of the Combustion Institute Vol. 25, pp. 1349-1356. 

Maas, U. (1998). Efficient Calculation of Intrinsic Low Dimensional Manifolds for Simplification of Chemical Kinetics, Comput. Visual Sci. Vol. 1, pp. 69-81. 

Schmidt, D., Blasenbrey, T. Maas, U. (1998). Intrinsic Low-Dimensional Manifolds of Strained and Unstrained Blames, Combustion Theory and Modelling Vol 2, pp. 135-152. 

Glassmaker NJ. Intrinsic low-dimensional manifold method for rational simplification of chemical kinetics 1999. p. 1-37. Consulted in July 13,2010, http //www.nd.edu/ powers/nick. glassmaker.pdfindexpdf. 

Among many techniques used to obtain skeletons of larger mechanisms, in what follows we discuss the Direct Relation Graph (DRG) technique, coupled with Depth First Search (DFS) technique, the Sensitivity Analysis of the Jacobian matrix for the chemical system, the Intrinsic Low-Dimensional Manifold (ILDM) technique, the Reaction Diffusion Manifolds (REDIM technique) and the flamelet technique. Among these, the flamelet technique is preferred for writing the simplified chemical system for premixed flames and diffusion flames presented in the following sections. 

Konig K, Maas U. Sensitivity of intrinsic low-dimensional manifolds with respect to kinetic... 

Simplifying Chemical Kinetics Using Intrinsic Low-Dimensional Manifolds U. Maas (With 4 Figures). 334... 

Simplifying Chemical Kinetics Using Intrinsic Low-Dimensional Manifolds... 

Thus, the fast chemical relaxation processes cause the existence of low-dimensional attractors (intrinsic low-dimensional manifolds) in the composition space. Depending on the time which is allowed for the relaxation to the lowdimensional manifold, we obtain different dimensions of the manifolds. If we allow a very long time (5ms in the example above), we obtain a point (0-dimensional manifold), namely the equilibrium value, if we allow 100 we obtain a line (1-dimensional manifold), if we only allow even smaller relaxation times, we obtain manifolds of higher dimension, and finally, if we do not allow any relaxation processes at all to take place instantanously, we end up with an n -dimensional manifold, namely the composition space itself, which corresponds to detailed chemical kinetics. 

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