Intrinsic low-dimensional manifold ILDM

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. 

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... 

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. 

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. 

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. 

An overview of the methods used previously in mechanism reduction is presented in Tomlin et al. (1997). The present work uses a combination of existing methods to produce a carbon monoxide-hydrogen oxidation scheme with fewer reactions and species variables, but which accurately reproduces the dynamics of the full scheme. Local concentration sensitivity analysis was used to identify necessary species from the full scheme, and a principle component analysis of the rate sensitivity matrix employed to identify redundant reactions. This was followed by application of the quasi-steady state approximation (QSSA) for the fast intermediate species, based on species lifetimes and quasi-steady state errors, and finally, the use of intrinsic low dimensional manifold (ILDM) methods to calculate the mechanisms underlying dimension and to verify the choice of QSSA species. The origin of the full mechanism and its relevance to existing experimental data is described first, followed by descriptions of the reduction methods used. The errors introduced by the reduction and approximation methods are also discussed. Finally, conclusions are drawn about the results, and suggestions made as to how further reductions in computer run times can be achieved. 

Maas and Pope developed an approach for the calculation of slow manifolds (Maas and Pope 1992a, b, 1994 Maas 1995, 1998, 1999 Maas and Thevenin 1998) utilising the approach of Roussel and Fraser, as well as the suggestion of Lam and Goussis, that timescales should be investigated pointwise via the eigenvalue-eigenvector decomposition of the Jacobian. Their approach was to tabulate these low-dimensional slow manifolds in phase space for several reaction systems in combustion. They called the slow manifolds intrinsic low-dimensional manifolds (ILDM). 

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... 

It was shown in the previous section that intrinsic low-dimensional manifolds of chemical systems can be used to simplify the chemical kinetics. However, the interesting case in practical applications is the coupling with flow and molecular transport. In principle the concept of the ILDM is still valid in systems with flow and molecular transport, but the physical processes act as perturbations of the chemical system, i.e. they tend to move the system off the manifold. If the perturbations occur with time-scales larger than the time scales of the relaxation towards the attracting manifold, then the fast chemical processes move the system back to the manifold instantaneously, and we can still use the manifold to simplify the kinetics. 

Fig. 6.5 (a) 2D and ID manifolds for the hydrogen flame example. Starting from any point in phase space, the trajectories (dotted lines) quickly approach the 2D manifold (mesh stffface) and then the ID manifold (bold line) and move along it towards the equilibrium point. Reprinted with permission from Davis and Tomlin (2(X)8b). Copyright (2008) American Chemical Society, (b) The collapse of reaction trajectories onto a 2D intrinsic low-dimensional manifold or ILDM (black mesh) for an wo-octane—air system plotted in a projection of the state space into CO2—H2O—H2 concentration coordinates. ID ILDM (purple symbols), OD ILDM (equilibrium, red circle). The coloured lines are homogeneous reactor calculations for different fuels. Reprinted from (Blasenbrey and Maas 2000) with permission from Elsevier... 

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