Method Of Intrinsic Low-dimensional Manifolds Research Articles

Computing the Steady state Dynamics: The Chemical Graphs of Complex Reaction

In a thermodynamically isolated system, in order to obtain numerical approximations of the complex models, different model reduction techniques are applied to reduce the complex chemical reactions from high dimensional to low dimensional manifolds. These techniques not only reduce the system but also provide the complete description of reaction kinetics. These techniques include Quasi Steady State Approximations, Partial Equilibrium Technique, Intrinsic Low Dimensional Manifold Method, etc. Among all these techniques, Spectral Quasi Equilibrium Manifold Method is one of the most convenient way to find the initial approximations of slow invariant manifolds. Initial invariant grids completely describe the slow invariant manifolds which are obtained from the dissipative system. This method is applicable for high dimensional complex chemical reactions.

Large-eddy simulation of direct injection natural gas combustion in a heavy-duty truck engine using modified conditional moment closure model with low-dimensional manifold method

This article describes the application of a modified first-order conditional moment closure model used in conjunction with the trajectory-generated low-dimensional manifold method in large-eddy simulation of pilot ignited high-pressure direct injection natural gas combustion in a heavy-duty diesel engine. The article starts with a review of the intrinsic low-dimensional manifold method for reducing detailed chemistry and various formulations for the construction of such manifolds. It is followed by a brief review of the conditional moment closure method for modelling the interaction between turbulence and combustion chemistry. The high computational cost associated with the direct implementation of the basic conditional moment closure model was discussed. The article then describes the formulation of a modified approach to solve the conditional moment closure equation, whose reaction source terms for the conditional mass fractions for species were obtained by projecting the turbulent perturbation onto the reaction manifold. The main model assumptions were explained and the resulting limitations were discussed. A numerical experiment was conducted to examine the validity the model assumptions. The model was then implemented in a combustion computational fluid dynamics solver developed on an open-source computational fluid dynamics platform. Non-reactive jet simulations were first conducted and the results were compared to the experimental measurement from a high-pressure visualization chamber to verify that the jet penetration under engine relevant conditions was correctly predicted. The model was then used to simulate natural gas combustion in a heavy-duty diesel engine equipped with a high-pressure direct injection system. The simulation results were compared with the experimental measurement from a research engine to verify the accuracy of the model for both the combustion rate and engine-out emissions.

Analysis of the approximate slow invariant manifold method for reactive flow equations

The Approximate Slow Invariant Manifold method of Singh, Powers and Paolucci is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. In this article, we present a mathematical analysis of the Approximate Slow Invariant Manifold method in the context of systems of reaction–diffusion equations with slow and fast reaction kinetics. Beginning with systems of two species (one slow and one fast), and then treating general systems with multiple slow and fast species, we explicitly determine the accuracy of the Approximate Slow Invariant Manifold method. We find that it is correct up to and including the terms of first order in the small parameter that measures the separation of the kinetics time scales, and that it captures many of the terms at second order, as well. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction–diffusion equation, as well as to the curvature of the critical manifold. We illustrate the results analytically on two prototypical examples, the Michaelis–Menten–Henri model with diffusion of the slow species and the Davis–Skodje model in which both the slow and fast species diffuse.

A Geometric Method for Model Reduction of Biochemical Networks with Polynomial Rate Functions

Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low-dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors.

Causes for “ghost” manifolds

The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239–64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol’dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353–74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1–2):66–93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108–23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called “ghost”-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol’dshtein V. “Ghost” ILDM – manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55–7; Borok S, Goldfarb I, Gol’dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029–38; Borok S, Goldfarb I, Gol’dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. “Ghost” ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin–Heidelberg–New York: Springer; 2006. p. 55–80; Borok S, Goldfarb I, Gol’dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125–50; Bykov V, Goldfarb I, Gol’dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006;71:359–82; Flockerzi D. Tutorial: intrinsic low-dimensional manifolds and slow attractors. Magdeburg: Max-Planck-Institut; 2001–2005. <www.mpi-magdeburg.mpg.de/people/flocke/tutorial-ildm.pdf>; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 1999;9:108–23; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 2006;16:048101]. The present work studies the causes for the “ghost” manifolds appearance for the case of a two-dimensional singularly perturbed system.

Turbulence-chemistry interactions in CFD modelling of diesel engines

The use of detailed mechanisms is indispensable in predicting intermediate species in practical combustion devices. Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions. Their size makes it prohibitive to use them in three-dimensional combustion simulations of real-life equipment. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. In such situations, the ILDM (Intrinsic Low Dimensional Manifold) method gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence. It is a method of automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time-scales identified by a local eigenvector analysis. The KIVA-III code is used to simulate the flow in a single cylinder of a DI diesel engine. A RNG (Renormalized Group Theory) k-ϵ model is added to the code to model the turbulent flow field. Turbulence–chemistry interactions are taken into account by integrating the chemical source terms over a presumed probability density function (PDF). A standard approach can lead to unphysical states and rather poor heat-release and temperature profiles. An improvised approach to eliminate this problem is outlined in this work. Since no equilibrium or partial-equilibrium approach for mechanism reduction is involved, accurate concentrations of intermediate species such as O radicals and soot-precursors (C2H2 and C3H3) can be obtained from the ILDM. This improves the accuracy of pollutant predictions. The goal of this work is to establish the methods used as industrially acceptable ways of coupling the chemistry with turbulence.

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.

The functional equation truncation method for approximating slow invariant manifolds: A rapid method for computing intrinsic low-dimensional manifolds

A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust. Flame 88, 239 (1992)] is frequently used as an approximation to the slow manifold. This approximation is based on a linearized analysis of the differential equations and thus neglects curvature. We present here an efficient way to calculate an approximation equivalent to the ILDM. Our method, called functional equation truncation (FET), first develops a hierarchy of functional equations involving higher derivatives which can then be truncated at second-derivative terms to explicitly neglect the curvature. We prove that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system. In higher-dimensional spaces, the ILDM and FETA manifolds agree to numerical accuracy almost everywhere. Solution of the FET equations is, however, expected to generally be faster than the ILDM method.

About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM)

The present paper contains an analysis of some aspects of a well known method of Intrinsic Low-Dimensional Manifolds (ILDM), which is regularly used for model reduction purposes in a number of combustion problems. One of these aspects relates to an existence of additional solutions (so-called “ghost”-manifolds), which represent intrinsic low-dimensional manifolds and do NOT represent any slow invariant manifold even for two-dimensional singularly perturbed systems (for a small but finite singular parameter). These “ghost”-manifolds are examples that contradict to the conjecture about the coincidence of ILDM and slow invariant manifolds published previously. Another aspect of the ILDM-method concerns the so-called transition zones (turning manifolds) between different invariant manifolds. It is shown that transition manifolds can not be correctly described by the ILDM-method. This statement is illustrated by an example taken from the mathematical theory of combustion.

Classical and Dynamic Analysis of Gas Phase Reactivity: Influence of Carbon Precursor in the CVD of SiC

The intrinsic low dimensional manifold method (ILDM), which is a reduction method of large gaseous chemical systems, is applied to the gas-phase chemistry used to deposit 4H or 6H silicon carbide (SiC). The 1-dimensional (1-D) manifold corresponding to the relaxation of all the fast reaction modes of the complex gaseous chemical system is calculated and represented according to different projections in the state space. The ILDM method is also used to compare trajectories corresponding to various initial gas-phase compositions. These initial gas-phase mixtures, which include different carbon precursors, all converge towards the same thermodynamic equilibrium. Long before thermodynamic equilibrium is reached, the trajectories are attracted by the 1-D manifold. © 2004 The Electrochemical Society. All rights reserved.

Simulation of finite-rate chemical kinetics in a turbulent bluff body flame

A turbulent combustion model using a Monte Carlo scalar PDF transport method with an adaptive number of PDF particles in each computational cell is coupled to an experimental version of a commercial CFD solver. While the molecular mixing is always modelled using a modified Curl model, the chemical source terms are calculated using a global 4-step reaction model and two different implementations of the method of intrinsic low-dimensional manifolds (ILDM), respectively. The quality of the chemistry representation by the three reduced chemical mechanisms is tested against detailed methane mechanisms using plug-flow reactor calculations. It is confirmed that the two ILDM implementations reproduce the chemical trajectories of the detailed reaction mechanism in scalar space quite well as expected, while the characteristic time scales of the global mechanism differ considerably. The full PDF transport CFD code is applied to a methane bluff-body diffusion flame stabilised by a recirculation zone. The calculations show a consistent improvement of predictions, particularly of temperature and CO concentrations of the PDF transport method compared to an presumed PDF calculation assuming equilibrium chemistry. Nevertheless, the CO concentrations are still considerably over predicted by all methods in such a highly strained flame, indicating that the mixing and turbulence models need to be improved.

Method of invariant manifold for chemical kinetics

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of the MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

A general algorithm for improving ILDMs

The method of intrinsic low-dimensional manifolds (ILDMs) has proven to be an efficient tool for the simplification of chemical kinetics. Nevertheless, there are still some open questions with respect to an efficient calculation and implementation of ILDMs. In this paper, we focus on the efficient calculation of ILDMs and present a refinement method for ILDMs. The method is based on an evolution equation of the manifolds towards a steady state solution which then represents the slow manifold. It has the property that it is continuous, differentiable, and in addition, it is an inertial manifold which represents the slow dynamics of the chemical system. In this way, many of the problems associated with the concept of ILDM are overcome.

On slow manifolds of chemically reactive systems

This work addresses the construction of slow manifolds for chemically reactive flows. This construction relies on the same decomposition of a local eigensystem that is used in formation of what are known as Intrinsic Low Dimensional Manifolds (ILDMs). We first clarify the accuracy of the standard ILDM approximation to the set of ordinary differential equations which model spatially homogeneous reactive systems. It is shown that the ILDM is actually only an approximation of the more fundamental Slow Invariant Manifold (SIM) for the same system. Subsequently, we give an improved extension of the standard ILDM method to systems where reaction couples with convection and diffusion. Reduced model equations are obtained by equilibrating the fast dynamics of a closely coupled reaction/convection/diffusion system and resolving only the slow dynamics of the same system in order to reduce computational costs, while maintaining a desired level of accuracy. The improvement is realized through formulation of an elliptic system of partial differential equations which describe the infinite-dimensional Approximate Slow Invariant Manifold (ASIM) for the reactive flow system. This is demonstrated on a simple reaction-diffusion system, where we show that the error incurred when using the ASIM is less than that incurred by use of the Maas-Pope Projection (MPP) of the diffusion effects onto the ILDM. This comparison is further done for ozone decomposition in a premixed laminar flame where an error analysis shows a similar trend.

Asymptotic analysis of two reduction methods for systems of chemical reactions

This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust. Flame 88 (1992) 239] and an iterative method due to Fraser [J. Chem. Phys. 88 (1988) 4732] and further developed by Roussel and Fraser [J. Chem. Phys. 93 (1990) 1072]. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The asymptotic expansions of these manifolds ( ε↓0, where ε measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M ε , the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M ε agree up to and including terms of O(ε) ; the former has an error at O(ε 2) that is proportional to the local curvature of M 0 . The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of M ε . Starting from M 0 , the ith application of the algorithm yields the correct expansion coefficient at O(ε i) , while leaving the lower-order coefficients invariant. Thus, after ℓ applications, the expansion is accurate up to and including the terms of O(ε ℓ) . The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis–Menten–Henri) and a model planar system due to Davis and Skodje.

Prediction of swirling confined diffusion flame with a Monte Carlo and a presumed-PDF-model

This work aims to compare numerical results obtained by using the Monte Carlo composition-PDF method and a presumed-β-PDF in order to reveal their effects on the prediction of flow and scalar fields in swirling confined methane diffusion flame. Using the intrinsic low dimensional manifolds method for modelling the chemistry and a second moment closure for the turbulence, it is shown that both PDF-methods provide a similar accuracy level of the prediction of mean quantities. While the presumed-β-PDF performs using reasonable computational efforts, the Monte Carlo-PDF allows to capture well the turbulence–chemistry interaction and strong finite-chemistry effects such as local extinction.

Intrinsic low-dimensional manifold method extended with diffusion

It is necessary to use chemical reduction techniques for the modeling of complex burner systems. The intrinsic low-dimensional manifold (ILDM) method has proved to be a useful method to simplify detailed reaction mechanisms. However, diffusion is neglected in the construction of the manifold, which results in less accurate manifolds in regions where both chemistry and diffusion are significant, which is generally the case in a large part of the reaction zone. This paper presents a new reduction method, which uses the basic principles of intrinsic low-dimensional manifolds. A manifold is constructed in the composition phase space instead of the composition space, so that the gradients of species concentrations and enthalpy can be used to include diffusion within the manifold. The composition phase space is spanned by the species mass fractions, the enthalpy, and their corresponding diffusive fluxes. The second-order conservation equations are transformed into first-order equations by introducing flux terms. The resulting set of first-order equations are used to construct the manifold. The procedure used to define the manifold is equivalent to the ILDM method. The new method, called phase space ILDM (PS-ILDM), is applied to a simple but illustrative example, and finally a manifold for a CO/H2 mechanism is presented. The results show that the new method gives more accurate manifolds than the ILDM method for parts of the flame where both reaction and diffusion are important. In addition, fluctuations of enthalpy and element fractions due to preferential diffusion are included in the manifold. This may result in a database with a lower dimension and therefore a more effective reduction.

Modeling of NO formation based on ILDM reduced chemistry

The reduction of NO formation is a challenging task in view of a clean operation of combustion processes. For the mathematical modeling of turbulent combustion processes, it is desirable to devise reduced schemes for the chemical kinetics which allow a reliable description of kinetically controlled processes such as NO formation. This work focuses on an extension of the method of intrinsic low-dimensional manifolds (ILDM) in order to include NO chemistry, which is well known to be governed by very slow chemical reactions. For the generation of simplified chemical reaction mechanisms, this behavior permits the assumption that these processes as nearly frozen in large parts of the reaction domain. At first, the chemical dynamics of the NO system are analyzed. Based on this analysis, a method is devised which allows the description of NO formation in the context of ILDM-reduced chemistry. Sample calculations of NO formation in CH 4 /air flames verify the approach, and the application in general computational fluid dynamics calculations is discussed.

Viscous detonation in H2‒O2‒Ar using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation

A standard ignition delay problem for a mixture of hydrogen-oxygen-argon in a shock tube is extended to the viscous regime and solved using the method of intrinsic low-dimensional manifolds (ILDM) coupled with a wavelet adaptive multilevel representation (WAMR) spatial discretization technique. An operator-splitting method is used to describe the reactions as a system of ordinary differential equations at each spatial point. The ILDM method is used to eliminate the stiffness associated with the chemistry by decoupling processes which evolve on fast and slow time scales. The fast time scale processes are systematically equilibrated, thereby reducing the dimension of the phase space required to describe the reactive system. The WAMR technique captures the detailed spatial structures automatically with a small number of basis functions thereby further reducing the number of variables required to describe the system. A maximum of only 300 collocation points and 15 scale levels yields results with striking resolution of fine-scale viscous and induction zones. Additionally, the resolution of physical diffusion processes minimizes the effects of potentially reaction-inducing artificial entropy layers associated with numerical diffusion.

Reaction mechanism reduction for higher hydrocarbons by the ILDM method

Because of its generality and simple concept, the intrinsic low-dimensional manifolds (ILDM) method is used to reduce the number of concentration variables from about 50–100 to only 1–3. Typical results for flame propagation and premixed flame-front structure are presented for n -heptane and n -dodecane combustion. The new features are that (1) ILDMs for higher hydrocarbons can be determined the first time, with sufficient accuracy to determine NO (which is sensitively dependent on the O-atom concentration) by use of a newly developed stiff-stable algorithm: (2) an effective in situ tabulation has been developed, which can save orders of magnitude of computer time and storage: and (3) use of a set of a key species leads to a very simple interface to (even commercial) codes where source terms determined from the ILDM simply replace source terms calculated from a reaction mechanism, so that the problematic projection of the conservations to the ILDM is no longer necessary. A secondary result is that flames at increasing pressure (1 bar, 20 bar, and 80 bar are considered in this paper) become more and more determined by a thermal rather than a chain-branching mechanism. One important consequence is that at elevated pressures, similar results for Lewis numbers Le =1 and Le ≠1 are produced, leading to a considerable simplification of the transport model to be applied. ILDMs generated in this way have been successfully applied to turbulent combustion simulation and pollutant formation in diesel and two-stroke engines.