Implementation of REDIM reduced chemistry to model an axisymmetric laminar diffusion methane–air flame

The goal of this work is to analyze the use of automatically reduced chemistry by the Reaction–Diffusion Manifold (REDIM) method in simulating axisymmetric laminar coflow diffusion flames. Detailed chemical kinetic models are usually computationally prohibitive for simulating complex reacting flows, and therefore reduced models are required. Automatic reduction model approaches usually exploit the natural multi-scale structure of combustion systems. The novel REDIM approach applies the concept of invariant manifolds to treat also the influence of the transport processes on the reduced model, which overcomes a fundamental problem of model reduction in neglecting the coupling of molecular transport with thermochemical processes. We have considered a previously well studied atmospheric pressure nitrogen-diluted methane–air flame as a test case to validate the methodology presented here. First, one-dimensional and two-dimensional REDIMs were computed and tabulated in lookup tables. Then, the full set of governing equations are projected on the REDIM and implemented in the object-oriented C++ Gascoigne code with a new add-on library to deal with the REDIM tables. The projected set of governing equations have been discretized by the Finite Element Method (FEM) and solved by a GMRES iteration preconditioned by a geometric multigrid method. Local grid refinement, adaptive mesh and parallelization are applied to ensure efficiency and precision. The numerical results obtained using the REDIM approach have shown very good agreement with detailed numerical simulations and experimental data.     

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.     

Reduced modeling of the NOx formation based on the reaction-diffusion manifolds method for counterflow diffusion flames

The accurate prediction of the NOx formation has gained great attention in view of clean combustion. In this direction, a reliable reduction technique for the chemical kinetics is important to capture the NOx formation accurately with reduced computational costs for the practical turbulent combustion processes. This work focuses on the hierarchical construction of Reaction-Diffusion Manifolds (REDIM) to include the NOx chemistry, which is well-known to be governed by very slow chemical reactions. Based on the hierarchical structure of the REDIM method, a two-dimensional (2D) and a three-dimensional (3D) REDIM model are generated, without any principle extension of the REDIM method. Sample calculations of NOx formation in methane/air non-premixed counterflow flames verify the REDIM method for both steady and transient processes.     

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.     

Comparative analysis of Reaction-Diffusion Manifold based reduced models for Head-On- and Side-Wall-Quenching flames

In this study, multi-dimensional molecular transport phenomena during Flame-Wall-Interactions (FWI) and their effects on model reduction strategies are investigated. In order to access the problem, the standard configurations of a two-dimensional Side-Wall Quenching (SWQ) flame and a one-dimensional Head-On Quenching (HOQ) flame are used and compared. In the case of the SWQ configuration it is shown that the gradients of the species scatter significantly both in the physical space and in the state space. Moreover, the gradient vector of the specific enthalpy describing energy losses towards the wall is not aligned with the gradient vectors of the species, which can be considered as a typical case while a flame in application might approach to the wall at any arbitrary transversal direction. This observation motivates to take the gradients’ scattering and multi-dimensional transport phenomena into account during model reduction to describe reliably the quenching process.The Reaction-Diffusion Manifold (REDIM) method is applied in this work. The method allows to take into account multi-dimensional transport in a very generic way. In order to generate the REDIM, gradient estimates are approximated by using a Singular-Value Decomposition (SVD) of SWQ detailed gradients fields. Two-dimensional REDIMs for both cases are constructed and compared to each other. Different transport (diffusion) models are implemented to compare quantitatively the manifolds with HOQ and SOQ gradients estimates. The comparison shows that the differences between reduced models with varying transport models is significantly larger than the differences for varying configurations (multidimensional gradient estimates). This justifies the use of a relatively simple REDIM for more complicated geometries and configurations. This simplifies the treatment and model reduction procedure significantly for such complicated transient phenomena.     

Reduced modeling of Flame-Wall-Interactions of premixed isooctane-air systems including detailed transport and surface reactions

In this work, the Reaction-Diffusion Manifold (REDIM) method, a method for model reduction, is applied to a premixed isooctane-air system with Flame-Wall-Interactions (FWI). In order to provide a highly accurate reduced kinetic model, a detailed model for the diffusive processes is applied and complex boundary conditions that account for heterogeneous wall reactions are implemented.The REDIM is constructed and validated by comparing results of detailed and reduced kinetics in the system state space. The results of the reduced computations are compared with those of the detailed computations. It is shown that the reduced kinetics reproduce the results of the FWI very accurately. In particular, the difference between detailed kinetics with and without wall reactions is larger than the difference between detailed and reduced kinetics with heterogeneous wall reactions, which demonstrates the quality of the model reduction.     

On transient behavior of non-premixed counter-flow diffusion flames within the REDIM based model reduction concept

In the present work non-stationary behavior of the counter-flow diffusion flame is examined in the context of the recently developed approach of model reduction called REaction–DIffusion Manifolds (REDIM) method. It is a natural extension of the ILDM approach which takes into account both the chemical reaction and the diffusion processes. It has been developed to treat both premixed and non-premixed regimes of combustion. In this work we investigate the ability of the concept to describe transient processes of extinction and re-ignition. A very simple flame configuration and transport model are considered in this current study for the sake of transparency because the main focus is on the transient and non-stationary behavior of flames. H₂/O₂/N₂ combustion system is considered in a non-premixed counter-flow diffusion 1D flame configuration. This study shows how the REDIM concept performs in the transient regimes; it interprets the effect of local extinction and reigniting phenomena using detailed and reduced models. It shows how the unstable/transient behavior of a detailed system can be successfully accounted with the help of the REDIM based reduced model.     

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.     

Modeling curvature effects in turbulent autoigniting non-premixed flames using tabulated chemistry

Turbulent flames are intrinsically curved. In the presence of preferential diffusion, curvature effects either enhance or suppress molecular diffusion, depending on the diffusivity of the species and the direction of the flame curvature. When a tabulated chemistry type of modeling is employed, curvature-preferential diffusion interactions have to be taken into consideration in the construction of manifolds. In this study, we employ multistage stage flamelet generated manifolds (MuSt-FGM) method to model autoigniting non-premixed turbulent flames with preferential diffusion effects included. The conditions for the modeled flame are in MILD combustion regime. To model the above-mentioned curvature-preferential diffusion interactions, a new mixture fraction which has a non-unity Lewis number is defined and used as a new control variable in the manifold generation. 1D curved flames are simulated to create the necessary flamelets. The resulting MuSt-FGM tables are used in the simulation of 1D laminar flames, and then also applied to turbulent flames using 2D direct numerical simulations (DNS). It was observed that when the curvature effects are included in the manifold, the MuSt-FGM results agree well with the detailed chemistry results; whereas the results become unsatisfactory when the curvature effects are ignored.     

Evaluation of chemistry models on methane/air edge flame simulation

The integration of chemistry into a numerical fully compressible solver is carried out in this study using three models: detailed chemistry, fully tabulated chemistry (CTC) and a model coupling both approaches called HTTC, for hybrid transported-tabulated chemistry. With HTTC major species are transported while most minor species are tabulated. As minor species are no longer transported with the flow, the time step is close to the values usually encountered for non-reactive flows, far beyond what is found in detailed chemistry. The performance of HTTC for reproducing the dynamics of a methane/air edge flame featuring a very strong mixture fraction gradient is also investigated. The results agree favorably with the reference case simulated with detailed chemistry unlike the CTC model which is unable to predict the topology of the flame. Finally, the shape of the flame, the flame speed and the flame stabilization height are reasonably well captured with HTTC with a calculation cost divided by about 5 compared to the reference case.     

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.     

Turbulent flame simulation taking advantage of tabulated chemistry self-similar properties

Predicting the flame shape, its stabilization process, and pollutant emissions in practical combustion devices requires to incorporate complex chemistry features. As detailed chemical schemes are too voluminous for practical numerical simulations, tabulated chemistry techniques have been proposed to account for the complexity of kinetics in turbulent flame simulations. Unfortunately, the size of these databases may become a crucial issue for efficient implementation on massively parallel computers. A reduction strategy that takes advantage of self-similar properties of tabulated chemistry is proposed for turbulent combustion modeling. A reduction of the database size by a factor of 1000 is achieved. This procedure is successfully applied to a RANS simulation of a turbulent jet flame.     

Method of invariant grid for model reduction of hydrogen combustion

The Method of Invariant Grid (MIG) is a model reduction technique based on the concept of slow invariant manifold (SIM). The MIG approximates the SIM by a set of nodes in the concentration space (invariant grid). In the present work, the MIG is applied to a realistic combustion system: an adiabatic constant volume reactor with H₂-air at stoichiometric proportions. By considering the thermodynamic Lyapunov function of the detailed kinetic system, the notion of the quasi-equilibrium manifold (QEM) is adopted as an initial approximation to the SIM. One- and two-dimensional discrete approximations of the QEM (quasi-equilibrium grids) are constructed and refined via MIG to obtain the corresponding invariant grids. The invariant grids are tabulated and used to integrate the reduced system. Excellent agreement between the reduced and detailed kinetics is demonstrated.     

Computational singular perturbation with non-parametric tabulation of slow manifolds for time integration of stiff chemical kinetics

This paper presents a novel tabulation strategy for the adaptive numerical integration of chemical kinetics using the computational singular perturbation (CSP) method. The strategy stores and reuses CSP quantities required to filter out fast dissipative processes, resulting in a non-stiff chemical source term. In particular, non-parametric regression on low-dimensional slow invariant manifolds (SIMs) in the chemical state space is used to approximate the CSP vectors spanning the fast chemical subspace and the associated fast chemical time-scales. The relevant manifold and its dimension varies depending on the local number of exhausted modes at every location in the chemical state space. Multiple manifolds are therefore tabulated, corresponding to different numbers of exhausted modes (dimensions) and associated radical species. Non-parametric representations are inherently adaptive, and rely on efficient approximate-nearest-neighbor queries. As the CSP information is only a function of the non-radical species in the system and has relatively small gradients in the chemical state space, tabulation occurs in a lower-dimensional state space and at a relatively coarse level, thereby improving scalability to larger chemical mechanisms. The approach is demonstrated on the simulation of homogeneous constant pressure H₂–air and CH₄–air ignition, over a range of initial conditions. For CH₄–air, results are shown that outperform direct implicit integration of the stiff chemical kinetics while maintaining good accuracy.     

Capturing multi-regime combustion in turbulent flames with a virtual chemistry approach

Multiple flame regimes are encountered in industrial combustion chambers, where premixed, stratified and non-premixed flame regions may coexist. To obtain a predictive tool for pollutant formation predictions, chemical flame modeling must take into account the influence of such complex flame structure. The objective of this article is to apply and compare two reduced chemistry models on both laminar and turbulent multi-regime flame configurations in order to analyze their capabilities in predicting flame structure and CO formation. The challenged approaches are (i) a premixed flamelet-based tabulated chemistry method, whose thermochemical variables are parameterized by a mixture fraction and a progress variable, and (ii) a virtual chemical scheme which has been optimized to retrieve the properties of canonical premixed and non-premixed 1-D laminar flames. The methods are first applied to compute a series of laminar partially-premixed methane-air counterflow flames. Results are compared to detailed chemistry simulations. Both approaches reproduced the thermal flame structure but only the virtual chemistry captures the CO formation in all ranges of equivalence ratio from stoichiometry premixed flame to pure non-premixed flame. Finally, the two chemical models combined with the Thickened Flame model for LES are challenged on a piloted turbulent jet flame with inhomogeneous inlet, the Sydney inhomogeneous burner. Mean and RMS of temperature and CO mass fraction radial profiles are compared to available experimental data. Scatter data in mixture fraction space and Wasserstein metric of numerical and experimental data are also studied. The analyses confirm again that the virtual chemistry approach is able to account for the impact of multi-regime turbulent combustion on the CO formation.     

Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of “minimal relaxation” of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to four test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.