This study outlines and discusses model reduction problems. The concept of Singularly Perturbed Vector Fields (SPVF) is revisited as a framework for handling the decomposition of motions and, in particular, for addressing the fundamental problem of initial-data consistency in reduced models. This issue arises because the initial conditions generally do not lie on the slow invariant manifold that approximates the reduced system. To address this, this study demonstrates how the method of Intrinsic Low-Dimensional Manifolds (ILDMs) can be employed both to approximate the slow manifold and to generate consistent initial data. Two classical benchmark examples in model reduction theory are examined to verify and discuss the approach. The Lindemann mechanism is used to illustrate how SPVF theory reconciles the standard asymptotic limits and thereby captures both pressure dependence and variations in reaction order. The Michaelis-Menten model serves as a key example where standard reduction approaches - such as QSSA and PEA - may fail under certain asymptotic conditions. The SPVF framework provides a systematic way to understand these limitations by clarifying the asymptotic structure and its parameter dependencies. Meanwhile the ILDM method not only offers an efficient way to approximate the slow manifold but also supplies consistent initial data projection procedure, ensuring a physically and mathematically sound reduced system.

The highly transient relaxational diffusive-thermal oscillations of flat burner-stabilized flames can be very attractive to probe the performance of detailed reaction mechanisms in the regimes close to ignition/extinction. In such regimes, certain reaction zones can travel over distances of the order of 10 mm and this raises an important question if one dimensional numerical models can be accurate in describing them. The question of quantitative comparison of modeling and experiments becomes crucial to study, to understand these regimes and to utilize them for validation. In this work, we experimentally investigate relaxational oscillations of methane-air flames on a flat porous burner with a surrounding nitrogen co-flow and perform fully resolved 2D numerical simulations of the same burner configuration, using a detailed reaction mechanism and molecular diffusion model, buoyancy and radiation, alongside corresponding experiments. The focus is on the effect of the nitrogen co-flow on the flame oscillations, which can only be studied numerically in 2D simulations due to the mutual effect of the complex flow field and flame dynamics. The results of both numerical and experimental approaches are found to be in quantitative agreement. They show that there is an optimal co-flow velocity that removes the secondary diffusion flame and extinguishes the edge flame settled in the stagnation flow region. This optimal regime makes the flame flatter and closer to a one-dimensional configuration and this is a most favorable condition for validation of kinetic mechanisms. The detailed data from the simulations will guide the design of the next generation of the burner configurations to study the kinetics and dynamics of complex fuels required for a sustainable energy transition.Novelty and Significance StatementThe novelty of this research lies in the synergy of these modeling, computations with experimental measurements, allowing both parametric studies of the oscillation regime and deeper insights into the flame structure. These results are significant because they allow to develop more accurate burner configurations for studying flames near extinction and ignition conditions, which will be an important task for more complex fuels from renewable sources required for a sustainable energy transition. Ultimately, our understanding of the interplay between chemistry and diffusion controlled combustion regimes under transient conditions can be approved and validation data for e.g. chemical reaction mechanisms can be generated. The latter becomes extremely important since efficiency and pollutant mitigation issues require lean and chemistry controlled combustion processes used in the combustion facilities. Thus understanding, optimization and control of such regimes has become a crucial point for further development of the sustainable combustion.

The limits of existence of a steady planar methane-air flame stabilized on a flat porous burner at normal and elevated pressure (2, 4 and 6 bar) have been experimentally and numerically investigated. In particular, the critical conditions for the blow-off and diffusive-thermal oscillations have been determined in the plane of parameters: mass flow rate vs. equivalence ratio. The Hopf frequency of oscillations is measured at the diffusive-thermal oscillation boundary. The results of numerical simulations, undertaken with the use of detailed reaction mechanisms, such as GRI, FFCM, USC II, SanDiego, and Aramco, show that, despite the good qualitative agreement with the experimental data, the relative quantitative difference between the numerical simulations and the experimental measurements is quite large. It is of the order of several tens of percent and is especially evident when the measurements are performed away from stoichiometry and under high pressures. In order to verify and validate detailed reaction mechanisms, in addition to the standard tests such as measurement of laminar burning velocity, ignition delay time and extinction strain rate, it is necessary to obtain a wider range of experimental data. It is especially important at elevated pressures and high temperatures. Determining the characteristics of the diffusion-thermal oscillations is a suitable way to achieve this. Novelty and Significance Statement For the first time, we experimentally found the critical conditions for the blow-off and onset of diffusive thermal pulsating instabilities, as well as the characteristics of diffusive thermal oscillations for the burner stabilized methane-air flames at elevated pressure. These data were compared for the first time with the predictions of several detailed reaction mechanisms to verify their performance under such conditions. The novel findings reported in this work on the regions of existence of stable combustion regimes are significant for the design of practical burners, while the data on the conditions and characteristics of the critical phenomena will facilitate the development of accurate and efficient mechanism of methane combustion.

This paper devises and applies a method for determining consistent, system -adapted reaction-diffusion manifolds (REDIM) for turbulent partially -premixed combustion. The method is an extension of a previous technique which was limited to laminar flames. It adapts a REDIM to the scalar gradients present in a given reacting flow by an iterative procedure, in which gradients from REDIM-reduced simulations are used to create a new and improved REDIM. An application of the method is demonstrated using large eddy simulations (LES) of a turbulent partially -premixed methane/air flame as the "gradient estimate generating"flame simulation. Compared to the previous laminar flame studies, the turbulent flame features less sharp gradient -scalar correlations because of the strong scattering of data. Methods for capturing the scatter in scalar -gradient correlations are applied, allowing us to assess the influence of the scatter on the resulting REDIM. Also, the fact that turbulent flame simulations often involve spatial filtering operations which tend to reduce the magnitude of the gradients is taken into account by comparing the REDIM output to quasi -direct numerical simulation (qDNS) data. It is found that the gradients devised from the filtered simulations are by a factor of 2-3 below those of the more fully resolved qDNS. Nevertheless, the REDIM devised from the LES predicts the states of the qDNS with good accuracy. Overall, observations show that the method can produce realistic reduced models also for systems with complex interactions of chemical reaction, molecular transport and flow, like partially -premixed turbulent flames.

In this paper, the dynamics of diffusion-thermal oscillations of burner-stabilized methane–air flames are numerically and experimentally investigated. High speed imaging and laser induced fluorescence of OH radicals are used as experimental techniques, while a hierarchy of models is employed in numerical simulations: a system of ordinary differential equations describing dynamics of flame location and temperature, models with one-step and detailed reaction mechanisms. It is demonstrated that for small (near the neutral stability boundary) and moderately relaxational oscillations, all three approaches are able to describe the flame dynamics however with different levels of detailization. This is due to the fact that in these regimes the flame front structure remains similar, although perturbed, to that observed in freely propagating combustion waves. However, this flame structure breaks in the regime of highly relaxational oscillations, which can only be described within the detailed reaction mechanisms. The combustion front splits into the high and low-temperature reaction zones, which are shown to be highly and weakly sensitive to the diffusive-thermal pulsations. The high temperature reaction zone is blown away and extinguishes in the course of flame pulsation, and the low temperature zone remains near the burner surface and causes re-ignition of the combustion front. Prospects of further investigations are discussed.

In this work, the overall activation energy of the combustion of lean hydrogen–methane–air mixtures (equivalence ratio φ = 0.7−1.0 and hydrogen fraction in methane α=0, 2, 4) is experimentally determined using thin-filament pyrometry of flames stabilised on a flat porous burner under normal conditions (p=1 bar, T = 20 °C). The experimental data are compared with numerical calculations within the detailed reaction mechanism GRI3.0 and both approaches confirm the linear correlation between mass flow rate and inverse flame temperature predicted in the theory. An analysis of the numerical and experimental data shows that, in the limit of lean hydrogen–methane–air mixtures, the activation energy approaches a constant value, which is not sensitive to the addition of hydrogen to methane. The mass flow rate for a freely propagating flame and, thus, the laminar burning velocity, are measured for mixtures with different hydrogen contents. This mass flow rate, scaled over the characteristic temperature dependence of the laminar burning velocity for a one-step reaction mechanism, is found and it can also be used in order to estimate the parameters of the overall reaction mechanisms. Such reaction mechanisms will find implementation in the numerical simulation of practical combustion devices with complex flows and geometries.

Partially-premixed flames (PPFs) incorporate effects of both premixed and non-premixed types of reaction zones. The modelling of PPFs using manifold-based model reduction methods faces some inherent difficulties due to the underlying assumptions of a-priori identification of the type of combustion system. In this work, the reaction-diffusion manifold (REDIM) model reduction method is applied to study PPFs. The REDIM method requires minimal prior knowledge about the type of combustion system, which makes it a suitable method for studying PPFs. It allows incorporating system-specific diffusion (gradients) terms in a generic way so that the manifold can evolve according to the diffusion related information provided by the combustion system. In this way, a prior identification of the type of combustion system is no longer needed.This work utilises an iterative methodology to generate REDIM chemistry tables so that the reduced manifold can be iteratively converged very close to the detailed manifold according to the gradients of the reduced coordinates provided by the physical combustion system in each iteration step. In addition, a new method is proposed to provide the gradient estimates of the reduced coordinates during the generation of REDIM from the scattered gradient data in REDIM reduced CFD calculations. Laminar triple flames, a special case of PPFs, with two types of mixture fraction gradients are selected as the target cases to assess the presented iterative methodology. REDIM reduced calculations are compared with simulations based on detailed finite-rate kinetics. It is found that in the final iteration steps, temperature and all considered major and minor species mass fraction profiles are very well predicted by the REDIM reduced calculations.

The global quasi-linearization (GQL) is used as a method to study and to reduce the complexity of mathematical models of mechanisms of chemical kinetics. Similar to standard methodologies, such as the quasi-steady-state assumption (QSSA), the GQL method defines the fast and slow invariant subspaces and uses slow manifolds to gain a reduced representation. It does not require empirical inputs and is based on the eigenvalue and eigenvector decomposition of a linear map approximating the nonlinear vector field of the original system. In the present work, the GQL-based slow/fast decomposition is applied for different combustion systems. The results are compared with the standard QSSA approach. For this, an implicit implementation strategy described by differential algebraic equations (DAEs) systems is suggested and used, which allows for treating both approaches within the same computational framework. Hydrogen–air (with 9 species) and ethanol–air (with 57 species) combustion systems are considered representative examples to illustrate and verify the GQL. The results show that 4D GQL for hydrogen–air and 14D GQL ethanol–air slow manifolds outperform the standard QSSA approach based on a DAE-based reduced computation model.

In this work the performance of various detailed reaction mechanisms of combustion of methane–hydrogen fuel mixtures is studied. The investigation is focused on the burner stabilized flame configuration and is undertaken for the normal pressure conditions, which is a starting point for the analysis of the combustion chemistry at elevated pressures met in rocket engine chambers. The study presents the experimental setup, computational approach and results of comparison, which allow us to access the properties of complex combustion systems in variety of dynamical regimes. The comparison of the predicted and measured characteristics of these regimes is used to test the performance of the reaction models. In particular, the critical behavior, e.g. for blow-up, onset of pulsations and quasi-steady flame front propagation and bifurcation between these regimes may be used to characterize and to study properties of combustion systems with mixed fuels. A neutral stability boundary for onset of pulsations is suggested to be used in this study. A method for automatic identification of the boundary is proposed and implemented with several detailed mechanisms. Although for pure methane the results look acceptable, they show gradual divergence with the increase of percentage of hydrogen addition. Significant quantitative differences between experiments and modeling with respects to frequencies of oscillations are reported even for 2:1 ratio of hydrogen to methane in the unburned mixture. The results of the work indicate that current reaction mechanisms need to be improved in order to gain the quantitative predictability of methane–hydrogen–air combustion at normal pressure before proceeding to the conditions fuel-oxygen high pressure combustion more relevant for rocket chamber.

The Global Quasi-linearization (GQL) approach has been developed for the dimension reduction of the chemical kinetics, which aim at speeding up the numerical simulation of reacting flows. GQL-RedChem is a MatLABbased package that integrates the homogeneous reacting systems, formulated mathematically as a system of Ordinary Differential Equations (ODEs). The package provides with an approximation of the fast/slow decomposition linear basis, describing a reacting source term by a system of Differential Algebraic Equations (DAEs). The GQL method can be applied for reacting systems with any complexity, and the GQL decomposition basis can be generated in a generic and automatic manner.

The paper analyzes numerically the development of the freely expanding flame before the transition to detonation in three-dimensional space. The risks related to the transition to detonation in unconfined space arise at launch places and near-Earth space during rocket engine accidents. Clear and adequate models for such scenarios are in demand for the elaboration of safety measures. Here it is shown that the process of transition to detonation involves multidimensional effects related to the generation of compression waves and their amplification when interacting with the developing flame. The necessary conditions for deflagration-to -detonation transition are flame acceleration up to near-sonic speed and generation of transversal compression waves (propagating along the flame front). So, while the subsonic stage of flame acceleration can be successfully described by the quasi one-dimensional model supplemented by a known self-similar law of the flame area growth (folding factor), the model for the pre-detonation stage should take into account considered multidimensional effects. The presented results imply that there is no unambiguous relation between the transition to detonation and the folding factor.

A chemical kinetic mechanism for ammonia decomposition and combustion from the CRECK modeling group is analyzed in this study with respect to the fast and slow chemical processes and the results are used to generate skeletal mechanisms for different applications. As examples two regimes of system parameters are chosen: values typical for engine exhaust gas systems (low temperature ammonia decomposition and NOX reduction) and a high temperature ignition problem studied by flow reactors and shock tube experiments. The mechanism is first tested and compared to existing experimental results. A time scale analysis is implemented to study the mechanism. A combination of local and global analysis based on the Global Quasi-Linearisation (GQL) concept is suggested and employed to estimate the number of relatively fast and slow processes governing the system dynamics. A GQL decomposition into invariant sub-spaces is used to determine the complimentary sub-sets of relatively fast and slow reactions. These are used to devise specific skeletal mechanisms for the chosen applications, which are verified in the considered parametric ranges. The suggested approach can be used to identify rate limiting processes, study how the coupling of relatively fast and slow (slaved) reactions proceed. The latter is extremely important for mechanism development and model reduction. The results show a very good agreement not only with respect to the ignition delay times, but also with the results from flow reactor experiments.

In this work the diffusive–thermal pulsations of the burner stabilized methane-air flames are investigated experimentally, by analysing the OH∗ chemiluminescence signal, and numerically within the model with various detailed reaction mechanisms. The employment of the nitrogen co-flow configuration to isolate the flame from the surrounding air allows us to obtain the experimental data of high fidelity such that the difference between the numerical data calculated with different reaction mechanisms is greater then the experimental uncertainty, demonstrating that the proposed technique can be used to verify the reaction mechanisms. Sensitivity analyses are carried out and it is shown that the steady and pulsating regimes of combustion has different although partly overlapping subsets of most important elementary reactions, especially for the case of relaxational oscillations in which the combustion front exhibits stages close to quenching and re-ignition. The suggested configuration can thus be used to access "transient combustion" and to gain an additional information to verify mechanisms.

In this work, the determination of the apparent activation energy of a global chemical reaction mechanism of the methane-air flames is revisited. The one-step formulation allows to derive the theoretical background for the method to measure the activation energy within the burner stabilised flame setup. The validity of this approach is demonstrated by using the numerical simulations with the detailed reaction model and direct thin filament pyrometry measurements of the temperature distribution in flame. The combination of numerical and experimental approaches allows us to find the activation energy for various mixture compositions. The prediction of numerical simulations and measured values of the activation energy is found to be in good agreement with each other and the data known from the literature. It is demonstrated that two critical phenomena need to be taken into account to obtain a reliable estimate of the activation energy: the flame blow-off and the onset of the diffusive-thermal instabilities. The effect of these critical events on the accuracy of the measurements is discussed as well as prospects of further investigation.

The Global Quasi-Linearization (GQL) method for model reduction of chemical kinetics is applied to describe the very sensitive regime of the onset of the thermal-diffusion oscillations of the rich hydrogen-air flames. This type of flame propagation is characterized by a complex interaction of chemical reaction, ordinary diffusion, as well as thermal diffusion. It represents a very challenging phenomenon with respect to both modeling and computation. Therefore, it is chosen as a test case for the GQL model reduction method. We show that the GQL matrix obtained for the auto-ignition problem can be applied to study the diffusive-thermal instabilities of premixed combustion systems. In order to demonstrate that the method is generic, the invariance of the GQL basis is tested on a number of well-known and established mechanisms. The results show that the GQL basis-based low-dimensional (4D) manifolds of slow motions can successfully describe rich hydrogen-air flames and reproduce all important characteristics for both the onset of pulsations and the properties of oscillatory regimes of the flame propagation. Moreover, estimations of critical parameters, e.g., critical pressure for the onset of pulsations, based on the GQL reduced models, show significantly better performance if compared to the differences between detailed mechanisms of chemical kinetics.

The problem of sudden flame acceleration in narrow channels and subsequent Deflagration to Detonation Transition (DDT) is revisited. The hydrogen-oxygen combustion system is considered both experimentally and numerically. The flame is initiated at one open end of a square narrow channel of different widths (4, 8, 10 and 20 mm) and propagates to another open end. Experimental results show a high sensitivity to the mixture composition: for higher mixture reactivity, an abrupt flame acceleration is reported. In 4 x 4 mm 2 channel, when the mixture composition is close to stoichiometry (2:1), the DDT is observed without any evidence of shock waves prior to the detonation transition. Two complementary sub-models are combined to account for the effect of walls on flow and on flame front geometry. A pseudo-spectral numerical scheme is used to integrate the system of equations both in time and in space. The simulation results show how the suggested model reproduces main features of the phenomenon. A simple criterion is suggested for the onset of the flame self-acceleration. There exists a critical flame folding factor for the DDT, which is an invariant of the system. The reported critical folding factor is compared to the theoretical estimation introduced by Gordon et al (2020). (c) 2021 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Due to strict environmental regulations in the automotive sector selective catalytic reduction with urea-water-solution is often used to reduce nitrogen oxide emissions. To improve and solve technical problems, like the formation of solid residuals, detailed simulations of the exhaust gas system are used. Such simulations have high computational needs due to the multitude of different processes involved at strongly differing spatial and time scales. For this reason, simplified or reduced models are applied and urea decomposition is often modeled with a vapor pressure curve fitted to experimental results. This method is incapable of describing the formation of solid residuals. A recently developed chemical mechanism for urea decomposition in the liquid phase is used in this work to simulate the decomposition of spherical droplets and planar wall films of urea-water-solution in exhaust gas. For droplets, the overall behavior with liquid chemistry is similar for the whole range of ambient temperatures. Less ammonia and nearly no isocyanic acid is produced compared to the evaporation model and residuals of liquid biuret and solid triuret remain. For wall films, the behavior as well as the composition of the residuals strongly depends on the temperature. The mechanism predicts the production of a similar amount of ammonia but less isocyanic acid compared to the evaporation model. The remaining mass loss is mostly composed of cyanuric acid. It is found that the process of urea decomposition is much slower with liquid chemistry and complete decomposition only happens at 673 K or above. The analysis of the chemical time scales results in a skeletal mechanism for the droplet decomposition with 6 out of 13 reactions among 7 out of 13 species that can describe the whole process with good accuracy, where the mass of the produced gases deviates by less than 5%.

Selective catalytic reduction (SCR) process with urea-water solution (UWS) is often used in automotive industry to decrease emissions of nitric oxides (NO x) in the exhaust gas. In this process the urea from UWS decomposes to isocyanic acid and ammonia, where the latter is needed to increase the efficiency of the NO xreduction on the catalyst surface. Along with the advantages of using UWS several drawbacks reduce the performance of a SCR system. Incomplete decomposition of urea leads to a formation of residuals affecting the efficiency of the exhaust gas systems. Therefore, the complete decomposition of urea and homogeneous distribution of the resulting ammonia in front of the SCR catalyst represent main challenges in improving the SCR technology. In order to investigate the process of the urea decomposition a detailed chemical kinetic mechanism in the liquid phase is employed. The results are compared with a commonly used approach to model urea decomposition as an evaporation with a following decomposition reaction in the gas phase. It is shown that by using such a mechanism, the decomposition of urea and the gas phase composition with the urea decomposition products can be described more accurately. However, implementing these mechanisms in computations (in CFD approaches) requires a large amount of computational (CPU) time and memory. The method of Reaction Diffusion Manifolds (REDIMs) is implemented for the reduction of the detailed chemical kinetics in the stage of urea decomposition such that the distribution of products of the urea decomposition can be captured accurately in the gas phase with only two reduced variables instead of the 7 gas phase species of the original model.

We apply ridge analysis for reaction zone characterisation in non-premixed combustion. Generalizing the topographical mountain-ridge archetype, ridges are low-dimensional regions where a scalar field has elevated values (local maxima) along intrinsically given directions. Ridge analysis of a chemical source term field can identify the reaction zone location, and also describe the spatial extent and shape of the zone. The concept is explored for non-premixed combustion by identifying ridges of a chemical source term field in three-dimensional Direct Numerical Simulation (DNS) data of a non-premixed, strongly diluted turbulent hydrogen/air flame. Properties of the ridge-based flame front are presented and discussed. The method reveals different reaction zone geometry types in the DNS data, including the conventional flame sheet, but also other, less common reaction zone structures like filaments or patches. A significant part of the combustion occurs in filament-shaped reaction zones, rather than in "classical" flame sheets. Comparison of the reaction zone derived from ridge analysis and the reaction zone derived from the stoichiometric iso-surface of mixture fraction shows profound differences in location, and also in reaction-front conditioned statistics of scalars. These findings, along with the intrinsic nature of the ridge concept — requiring no definitions of thresholds, iso-values or similar — confirm the usefulness of the method for the analysis of reacting flows.