A neural network (NN) aided model is proposed for the filtered reaction rate in moderate or intense low-oxygen dilution (MILD) combustion. The framework of the present model is based on the partially stirred reactor (PaSR) approach, and the fraction of the reactive structure appearing in the PaSR is predicted using different NN's, to consider both premixed and non-premixed conditions while allowing the use of imbalanced training data between premixed and non-premixed combustion direct numerical simulation (DNS) data. The key ingredient in the present model is the use of local combustion mode prediction performed by using another NN, which is developed in a previous study. The trained model was then assessed by using two unknown combustion DNS cases, which yields much higher dilution level (more intense MILD condition) and higher Karlovitz number than the DNS cases used as training data. The model performance assessment has been carried out by means of the Pearson's correlation coefficient and mean squared error. For both the present model and zeroth-order approximated reaction rate, the correlation coefficient with the target values shows relatively high values, suggesting that the trend of predicted field, by the present model and zeroth-order approximation, is well correlated with the actual reaction rate field. This suggests that the use of PaSR equation is promising if the fraction of the reactive structure is appropriately predicted, which is the objective in the present study. On the other hand, substantially lower mean squared error is observed for a range of filter sizes for the present model than that for the zeroth-order approximation. This suggests that the present filtered reaction rate model can account for the SGS contribution reasonably well.

Multi-scale analysis of turbulence-flame interaction is performed using experimental data sets from three methane- and propane-fired premixed, turbulent V-flames, at an approach flow turbulent Reynolds number of 450 and a ratio of r.m.s. fluctuating velocity from the mean to laminar flame speed of between 2.1 and 3.0, straddling the border between corrugated flamelets and thin reaction zone in the Borghi-Peters diagram. The measurements were made in the plane of a single laser sheet using stereo particle image velocimetry SPIV and planar laser-induced fluorescence to measure three orthogonal components of velocity and flame OH. Methods to approximate the remaining, unmeasured, out of plane derivatives are described. The instantaneous SPIV images were bandpass filtered at user-specified characteristic length scales L_ω and L_s (for vorticity and strain rate, respectively) resulting in instantaneous bandpass-filtered velocity fields, u̲_b^L_ω and u̲_b^L_s, which were further analysed to give the bandpass filtered vorticity field, ω̲^L_ω=∇×u̲_b^L_ω, the strain-rate field, e_ij^L_s, and the tangential strain rate field a_T^L_s. This work quantifies two aspects of turbulence-flame interaction. The first aspect is that of the flame interaction of eddies of size L_s on the turbulence, as found by the statistics of the alignment of vorticity with strain rate. We find that vortical eddies with scale about L_ω=2δ_th (where δ_th is the flame thickness) are stretched by L_s structures which are larger than about 2 L_ω, with this factor broadly true also for vortical eddies of scales L_ω=4δ_th and L_ω=6δ_th. Within the limitations of the data set, these findings are consistent with those in the literature on reacting and non-reacting flows, suggesting that the premixed flame has had little influence on the vortex stretching mechanism. The second aspect of turbulence-flame interaction examined is that of flame surface-averaged tangential strain rate imparted by eddies. Eddies with length scales L_s smaller than 3 or 4δ_th are likely to have the strongest individual contribution but eddies of this length scale and smaller may contribute only about 1/5th of the total tangential strain rate. This is to be compared with the value of 10% that has been reported in the literature based on analysis of DNS predictions of premixed flames at turbulent Reynolds numbers up to 110. Eddies with length scale L_s larger than about 20δ_th contribute a negligible amount to the total tangential strain rate. We have found no evidence that the Lewis number up to about 1.8 has an observable effect, but this may reflect the limitations of the current experiment. In the context of large eddy simulations (LES) of premixed combustion, these results are preliminary experimental evidence supporting the suggestion that resolving turbulence scales down to a few multiples of δ_th might be adequate to capture much of the flame straining caused by turbulence.

In this chapter, the research dedicated to moderate or intense low-oxygen dilution (MILD) combustion (also called flameless combustion) that relied on direct numerical simulations (DNS) is summarized. In particular, the various DNS carried out are detailed and three different configurations are considered: the autoigniting mixing layer between fuel and hot and diluted oxidizer, the premixed MILD combustion resulting from internal exhaust gas recirculation, and the nonpremixed MILD combustion with internal exhaust gas recirculation. Focus is placed here on different aspects of MILD combustion. First, works that relate to the onset of MILD combustion and the apparition of the initial ignition kernels are discussed, in particular, a summary is provided on the findings that show the particular physics of MILD combustion, where the initial ignition kernels are mostly related to the distribution of mixture fraction and recirculating radicals. Subsequently, the identified physical mechanisms involved in the development of those ignition kernels are summarized. In particular, focus is placed on the balance between ignition and deflagrative mechanisms. Using different analysis methods, the works summarized here show that, while there is a coexistence between ignition and deflagration, ignition is the main contributor to the overall heat release. Finally, the implications of these findings on the modeling of MILD combustion are discussed through various studies that assessed a priori different modeling frameworks for MILD combustion. In those, models that capture this essential and dominant ignition behavior of MILD combustion were shown to be more accurate.

We propose a physics-constrained machine learning method - based on reservoir computing - to time-accurately predict extreme events and long-term velocity statistics in a model of chaotic flow. The method leverages the strengths of two different approaches: empirical modelling based on reservoir computing, which learns the chaotic dynamics from data only, and physical modelling based on conservation laws. This enables the reservoir computing framework to output physical predictions when training data are unavailable. We show that the combination of the two approaches is able to accurately reproduce the velocity statistics, and to predict the occurrence and amplitude of extreme events in a model of self-sustaining process in turbulence. In this flow, the extreme events are abrupt transitions from turbulent to quasi-laminar states, which are deterministic phenomena that cannot be traditionally predicted because of chaos. Furthermore, the physics-constrained machine learning method is shown to be robust with respect to noise. This work opens up new possibilities for synergistically enhancing data-driven methods with physical knowledge for the time-accurate prediction of chaotic flows.

We explore the possibility of combining a knowledge-based reduced order model (ROM) with a reservoir computing approach to learn and predict the dynamics of chaotic systems. The ROM is based on proper orthogonal decomposition (POD) with Galerkin projection to capture the essential dynamics of the chaotic system while the reservoir computing approach used is based on echo state networks (ESNs). Two different hybrid approaches are explored: one where the ESN corrects the modal coefficients of the ROM (hybrid-ESN-A) and one where the ESN uses and corrects the ROM prediction in full state space (hybrid-ESN-B). These approaches are applied on two chaotic systems: the Charney-DeVore system and the Kuramoto-Sivashinsky equation and are compared to the ROM obtained using POD/Galerkin projection and to the data-only approach based uniquely on the ESN. The hybrid-ESN-B approach is seen to provide the best prediction accuracy, outperforming the other hybrid approach, the POD/Galerkin projection ROM, and the data-only ESN, especially when using ESNs with a small number of neurons. In addition, the influence of the accuracy of the ROM on the overall prediction accuracy of the hybrid-ESN-B is assessed rigorously by considering ROMs composed of different numbers of POD modes. Further analysis on how hybrid-ESN-B blends the prediction from the ROM and the ESN to predict the evolution of the system is also provided.

We present an Auto-Encoded Reservoir-Computing (AE-RC) approach to learn the dynamics of a 2D turbulent flow. The AE-RC consists of an Autoencoder, which discovers an efficient manifold representation of the flow state, and an Echo State Network, which learns the time evolution of the flow in the manifold. The AE-RC is able to both learn the time-accurate dynamics of the flow and predict its first-order statistical moments. The AE-RC approach opens up new possibilities for the spatio-temporal prediction of turbulence with machine learning.

We extend the Physics-Informed Echo State Network (PI-ESN) framework to reconstruct the evolution of an unmeasured state (hidden state) in a chaotic system. The PI-ESN is trained by using (i) data, which contains no information on the unmeasured state, and (ii) the physical equations of a prototypical chaotic dynamical system. Non-noisy and noisy datasets are considered. First, it is shown that the PI-ESN can accurately reconstruct the unmeasured state. Second, the reconstruction is shown to be robust with respect to noisy data, which means that the PI-ESN acts as a denoiser. This paper opens up new possibilities for leveraging the synergy between physical knowledge and machine learning to enhance the reconstruction and prediction of unmeasured states in chaotic dynamical systems.

Direct numerical simulations (DNS) of statistically planar flames at moderate and high Karlovitz number (Ka) have been used to perform an a priori evaluation of a presumed-PDF model approach for filtered reaction rate in the framework of large eddy simulation (LES) for different LES filter sizes. The model is statistical and uses a presumed shape, based here on a beta-distribution, for the sub-grid probability density function (PDF) of a reaction progress variable. Flamelet tabulation is used for the unfiltered reaction rate. It is known that presumed PDF with flamelet tabulation may lead to over-prediction of the modelled reaction rate. This is assessed in a methodical way using DNS of varying complexity, including single-step chemistry and complex methane/air chemistry at equivalence ratio 0.6. It is shown that the error is strongly related to the filter size. A correction function is proposed in this work which can reduce the error on the reaction rate modelling at low turbulence intensities by up to 50%, and which is obtained by imposing that the consumption speed based on the modelled reaction rate matches the exact one in the flamelet limit. A second analysis is also conducted to assess the accuracy of the flamelet assumption itself. This analysis is conducted for a wide range of Ka, from 6 to 4100. It is found that at high Ka this assumption is weaker as expected, however results improve with larger filter sizes due to the reduction of the scatter produced by the fluctuations of the exact reaction rate.

Subgrid correlation of mixture fraction, Z, and progress variable, c, is investigated using direct numerical dimulation (DNS) data of a hydrogen lifted jet flame. Joint subgrid behaviour of these two scalars are obtained using a Gaussian-type filter for a broad range of filter sizes. A joint probability density function (JPDF) constructed using single-snapshot DNS data is compared qualitatively with that computed using two independent β-PDFs and a copula method. Strong negative correlation observed at different streamwise locations in the flame is captured well by the copula method. The subgrid contribution to the Z–c correlation becomes important if the filter is of the size of the laminar flame thickness or larger. A priori assessment for the filtered reaction rate using the flamelet approach with independent β-PDFs and correlated JPDF is then performed. Comparison with the DNS data shows that both models provide reasonably good results for a range of filter sizes. However, the reaction rate computed using copula JPDF is found to have a better agreement with the DNS data for large filter sizes because the subgrid Z–c correlation effect is included.