Data Analysis for Direct Numerical Simulations of Turbulent Combustion

Datasets generated using Direct Numerical Simulation (DNS) are used to investigate the influence of local flame surface topology on global flame propagation. A mathematical framework based on Morse theory is presented and is shown to lead to a classification of all possible types of flame surface topology. A similar mathematical approach is shown to provide insight into the behaviour of the surface density function (SDF) and the displacement speed in the vicinity of flame pinch-off and pocket burnout events. DNS data for a pair of colliding premixed turbulent hydrogen–air flames is used to identify and locate topological points of interest and to determine their frequencies of occurrence on the flame surface. Further analysis of the dataset is carried out to evaluate terms of the SDF balance equation and the displacement speed in the presence of flame–flame interactions. Considerable insight is gained into the underlying mechanisms of flame propagation.

Dissipation elements provide a procedure for compartmentalizing scalar fields into physically meaningful sub-units which provides a direct measure for turbulent scales. Furthermore, dissipation elements enable a variety of additional ways of investigating non-local effects in reacting and non-reacting turbulent flows. After the underlying physical ideas of dissipation elements are explained and a parameterization of dissipation elements is defined, the method of detecting dissipation elements with gradient trajectories is explained and physical and numerical prerequisites are presented. Common characteristics of dissipation elements are interpreted and compared for a large range of selected reacting and non-reacting flow configurations. To provide the reader with a degree of familiarity, dissipation element statistics are then related to more commonly used methods of obtaining statistics. The additional benefit of using the dissipation element analysis in free shear flows is highlighted by using it as an alternative way of identifying turbulent core regions. Next, a dissipation element-based procedure for the local investigation of the turbulence–combustion interaction in the context of non-premixed flames is presented. The chapter is concluded with the application of a dissipation element statistics-based modeling procedure for computational fluid dynamics of a passenger car diesel engine, employing the previously gained insight into the structure of turbulent scalar fields.

This chapter provides a review of the basic ideas at the core of the Computational Singular Perturbation (CSP) method and the Tangential Stretching Rate (TSR) analysis. It includes a coherent summary of the theoretical foundations of these two methodologies while emphasizing their mutual interconnections. The main theoretical findings are presented in a systematic fashion. Their virtues and limitations will be discussed with reference to auto-ignition systems, laminar and turbulent premixed flames, and non-premixed jet flames. The material presented in the chapter constitutes an effective guideline for further studies.

Chapter 3 summarized the highlights of the concepts behind the CSP method and the TSR analysis. In this chapter, we will discuss a few applications of these techniques.

Direct numerical simulation (DNS) has become an important tool to predict and understand complex structures and behaviors of turbulent flames over the last two decades, enabled by the rapid growth of supercomputer power and development of more efficient and accurate Navier–Stokes equation solvers [1]. To predict the strongly nonlinear chemical kinetic processes and their interactions with the flow, detailed chemistry is typically employed in DNS while the computational cost is high even after aggressive mechanism reduction [2]. DNS on today’s supercomputer is capable to generate massive datasets, say tens or hundreds of terabytes, even in cleaned forms, such that systematic computational diagnostic tools need to be developed to extract salient information from the massive raw data. Canonical diagnostic methods based on individual scalars, such as temperature or a species concentration and their combinations (e.g., progress variable and mixture fraction) have been widely employed in previous studies. However, the use of such scalars typically requires semi-empirical criteria that need to be adjusted for different flame types and conditions, rendering them difficult to be automated for the processing of large flame data. Tools universally applicable to different flames and suitable for DNS data diagnostics are scarce and need to be developed. To address this need, a method of chemical explosive mode analysis (CEMA) was recently developed to systematically detect critical flame features for general reacting flows, particularly when local ignition, extinction, and premixed flame fronts are involved [3‐6]. CEMA has been demonstrated in elementary reactors, laminar flames and a variety of turbulent flames [3‐9]. It was found that CEMA-based criteria are rather robust and reliable for limit phenomena detection for both premixed and partially premixed flames, and the use of CEMA in computational diagnostics of different types of flames is discussed in the present chapter.

We propose the use of higher order tensors, and their decompositions, for efficient analysis of combustion direct numerical simulation (DNS) data. Turbulent combustion DNS data, being inherently multiscale and multivariate, pose many challenges and higher order tensors are a natural abstraction to organise, probe and analyse them. The chapter gives a high-level overview of prominent tensor decomposition methods, their interpretation, algorithmic challenges and desirable properties. Two examples of DNS analysis employing tensor decompositions are then presented. The first analysis, based on truncated higher order singular value decomposition (truncated HOSVD), also known as Tucker decomposition, allows significant, albeit lossy, compression of DNS data, which may be inevitable in the exascale computing era. The factors aiding, and impeding, compression and the implications in terms of element-wise error distributions are presented using three candidate DNS data sets. The second analysis is centred on higher order joint moment tensors, which are richly informative for multivariate non-Gaussian variables. An anomaly detection algorithm based on the decomposition of the fourth moment tensor is presented, and its ability in detecting localised auto-ignition kernels in a homogeneous charge compression ignition (HCCI) data set is examined.

Data-driven decomposition techniques are presented for the analysis and development of reduced-order models of complex flow dynamics. The Proper Orthogonal Decomposition (POD) produces optimal representations of the dynamics in the sense of the energy norm. Alternatively, Dynamic Mode Decomposition (DMD) efficiently extracts coherent dynamics based on eigendecompositions of linearized dynamics. An extension to the latter, the Higher Order Dynamic Mode Decomposition (HODMD) method uses time delays to develop efficient reduced models to represent complex dynamics in a nonintrusive manner. High-fidelity simulation results of a laboratory-scale single-element gas turbine combustor are used to demonstrate and evaluate the capabilities of the aforementioned decomposition techniques.

Dynamic Mode Decomposition (DMD) is able to decompose flow field data into coherent modes and determine their oscillatory frequencies and growth/decay rates, allowing for the investigation of unsteady and dynamic phenomena unlike conventional statistical analyses. The decomposition can be applied for the analysis of data having a broad range of temporal and spatial scales since it identifies structures that characterize the physical phenomena independently from their energy content. In this work, a DMD algorithm specifically created for the analysis of massive databases is used to analyze three-dimensional Direct Numerical Simulation of spatially evolving turbulent planar premixed hydrogen/air jet flames at varying Karlovitz number. The focus of this investigation is the identification of the most important modes and frequencies for the physical phenomena, specifically heat release and turbulence, governing the flow field evolution.

High-fidelity simulations of turbulent reacting flows enable scientific understanding of the physics and engineering design of practical systems. Whereas Direct Numerical Simulation (DNS) is the most suitable numerical tool to understand the physics, under-resolved and large-eddy simulations offer a good compromise between accuracy and computational effort in the prediction of engineering flows. This compromise speeds up the computations but reduces the space-and-time accuracy of the prediction. The objective of this chapter is to (i) evaluate the predictability horizon of turbulent simulations with chaos theory, and (ii) enable the space-and-time-accurate prediction of rare and transient events using a Bayesian statistical learning approach based on data assimilation. The methods are applied to DNS of Moderate or Intense Low-oxygen Dilution (MILD) combustion. The predictability provides an estimate of the time horizon within which the occurrence of ignition kernels and deflagrative modes, which are considered here as rare and transient events, can be accurately predicted. The accurate detection of ignition kernels and their evolution towards deflagrative structures are well captured on a coarse (under-resolved) grid when data is assimilated from a costly refined DNS. Physically, such an accurate prediction is important to understand the stabilization mechanism of MILD combustion. These techniques enable the space-and-time-accurate prediction of rare and transient events in turbulent flows by combining under-resolved simulations and experimental data, for example, from engine sensors. This opens up new possibilities for on-the-fly calibration of reduced-order models for turbulent reacting flows.

For new combustion control concepts such as Combustion Rate Shaping, a crank angle resolved model of the compression ignition (CI) combustion process is necessary. The complex CI combustion process involving fuel injection, turbulent flow, and chemical reactions has to be reproduced. However, to be suitable for control, it has to be computationally efficient at the same time. To allow for learning-based control, the model should be able to adapt to the current measurement data. This paper proposes two algorithms that model the CI combustion dynamics by learning a crank angle resolved model from past heat release rate (HRR) measurement data. They are characterized by short learning and evaluation times, low calibration effort, and high adaptability. Both approaches approximate the total HRR as the linear superposition of the HRRs of individual fuel packages. The first algorithm approximates the HRR of a single fuel package as a Vibe function and identifies the parameters by solving a nonlinear program having the squared difference between the measured HRR and the superposition as cost. The second algorithm approximates the individual packages’ HRRs as Gaussian distributions and estimates the parameters by solving a nonlinear program with the Kullback–Leibler divergence between the measurement and the superposition as cost function using the expectation–maximization algorithm. Both algorithms are validated using test bench measurement data.

Following the rapid and continuous progress of computing power, allowing for increasing the mesh resolution in large eddy simulation (LES), new modeling strategies appear which are based on a direct treatment of the now well resolved, but still not fully resolved scalar signals. Along this line, deconvolution or inverse filtering, either based on discrete or iterative operators, is first discussed. Recent results obtained from a direct numerical simulation (DNS) database and LES of a premixed turbulent jet flame are presented. The analysis confirms the potential of deconvolution to approximate the unclosed non-linear terms and the SGS fluxes. Then, the introduction of machine learning in turbulent combustion modeling is illustrated in the context of convolutional neural networks.

The interpretation of high-dimensional data, like those obtained from Direct Numerical Simulations (DNS) of turbulent reacting flows, constitutes one of the biggest challenges in science and engineering. Although these simulations are a source of key information to advance the knowledge of turbulent combustion, as well as to develop and validate modeling approaches, the dimensionality of the data often limits the full opportunity to leverage the detailed and comprehensive information stored in datasets. The Principal Component Analysis (PCA) and its local formulation (LPCA) are widely used in many fields, including combustion. During the last 20 years, they have been used in combustion for the identification of low-dimensional manifolds, data analysis, and development of reduced-order models. Lower dimensional structures, either global or local, can provide better insights on the underlying physical phenomena, and lead to the formulation of high-fidelity models. This chapter aims to offer to the reader a comprehensive introduction of the PCA potential for data analysis, firstly introducing the main theoretical concepts, and then going through all the required computational steps by means of a MATLAB® code. Finally, the methodology is applied to data obtained from a DNS of a turbulent reacting non-premixed n-heptane jet in air. The latter can be regarded as an optimal case for data analysis because of the complex physics characterized by turbulence–chemistry interaction and soot formation.

Gene Expression Programming (GEP) has been used successfully for modelling the unclosed terms in the context of Reynolds Averaged Navier–Stokes (RANS) and Large Eddy Simulation (LES)-based turbulence modelling. In contrast to deep-learning-based methodologies, this approach has the advantage that the model can be documented in the form of a mathematical expression; it can be interpreted and easily implemented in existing solvers. Recently, application of GEP to a priori LES modelling has demonstrated the efficiency of the approach to find high fidelity LES closures. The present contribution explains the methodology, reviews recent work in the field and focuses on the robustness of the method and the scope for future efficiency improvements, by applying it to the modelling of the unclosed stress tensor in turbulent premixed statistically planar flames.

In this chapter we demonstrate how supervised deep learning techniques can be used to construct models for the filtered progress variable source term necessary for large eddy simulation (LES). The source data for the model is a direct numerical simulation (DNS) of a reacting flow in a low swirl burner configuration. Filtered quantities taken from the DNS data are used to train a deep neural network (DNN)-based model. An efficient data sampling strategy was devised to ensure that a uniform representation of all the states observed in the filtered DNS data are equally present in the training dataset. A-priori testing of the DNN-based model highlights the representative power of DNN to accurately reproduce the filtered reaction progress variable source term over a range of scales and various flame regimes as seen in an industrial burner.