Abstract

Intrinsic combustion instabilities manifest as wrinkles or cellular structures on spherically expanding flame fronts. They are a fundamental aspect of the physics of flames and can be practically significant for some mixtures, such as those containing hydrogen (H2). In the case of spherical flames, they are studied visually using high-speed cameras, while the use of pressure traces is significantly less explored. This work investigates the use of time series classification (TSC) algorithms for detecting flame wrinkling in spherical flames using pressure traces recorded in a constant volume combustion chamber (CVCC). Some algorithms are observed to have very high classification accuracies (>90%), with 94.5% for the ROCKET (Random Convolutional Kernel Transform) classifier, 92.5% for HIVE-COTE (Hierarchical Vote Collective of Transformation-based Ensembles), and 96.1% for 1-nearest neighbor (1-NN) with DTW (dynamic time warping). The approach potentially provides an alternative methodology for instability detection, particularly where optical access is unavailable.

1 Introduction

Combustion instabilities amplify small perturbations in a flame into larger disturbances. They are generally categorized as intrinsic, chamber, and system instabilities [1]. Intrinsic instability is the growth of flame front perturbations due to mechanisms that are inherent to the flame, regardless of its interactions with the combustion chamber or other systems. Intrinsic instabilities are most relevant when the focus is on the combustion properties of a mixture without reference to a specific application. They have been the subject of experimental and theoretical studies for decades due to their fundamental nature and impact on practical systems [2]. While turbulence can dominate the effects of intrinsic combustion instabilities for most mixtures, flame wrinkling due to instabilities still has a significant enough effect in hydrogen (H2) combustion that it has to be taken into account in addition to turbulence [3,4]. Several theoretical models explaining the wrinkling of flames due to intrinsic instabilities have been proposed over the years, seeking to describe the underlying physical mechanisms and the relation between mixture and flame properties with wrinkling. Starting with the work of Darrieus [5] and Landau [6] on hydrodynamic instability, models have grown over the years to account for the effects of thermal, molecular, and viscous diffusion [7,8]. The nature and extent of instabilities depend on several factors including thermal expansion factor, flame thickness, Lewis number (Le, ratio of thermal diffusivity to mass diffusivity), and global activation energy [9]. With the dramatic increase in computing power in recent years, flame wrinkling has also been studied using direct numerical simulations (DNS) [10,11]. In numerical and analytical studies, the propensity for instabilities is commonly characterized by a dispersion relation [12], which relates the growth rate of perturbations (often denoted by ω) with their wavenumbers (k). It could be in the form of a plot derived from a DNS study, or it may be written in the form of an equation to explain the growth rate in terms of the underlying physical mechanisms [8], where it can also shed light on how different mixture properties (such as expansion factor, thermal diffusivity, mass diffusivity, and viscosity) affect the formation of intrinsic instabilities. The mechanisms for intrinsic instabilities are now usually classified as being hydrodynamic, diffusive-thermal, or body-force (Rayleigh–Taylor) instabilities [13]. The hydrodynamic mechanism is always destabilizing due to the density difference between the two sides of the flame, while the thermal-diffusive mechanism may be stabilizing or destabilizing, depending on the Lewis number.

A constant volume combustion chamber (CVCC) is a canonical device used to study fundamental combustion behaviors, including changes in morphology due to intrinsic instabilities such as flame wrinkling and cellularization. In its most common configuration, a CVCC is filled with the mixture of interest and ignited in the center using electrodes, producing a spherically expanding flame. Such flames enable combustion studies in a controlled, quiescent environment without the effects of fluid flow. CVCCs with optical access can be used to record high-speed images of the flame, often in combination with imaging techniques such as Schlieren. The images from such experiments (such as those in Fig. 1) show the flame fronts due to density gradients and the perturbations appear as wrinkles or cracks, which may form cells [1416]. This provides a direct and intuitive way of studying intrinsic instabilities. However, this method is limited to systems with optical access.

Fig. 1
Illustrative comparison of high-speed Schlieren images of (a) a stable spherically expanding flame with no wrinkles, and (b) a spherical flame with extensive wrinkling, showing an intrinsically unstable flame
Fig. 1
Illustrative comparison of high-speed Schlieren images of (a) a stable spherically expanding flame with no wrinkles, and (b) a spherical flame with extensive wrinkling, showing an intrinsically unstable flame
Close modal

Theoretical models of intrinsic instabilities in spherically expanding flames [1719] can generally predict trends of such experiments very well and have been very useful in providing explanations for the influence of different mixture properties on flame wrinkling. Spherical flames of mixtures deficient in light reactants (low values of Le, below a certain critical value Lec) tend to be thermal-diffusively unstable and exhibit wrinkling with cellular structures from very short flame radii. If Le > Lec and the flame does not have thermal-diffusive instability, it appears as a smooth sphere until it is large enough for the hydrodynamic instability to manifest [18]. The radius at which this happens is often characterized by the dimensionless Peclet number (Pe). The change in pressure is a more prominent feature in the prediction and detection of other types of instabilities, most notably thermoacoustic instabilities [20,21]. However, the relationship between pressure and intrinsic instabilities of a spherical flame is relatively less explored experimentally. This study investigates the use of pressure data from spherically expanding flames in a CVCC to detect intrinsic instabilities. This work can be viewed as an initial step towards identifying distinguishing patterns or features hidden in the pressure data of spherically expanding flames for the purpose of detecting intrinsic instabilities.

2 Methodology

This study had two major parts; the collection and labeling of time series data from CVCC experiments, followed by the use of time series classification (TSC) algorithms for the detection of instabilities from the pressure-time data.

The pressure traces and high-speed images were recorded for several independent combustion studies over the course of three years, with disparate experimental focuses as presented in previous literature [2225]. The data covers 7 different fuel-oxidizer-working fluid combinations over different equivalence ratios, fuel fractions (in the case of multi-fuel mixtures), and initial pressures (summarized in Table 1). Each trial's pressure data was recorded as a time series of 3250 time-steps (1 ms per time-step). The combustion event itself corresponds to a much shorter sequence, up to a few hundred time-steps for the slower flames. Ten pressure traces included in this study are shown in Fig. 2 for illustration, with further discussion included in Sec. 3.

Fig. 2
Illustration of pressure traces from the CVCC. The blue lines are from flames without significant wrinkling, while the experiments represented by orange lines did show extensive evidence of instabilities. The pressure traces do not provide any features that allow the naked eye to distinguish between wrinkled and non-wrinkled flames.
Fig. 2
Illustration of pressure traces from the CVCC. The blue lines are from flames without significant wrinkling, while the experiments represented by orange lines did show extensive evidence of instabilities. The pressure traces do not provide any features that allow the naked eye to distinguish between wrinkled and non-wrinkled flames.
Close modal
Table 1

Pressure time series dataset and its classes (before data augmentation)

MixtureTotal samplesInstabilitiesNo instabilities
NH3 – H2 – O2 – N21222597
H2 – O2 – CO259590
CH4 – C2H4 – C2H6 – O2 – N2909
CH4 – H2 – NH3 – O2 – N228028
CH4 – O2 – N231031
CH4 – O2 – Ar29524
CH4 – O2 – Kr20164
Total298105193
MixtureTotal samplesInstabilitiesNo instabilities
NH3 – H2 – O2 – N21222597
H2 – O2 – CO259590
CH4 – C2H4 – C2H6 – O2 – N2909
CH4 – H2 – NH3 – O2 – N228028
CH4 – O2 – N231031
CH4 – O2 – Ar29524
CH4 – O2 – Kr20164
Total298105193

Each time series sample was assigned a binary label as either having instabilities (1) or no instabilities (0) based on a combination of visual inspection and automated analysis of the high-speed images based on edge and crack detection methodologies discussed in an earlier study [22]. To keep labeling consistent, only flames where wrinkles multiplied into several other wrinkles before getting close to the chamber walls were labeled as having instabilities. In other words, singular wrinkles that did not develop into more wrinkles were not labeled “1.”

Class imbalance can introduce a bias towards the majority class [26] in classification accuracy, so data augmentation was used, and 88 samples from the minority class (instabilities) were duplicated into the data set from NH3–H2, H2–CO2, and CH4–Ar mixtures. This can increase the risk of overfitting, so all of the ML algorithms were also tested with the non-augmented dataset. Additionally, as some algorithms for time series data perform better with normalized data [27], the algorithms were also studied with min-max normalized and z-score normalized copies of the dataset.

Before any learning algorithms were used, the pressure traces were explored in terms of more direct and interpretable parameters. These were the pressure trace itself, the peak pressures (Pmax) the maximum of the first derivative of pressure ((dP/dt)max), and the second derivative of the pressure ((d2P/dt2)max). After that, the following time series classification algorithms were tested, using their implementations in the scikit-learn-compatible python libraries for time series data, “aeon” [28] and “pyts” [29].

  1. ROCKET (Random Convolutional Kernel Transform) classifier [30];

  2. HIVE-COTE v2 (Hierarchical Vote Collective of Transformation-based Ensembles version 2) classifier [31];

  3. 1-nearest neighbor (1-NN) [32] with dynamic time warping (DTW)[33] as the metric;

  4. SAX-VSM (Symbolic Aggregate approXimation – Vector Space Model) [34];

  5. BOSSVS (Bag-of-SFA (Symbolic Fourier Approximation) in Vector Space) [35].

The train-test split was 80-20 for all runs. The conventional 80-20 data split was chosen to utilize an appropriate quantity of data for training despite the small dataset, while still having a reasonably representative test set.

3 Results and Discussion

As can be seen in Fig. 2 (5 pressure-time series for each class), there is no clear pattern to distinguish between flames with or without instabilities. Similarly, the global pressure trace parameters (the distributions of maximum pressure (Pmax)), the first derivative of the pressure ((dP/dt)max), and the second derivative of the pressure ((d2P/dt2)max) are not indicative of the presence of instabilities, as shown in Figs. 3(a)3(c). Although Fig. 3(a) shows that the peak pressures for the instabilities samples seem to be more evenly distributed compared to those without instabilities, and Fig. 3(c) shows the instabilities class is the only one where (d2P/dt2)max exists above than the 1 bar/ms2 range, there is still no clear demarcation between the two classes. Similarly, for (dP/dt)max, while instabilities tend to have higher values, most of its samples still exist in the 0–2 bar/ms range together with the no instability samples.

Fig. 3
Distribution of different pressure parameters for the original dataset before augmentation, consisting of 298 samples: (a) distribution of maximum pressure Pmax for each class; (b) distribution of maximum pressure rise rate (dP/dt)max for each class; (c) distribution of maximum second derivative pressure (d2P/dt2)max for each class.
Fig. 3
Distribution of different pressure parameters for the original dataset before augmentation, consisting of 298 samples: (a) distribution of maximum pressure Pmax for each class; (b) distribution of maximum pressure rise rate (dP/dt)max for each class; (c) distribution of maximum second derivative pressure (d2P/dt2)max for each class.
Close modal

As DTW [33] is a measure of similarity for time series data, pairwise “distances” were calculated between all samples. If DTW can capture the difference between samples well, the distance between samples of different classes would be higher, whereas that of the same class would be lower. However, it does not seem to be the case here. When viewed as a heatmap, this pattern only appears to partially hold for the no instability class (Fig. 4(a)) and even less so for the instability class. Normalization operations (both z-normalization in Fig. 4(b) and min-max normalization, Fig. 4(c)) seem to reduce the magnitude of “distance” between the samples, but the lack of distinguishing features between the two classes in the heatmaps stays the same.

Fig. 4
Pairwise DTW distances between the samples in the dataset, (a) DTW distances without normalization, (b) DTW distances with z-normalization, and (c) DTW distances with min-max normalization. Samples 0–192: with instabilities, 193–385: no instabilities (separated by green lines). DTW distance seems unable to differentiate between the two classes.
Fig. 4
Pairwise DTW distances between the samples in the dataset, (a) DTW distances without normalization, (b) DTW distances with z-normalization, and (c) DTW distances with min-max normalization. Samples 0–192: with instabilities, 193–385: no instabilities (separated by green lines). DTW distance seems unable to differentiate between the two classes.
Close modal

3.1 Time Series Classification Results.

The 5 TSC algorithms listed above were trained and tested on the non-augmented dataset, augmented dataset, as well as their normalized versions. The models trained on non-augmented data achieved similar or better results compared to the augmented dataset-trained models. This indicated that the imbalance in the test set could be inflating the accuracy more compared to any potential overfitting. So the results reported here are from the augmented dataset models.

The classification performance of HIVE-COTE, 1-NN, and BOSSVS is affected by normalization. It is possible that these specific algorithms were learning features that relied more on an aspect of amplitudes, which are affected by normalization. 1-NN has the highest accuracy with z-normalized data, which is in line with what is reported about it in literature for other datasets. HIVE-COTE appears to perform the best with min-max normalized data, while BOSSVS has the best performance with raw pressure data, although it is still the lowest among all the algorithms. Of the algorithms that depend on an initial random state, the classification accuracy for ROCKET varied between 91–97% for 50 different states, while for HIVE-COTE it varied between 87% and 95%. Both the mean and mode of the classification accuracies from the 50 initial states were within 0.03% of each other for ROCKET, and 2% for HIVE-COTE. The values reported here (Table 2) are the mean classification accuracies of those 50 runs. ROCKET, HIVE-COTE, and 1-NN yielded the highest classification accuracies (>90%).

Table 2

Classification accuracies of the different TSC algorithms compared in this study

AlgorithmDataset
No normalizationZ-NormalizedMin–max normalized
ROCKET94.5%94.5%94.5%
HIVE-COTE v291.2%91.8%92.5%
1-NN (with DTW)94.9%96.1%88.5%
SAX-VSM75.6%75.6%75.6%
BOSSVS69.2%57.7%58.9%
AlgorithmDataset
No normalizationZ-NormalizedMin–max normalized
ROCKET94.5%94.5%94.5%
HIVE-COTE v291.2%91.8%92.5%
1-NN (with DTW)94.9%96.1%88.5%
SAX-VSM75.6%75.6%75.6%
BOSSVS69.2%57.7%58.9%

The confusion matrices in Figs. 5(a)5(c) for ROCKET, HIVE-COTE, and 1-NN (with DTW as the distance metric) showcase their performance in terms of true positives (upper left), false positives (upper right), false negatives (lower left), and true negatives (lower right).

Fig. 5
Confusion matrices for the three models with the highest classification accuracy. Label 0 represents no instabilities, 1 instabilities. They show that the high classification accuracy is not just due to good performance in a single class. (a) ROCKET classifier, (b) HIVE-COTE classifier, and (c) 1-NN classifier.
Fig. 5
Confusion matrices for the three models with the highest classification accuracy. Label 0 represents no instabilities, 1 instabilities. They show that the high classification accuracy is not just due to good performance in a single class. (a) ROCKET classifier, (b) HIVE-COTE classifier, and (c) 1-NN classifier.
Close modal

As the ROCKET classifier works by the convolution of thousands of different kernels, it is likely that some of the randomly generated kernels are able to match up with underlying patterns at scale that are not apparent in the pressure trace as a whole. Furthermore, the fact that the classification is done on a combination of these kernels makes it even more difficult to ascertain these traits. HIVE-COTE which uses other models in its ensemble in addition to ROCKET-based classifiers performs slightly worse. This could mean that one of the other classifiers in the HIVE-COTE ensemble is extracting patterns that can predict intrinsic instabilities in the pressure traces of CVCCs. However, due to the inherent lack of explainability in both these models (ROCKET and HIVE-COTE), it is difficult to say with certainty. Although 1-NN has been shown to generally perform well compared to more complicated methods across many different datasets [27], the classification accuracy seen here is still unexpected when taking into account the pairwise DTW distance discussed earlier. It is worth noting though that 1-NN would have a high variance due to the fact that it makes its classification based on the single closest neighbor. The performance of these classification algorithms is encouraging and has room to be further explored in terms of fine-tuning these algorithms and larger datasets, and investigating the trained models for the patterns that they are using for classification.

4 Conclusions

This study explored the use of several time series classification algorithms for predicting flame wrinkling in spherically expanding flames in a CVCC from their pressure data. Multiple combustible mixtures across varying conditions were tested. Some algorithms, namely the ROCKET, HIVE-COTE, and 1-NN classifiers, were remarkably found to perform well with greater than 90% classification accuracies, even though the pressure traces were essentially indiscernible to the naked eye. This is a promising first step towards identifying patterns for the detection of intrinsic instabilities using pressure traces. These classifiers could be used to detect flame wrinkling in CVCCs or other systems that do not have optical access. This work may be extended in several different directions. The tuning of the classification algorithms' parameters could be explored in more detail to improve performance. Further exploration of the data in conjunction with the algorithms would also be important to determine what features the algorithms are picking out to classify the time series and whether these features could be generalized to get a better understanding of the direct relation of the pressure in the chamber with the intrinsic instabilities. Another interesting, but challenging, proposition could be detecting the extent of flame wrinkling and cellularization from pressure traces, as has been done with images of spherical flames. Overall, the presented approach to robustly detecting the presence of intrinsic instabilities in a constant volume combustion chamber provides a new, accurate methodology relevant to multiple combustion systems ranging from canonical experiments to practical devices.

Acknowledgment

This work relates to the Department of Navy award N00014-22-1-2001 issued by the Office of Naval Research. Additional support for the corresponding author is provided by the Fulbright Foreign Student Program.

Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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