Investigation of Thermal Characteristics of Turbulent Non-Premixed Flames in a Conical Bluff Body Using Ridge and Support Vector Regressor Algorithms

Non-premixed combustion is generally valued for its simplicity and reliability, therefore, finds extensive applications in heating and industrial processes. Yet, rising environmental standards are driving innovations focused on reducing emissions and improving energy efficiency [1]. The pivotal element seems to lie in flame stability, which is determined by the correlation between turbulent mixing and chemical reactions. To augment mixing and regulate the intensity of the reaction, the design and optimization of the burner become primary concerns. For quite some time, studies have demonstrated the effectiveness of flame stabilization [2,3] using a central fuel jet and an oxidizer to create specifically tailored mixing zones [4–6]. Within this framework, various bluff body shapes have been investigated in this research, including cylindrical structures [7,8], circular disks [9,10], and tulip-shaped bluff bodies [11,12].

Previous investigations into the flame holders revealed that the bluff body flames develop a recirculation zone that houses two opposing toroidal vortices. This phenomenon entails the inner vortex propelled by the fuel jet and the outer vortex driven by co-flow. These vortices interact within a radial zone lying between the fuel jet and co-flow, fostering a mix of combustible qualities. This location marks the point of peak flame temperature, which fluctuates based on the ratio of momentum between air and fuel streams [13]. Understanding the impact of bluff body shapes is crucial in devising optimal burner designs. Comparative analyses are limited to cold flows [14], providing insight into the wake flow characteristics of the disk and conical burners. It is well established that conical burners exhibit a smaller wake region compared to disk burners, leading to the formation of reduced recirculating mass, thereby promoting improved combustion efficiency and more uniform temperature distribution. However, these studies lacked the impact of combustion, which was addressed in a numerical investigation by Ma and Harn [15] concerning cone angles and cylindrical flame holders. Their comparative study indicated that the larger cone angles amplified the circulation in the flame holder's wake, enhancing the air–fuel mixture and promoting more stable flames. Furthermore, they demonstrated that the increased cone angles reduced the length of the recirculation zone, widening its span. Rowhani et al. [16] discovered that enlarging the burner head diameter leads to longer residence time and larger recirculation zones, resulting in shorter flames with higher soot content, thereby reducing the overall temperature, making them less efficient. Studies comparing different baffle shapes in non-premixed flames, supported by conical flame holders [17], examined flame behavior based on Reynolds number (Re). The Re of fuel influenced extinction, while the Re of air influenced flame configuration and temperature. Research involving conical flame holders surrounded by rifling [18] demonstrated that an increase in turbulence led to amplified combustion intensity and reduced flame length, resulting in higher flame temperatures due to efficient oxidation and heat release.

Christo et al. [19] were the first to pioneer the use of artificial neural networks (ANNs) in combustion chemistry. They modeled the turbulent H₂/CO₂ flame by implementing a reduced reaction scheme that utilized a joint velocity-scalar probability density function (PDF) method. This approach demonstrated good compliance with the experimental data while simultaneously controlling major variations in CPU time and memory storage. Following that, Blasco et al. [20] used ANNs to model a diminished CH₄ combustion process with 13 species. Their method used three multilayer perceptrons (MLPs) to monitor the development of responsive scalars and an extra MLP to compute density and temperature data. Building on previous work, the same team of researchers in Ref. [21] used the self-organizing map to organize and split composition regions into subdomains. All the subdomains were subsequently trained using a different MLP. An excellent correlation between the ANN and target data was achieved, accompanied by significant reductions in CPU processing time and memory usage. While the mentioned research utilized ANNs to construct mechanisms for partially stirred reactor simulations, other studies explored different applications. Kempf et al. [22] and Ihme et al. [23] utilized ANNs within steady-state laminar flamelet models as substitutes for traditional tabulation methods in the realm of large-eddy simulation.

Moreover, Sen and Menon [24,25] successfully captured the effects of unsteady flame–turbulence–vortex interactions by creating an ANN database derived from direct numerical simulation (DNS) of flame–vortex interactions and individual linear eddy mixing model computations. Lapeyre et al. [26] employed DNS to premixed turbulent flames to train a convolutional neural network (CNN) aiming to predict the sub-grid flame surface density. CNN effectively captured the flame's structure and estimated the smaller-scale wrinkling. Comparing the results, the CNN outperformed algebraic models that relied on both power-law expressions [27] and fractal surfaces [28]. In a different study, Barwey et al. [29] utilized a CNN model to deduce the three velocity components within premixed flames in a swirl combustor from sequences of time-resolved planar laser-induced fluorescence images of OH. These images encompassed both attached and detached flame configurations. The authors noted that, specifically in the attached flame scenario, both CNN models offered more accurate estimations of the flame surface density compared to fractal and power-law models, exhibiting an almost negligible absolute error

In summary, while there may not be numerous studies utilizing ANNs in combustion, they exhibit great promise for addressing the time-intensive calculations inherent in combustion chemistry. This aspect poses a significant challenge when it comes to accurately modeling turbulent combustion, especially with PDF methods. To the best of the authors' knowledge, no research has reported the use of machine learning (ML) and artificial intelligence techniques in exploring the influence of diverse geometries and flowrates on the temperature of turbulent flames. Consequently, our comprehension of flame behavior for various input configurations and levels is limited. Our goal is to develop a method wherein ANNs are trained using data from an abstract problem, enabling the creation of a model capable of accurately simulating various real combustion scenarios, like non-premixed turbulent flames. To achieve this, this study systematically examines the influence of flow and geometric parameters, including burner head diameter (D), half cone angles (α), and co-flow air velocity (U_cf), in the prediction of the temperature of turbulent non-premixed flames. The Taguchi-based design of experiments (DOE) [30,31] and regression [31] approach are implemented to develop the regression equation for temperature. The regression equation developed from DOE was utilized to generate 250 output data points, which were then deployed to train the ML models.

The experimental setup was detailed in Ref. [1]. It consisted of a non-premixed burner supplied by a system delivering fuel and air, expelling combustion byproducts through an exhaust vent. The setup, housed within a cubicle with black walls, included tools for measurement and visualization. Test-grade methane (G20, with a concentration of 99.5% CH₄) was delivered to the fuel inlet at a constant pressure of 25 mbar. Brass-made conical bluff bodies were affixed to the fuel pipe. While maintaining a constant height (H) of 12 mm for the burner, its diameter (D) was varied at 12, 18, and 27 mm, corresponding to half cone angles (α) of 14, 26, and 41 deg, respectively. These bluff bodies are denoted as BH#1, BH#2, and BH#3, respectively. The schematic of the burner is shown in Fig. 1. The aluminum fuel pipe of outer diameter d = 6 mm and inner diameter d_f = 4 mm ended the conical bluff body made of brass. A fuel flow with a velocity of 15 m/s, resulting in a Re of 3553, was directed into an air stream with velocities ranging from 2.0 m/s to 5.9 m/s, measured at 6 mm above the edge of the air tube. The dimensions of the bluff body and the experimental factors in the present study are outlined in Table 1.

A specific thermocouple of type B was employed for measuring temperatures up to 1973 K. Data acquisition involved sampling its output at 50 Hz for a duration of 10 s through a card possessing an accuracy of ±1.8 K at 523 K and ±3.9 K at 2073 K. The temperature probes covered identical measurement points via a precision traverse mechanism. This mechanism was calibrated using a caliper and a laser sheet aligner (Bosch GCL 2–50 C). A step motor, driven by associated electronics, was programmed to precisely position the probes, achieving a spatial resolution of 1 mm.

The Taguchi DOE is integral in pinpointing crucial variables across diverse industrial endeavors, like process preparation and experimentation [31]. These methods offer a broad spectrum of options by incorporating various factors at multiple levels, with orthogonal arrays such as L₉, L₁₆, L₂₇, and L₈₁. Selecting a specific array facilitates conducting experiments to derive a response function reliant on the input variables. This technique is instrumental in addressing prevailing challenges and attaining optimal temperature data through the appropriate selection of parameters. Hence, employing DOE to explore various combinations of input parameters enables the acquisition of input–output data that could be used to train the ML models [34].

The performance of the computational models is purely based on the quantity and the quality of the data used. For our study, 250 sets of data were generated, out of which 27 were generated using the conventional L₂₇ orthogonal array, and the remaining 223 were generated using the regression equation for different input parameter combinations. The computational models typically demand extensive data to fine-tune and optimize parameters during the training process. Gathering such voluminous data solely through experiments would be unfeasible owing to the considerable time and cost implications. Hence, leveraging the response equations derived from DOE designs via actual experiments proved to be a practical approach for generating substantial input–output data. This was achieved by selecting input parameters with their designated levels and ranges.

Furthermore, the nonlinear regression models used to estimate temperature were verified for statistical reliability. For this purpose, 25 arbitrary test cases were executed with each test case consisting of a different mix of input parameters than those utilized in the L₂₇ orthogonal array however, were within the range of the levels listed in Table 1. It shows the error between experimental temperature (T_Exp) and regression-predicted temperature (T_Pre) values. The regression equation was shown to be accurate in predicting temperature, with a mean absolute error (MAE) of 1.671%. This suggests that the regression equation was predicted highly accurate readings, therefore was found to be reliable to gather more data for a pre-defined set of parameters and levels. More crucially, the regression equations yielded a large database. The regression (Eq. (2)) was used to generate 223 additional output temperature values for different combinations of input variables within the appropriate range of levels.

Support vector regression (SVR) is a regression algorithm capable of handling both linear and nonlinear regression tasks. It is based on the principles of support vector machine (SVM), a classifier used for discrete categorical outcomes. In contrast, SVR is designed for continuous numerical variables. SVR employs the SVM concept for regression problems, aiming to approximate a function that maps input domains to real numbers based on training data. Its core revolves around finding a hyperplane that optimally segregates features into distinct domains. The premise is that the further the support vector points are from the hyperplane, the more accurately they represent points in their respective regions or values, influencing the hyperplane's positioning due to their crucial role in its computation (Fig. 1) [37].

$‖A1−A2‖$ is the Euclidean distance between $A1$ and $A2$⁠. The increasing value of $σ$⁠, which is the RBF kernel, indicates that the model is overfitting, and a decrease in its value indicates an underfit.

The performance metrics were selected individually. The metrics are trained and tested for the numerical values provided by the model to determine the effectiveness of testing predictions [38–41].

The $R2$ expression includes variables such as n (data points) and $Vot$ and $Vop$⁠, (predicted and actual values, respectively in a regression analysis of output data). $R2$ is a statistic that spans from 0 to 1. Higher values, i.e., values closer to 1 indicate a better model fit, whereas lower values indicate significant flaws with the model's performance or applicability.

The modeling outcomes were generated through the application of various regression algorithms, encompassing RR with different α values, linear regression (LR), support vector regressor-based LR (SVR-LR), SVR employing radial basis function (SVR-RBF), and SVR employing polynomial regression (SVR-Poly). The evaluation of performance metrics such as MAE, R², MAPE, and MPD were conducted for comparative analysis. It's noteworthy that a detailed examination was conducted specifically for RR, while the ridge model was exclusively utilized in the latter part of this section. The predictor variable in focus is the flame temperature (T) influenced by variables, such as burner head diameter (D), burner height (H), half cone angles (α), and co-flowing air velocity (U_cf). These temperature measurements were collected through experimental recordings and Taguchi-based DOE regression equations, which were compiled as part of the research.

Figures 2(a) and 2(b) show that the flame temperature data modeled using the ridge regressor for $α=0$⁠, for the training and testing output matched conveniently with the DOE-based experimental data. The trendline indicates that the data obtained from the regressor was in line with the recorded data. The indicator showed an $R2=0.9219$⁠, which suggested that the training was successful, as this value was close to unity. Upon this, an attempt was made to assess the predictability of the regressor model using the testing data. As shown in Fig. 2(b), the regressor made excellent predictions with the DOE-based experimental findings. The testing results are even better than the training results. The closeness of the data points with the trendline gives an indication of this algorithm's ability to predict the flame temperature values based on burner and flame parameters. Even though the data were highly nonlinear, the ridge model showed considerable success in its predictions. The $R2=0.9284$ obtained for testing is a testimony to its predictability, as this value nears unity. For $α=0.2$–1.0, due to brevity issues, the testing and training results are avoided.

The predictability of the ridge regressor for better interpretation is determined for a specific set of data from Fig. 3. A comparison was made between the initial 30 data points of the DOE-based experimental data and the training results of the ridge regressor across $α$ values ranging from 0 to 1 in steps of 0.5. The representation displays a close alignment between the trendline of the experimental data along with the data points and the trends exhibited by ridge regressor. At times, the trendlines overlapped, posing a challenge in discerning differences. A closer examination reveals that the best alignment between the experimental data and the ridge regressor outputs was for $α=0$⁠, specifically during high and low peaks. As the value of $α$ increased, a deviation in the prediction of flame temperature values from the actual data was observed. This deviation occurred because of the higher $α$ values, which led to increased data shrinkage, propelling coefficients toward zero [35]. The testing data for the ridge regressor against the experimental data for different values of $α$ are shown in Fig. 4. For different $α$ values, scattered data points show predictions along each vertical line between the experimental data and ridge regressor outputs. It is clearly observed that the empty squares representing the experimental data and the empty circles representing $α=0$ were very close to each other.

Figure 5 shows the performance metric (MAE, R², MAPE, and MPD) for outputs developed from the ridge regressor algorithm for different values of $α$⁠. This performance is obtained from the trained data of the ridge regressor model. It is clearly observed that the MAE increased linearly with the increasing $α$ values. This increase was mainly caused due to the growth in data shrinkage, as $α$ increased. As far as the value of $R2$ is concerned, an increase in $α$ decreased the accuracy of the ridge regressor. The best and the closest value of $R2$ was obtained at $α=0$⁠. Although the $R2$ looks to be constant, a closer look at it clearly shows that its value is decreasing, which is not acceptable. The MAPE and MPD metrics showed an increase with an increase in the $α$ values with the training ability of the regressor deteriorate. Figure 6 shows the variation of all the metrics for testing data for the increasing value of $α$⁠. It can be clearly seen that the performance of the ridge regressor diminished with increasing $α$ values, in comparison to the training data set. Apparently, a linear increase in the metrics is observed as compared to the previous training data.

Figure 7 shows the flame temperature data modeled using the LR. A comparison of the outputs obtained from the LR with the DOE-based experimental data clearly shows a perfect match. Except for a few data lying beyond the range, the training was successful and appropriate as most data points were compact. The outliers are pretty common in data modeling and are acceptable as the values of $R2$ are equal to 0.912. The training is conducted for 65% of the data available, and the remaining 35% was used for testing purposes. Figure 8 shows the comparison of the experimental data with the linear regressor outputs for testing data. Most of the data points lie very close to the trendline. The evaluation of the linear regressor model reveals its suitability comparable to the ridge model predicting the flame temperature in the burner. With an $R2$ of 0.921 in testing, the linear regressor demonstrates close results, suggesting its efficiency. This eventually results in substantial computational cost and time savings compared to the other models. Figure 9 illustrates the performance of the trained and tested linear regressor model results, emphasizing its consistency of metrics between training and testing. Apparently, the majority of cases exhibit similar values for MAE, R², MAPE, and MPD in the modeling process. The proximity to zero in MAPE and MPD indicated minimal errors and significantly reduced the differences between predicted and actual values, regardless of the numerical magnitude of the data points.

In Fig. 10, the application of SVR (LR) algorithm for training and testing temperature data using SVR is presented. The $R2$ values for both training and testing cases, displayed, surpass 0.92, indicating successful regression of the temperature data. Densely packed data points signify effective training computations, and the testing results demonstrate reliable predictions from the SVR-LR model. Figure 11 showcases the training of the dataset using the SVR-RBF algorithm, proving to be highly successful in this study. The $R2$ value for SVR-RBF-based training exceeds 0.955, the highest and closest to unity (Fig. 11). Results as per Fig. 11 also indicate excellent predictions, with the $R2$ value during SVR-RBF testing approximately 0.940, dominating over other algorithms.

The remarkable performance of SVR-RBF could be attributed to its generalization capabilities. This algorithm demonstrates robustness to outliers and a high tolerance toward the input data noise, mainly dependent on mapping. Its performance is comparable to that of neural networks and neuro-inference systems. Figure 12 presents the training results of the SVR (Poly) algorithm based on polynomial regression. It is observed that the training of the SVR (Poly) model wherein data points align well with the provided actual data. Its competence is on par with the SVR-RBF models, as evidenced by the R² of 0.930. A notable drawback observed during training is the excessive computational time required by this model for a 1-degree polynomial function. For 3 deg of polynomial function, the time was intolerable, leading to the discontinuation of the training process. Interestingly, during testing, the SVR (Poly) algorithm exhibited slightly better proximity to the actual values predicted compared to training with an $R2=0.932$⁠.

Figure 13 depicts the variation of flame temperature data, comparing the DOE-based experimental readings with the computational/predicted readings obtained using SVR-LR, SVR-RBF, and SVR(Poly) algorithms. Only 20 data points are chosen arbitrarily for demonstration. The black squares represent the experimental readings, while the remaining colored circles and triangles represent the corresponding readings computed during the training session of the three SVR algorithms. SVR-RBF reading for the majority of the data points overlap completely, whilst SVR-LR and SVR(Poly) deviate nut still exhibit a strong match with the actual values. Similarly, predictions for the SVR algorithms are conducted for testing data. The SVR-RBF predicted values were found to be closest to the experimental. Figures 14(a) and 14(b) depict the training output computed and the predictions made by the trained and tested algorithms, respectively. Comparative analysis involves SVR-LR, SVR-RBF, SVR(Poly), and Ridge CV algorithms. The bar plots in Fig. 14(a) highlight that the SVR-RBF yields the least error. However, during testing in Fig. 14(b), the Ridge CV and SVR-RBF algorithms exhibit nearly identical error values.

The computational results obtained from the best algorithm, i.e., SVR-RBF were compared with the experimental results for further validation. The computational temperature results were compared with the experimental temperatures obtained from Ata and Ozdemir [44] for a different set of input parameter combinations. The time-averaged temperature profiles are determined for a cone height of Z = 0.5D and Z = D, above the bluff body, co-flow velocity of U_cf = 3.9 m/s, and conical bluff bodies of dimensions having constant D = 18 mm, and varying α = 18.4, 15.5, 12.5, and 9 designated as BB#1, BB#2, BB#3, and BB#4, respectively. The temperatures are shown along (r/0.5df) which is the radial location of the bluff body from the inlet to the exit plane of the burner. The accuracy was assessed by using only two criteria, namely, MAE and R². MAE was chosen as it showed the average difference between experimental and model predicted values, and the $R2$ gives the picture of the overall model fit [41–43]. Co-flow velocity variation is one way to differentiate between the flames obtained from all four bluff bodies [44]. As can be seen from Figs. 15(a) and 15(b), the temperature profiles of BB#1 and BB#2 do not change much except for the peak values. Additionally, the temperature ranges over a relatively small radial distance, as the jet flames do not spread radially. For BB#1 and BB#2, the temperature profiles see a very good agreement with the SVR (RBF) computations. The MAE was 0.86%, and $R2$ was 0.9824 for BB#1, whereas, for BB#2, it was 0.76% and 0.9882, respectively. This clearly showed that the SVR-RBF accurately predicted for BB#2. For the second group of bluff bodies, i.e., BB#3 and BB#4 shown in Figs. 15(c) and 15(d), temperature profiles widen and the location of the peak moves toward the co-flow [4,13]. Here, the temperature profiles see a drastic change in their skewness due to an increase in the separation between the fuel jet and the co-flow [1,4]. As for the computational values for BB#3 and BB#4, the SVR-RBF computations observe a slight deviation from the experimental temperature profiles. The MAE and $R2$ for BB#3 were 2.105% and 0.9636, respectively, and subsequently, for BB#4, the values were 1.875% and 0.9712, respectively. It is also observed that the computations for temperature profiles of BB#3 and BB#4 show an excellent affiliation with the experimental temperature profiles, resulting in very high accuracy. For BB#3 and BB#4, the computations show a slight deviation mainly due to the mixing characteristics, which are primarily controlled by the wall effects that lead to different radial spreading of the jet flames further downstream. It is also worth noting that in the vicinity of the burner, the flame temperatures in between the fuel jet and the co-flow exhibit a rather flat distribution, where the vertical velocity takes very low or zero values. As a result, the peak flame temperatures at the first measurement point z = 0.5D remain nearly the same as in the corresponding no co-flow cases [4,44]. However, away from the burner, while the temperature profiles on the fuel jet side develop very similarly to those in the no co-flow cases, the skewness of the profiles. Notably, for BB#3 and BB#4, the skewness of the profiles changes drastically on the co-flow side, with the tail extending radially outwards [44]. Therefore, the flat profiles are replaced by the peaked distributions.

The flames of all four burners are visualized at a co-flow velocity of 3.9 m/s (see Fig. 16). For the burner BB#1 (Fig. 16(a)), there is poor interaction between the fuel jet and oxidizer co-stream, resulting in a highly stretched non-premixed flame that exhibits a coloration [4]. This indicates that the local residence times are insufficient for soot formation, leading to BB#1 being classified as jet-dominated (Fig. 17(a)). For BB#2, the co-flow velocity results in heightened shear forces between the co-flow and wake zone. The improved stretching precipitates a ring of flame encircling the wake area (Fig. 16(b)). This peripheral flame subsequently detaches and swiftly moves away from the burner, displaying a dual flame configuration. This dual flame generates soot primarily in the proximity to the burner due to enhanced particle combustion (Fig. 17(b)). As the co-flow velocity is on the higher side for these burners, BB#3 and BB#4 remain stable up to the distance of the bluff body exit plane diameter (Figs. 16(c) and 16(d)). In these burners, the combustion becomes increasingly unstable and sooty further away from the burner. With a further increase in the co-flow velocity, the stable region disappears, and flames become attached but remain unstable, as per Ref. [44] (Fig. 17(c)).

The comparative analysis of various regression algorithms, including RR and SVR-based approaches, has been demonstrated in modeling the flame temperature influenced by burner and flame parameters. The RR exhibited promising results, particularly with α = 0, showcasing successful training and excellent predictability; however, its performance diminished as α increased. Conversely, SVR algorithms, notably SVR-RBF, outperformed the other SVR algorithms, displaying robustness to outliers and noise while achieving high accuracy in both training and testing phases. The validation against experimental data further reinforced the accuracy of SVR-RBF, particularly noticeable in cases involving varied input parameters. Although slight deviations were observed, these were attributed to the mixing characteristics of the burner. Overall, SVR-RBF emerges as the most reliable model, offering predictions closely aligned with experimental data under various conditions. This highlights its potential for predicting complex flame behaviors in practical scenarios. These findings underscore the significance of SVR-based approaches, especially SVR-RBF, in enhancing predictive capabilities and deepening the understanding of the intricate flame dynamics. Consequently, they contribute to advancements in the fields of combustion science and engineering.

There are no conflicts of interest.

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