Research on Flow Field Prediction in a Multi-Swirl Combustor Using Artificial Neural Network

Abstract

In aero-engine combustion research, the pursuit of cost-effective and rapid methods for acquiring precise flow fields across various operating conditions remains a significant challenge. This study offers novel insights into the rapid modeling of complex multi-swirling flows, introducing flow-field-based analytical methods to evaluate flow topologies, spray dispersion, ignition dynamics, and flame propagation patterns. A data-driven model is proposed to predict the swirling velocity field inside a multi-swirl combustor, using spatial coordinates and air pressure drops as input features. Particle Image Velocimetry (PIV) experiments under different air pressure drops are performed to generate the necessary flow field dataset. A fully connected deep neural network is designed and optimized with a focus on prediction accuracy, training efficiency, and mitigation of over-fitting. The predicted flow characteristics, including swirling jets, shear layers, recirculation zones, and velocity profiles, align closely with the PIV experimental results. This demonstrates the model’s capability to effectively capture the intricate multi-swirling flow structures and the complex relationships between input parameters and the resulting flow field. Furthermore, the trained model shows excellent generalization capability, accurately predicting flow fields under previously unseen operating conditions. Finally, combustion-relevant characteristics, such as ignition and flame propagation, are successfully extracted and analyzed from the predicted flow fields using the proposed deep learning framework.

1. Introduction

Aero-engine combustion is an interdisciplinary field involving fluid dynamics, fuel atomization and evaporation, chemical reactions, and thermodynamics [1]. Flow aerodynamics are critical to combustor performance, as they influence the stability and efficiency of combustion. The combustor’s flow fields typically contain complex structures, including swirling flows, shear layers, recirculation zones, and jet penetration [2]. These intricate flow patterns govern the transport and spatial distribution of fuel droplets, impacting key factors such as flame stabilization, temperature uniformity, and emissions control. Furthermore, the swirling flow topology plays a pivotal role in the ignition process and in managing flame propagation dynamics [3]. Consequently, accurately capturing these velocity fields provides a valuable basis for improving combustor performance.

The commonly used techniques for acquiring flow morphology in swirl combustors fall into two main categories: experimental measurements and computational fluid dynamics (CFDs) simulations. Traditional intrusive velocity measurement methods, such as the pitot tube [4] and hot-wire anemometry [5], suffer from low measurement efficiency and can disturb the flow field, compromising accuracy. With advances in optical, photoelectric, and image processing technologies, non-intrusive laser diagnostics have become prevalent in flow measurements. Techniques such as Laser Doppler Velocimetry (LDV), Particle Image Velocimetry (PIV), and Molecular Tagging Velocimetry (MTV) offer high-resolution, detailed insights into complex flow structures without disturbing the flow. PIV has become a widely adopted technique in engineering applications [6] due to its capability of capturing 2D and 3D velocity fields by analyzing double-frame tracer images without disturbing the flow. Despite its advantages, PIV faces limitations when applied to boundary layers near the liner wall and upstream flow fields close to the fuel nozzle. These challenges arise from its relatively low spatial resolution (approximately 1 mm) and the potential disturbance caused by fuel droplets. Additionally, imaging the scattered light from PIV tracers through optical windows presents further challenges, including reduced mechanical strength of the window compared to real combustor conditions, contamination from seed particles or soot, and optical aberrations or distortions [7]. Numerical simulations, particularly large eddy simulations (LESs) and direct numerical simulations (DNSs), have demonstrated the ability to capture high-fidelity flow patterns and accurately represent turbulence levels. However, simulations in combustion systems depend heavily on precise turbulence, atomization, and combustion models, along with their complex couplings [8]. These requirements result in high computational costs and long simulation times, posing challenges for practical applications.

Recent advancements in machine learning (ML) applied to combustion flow have introduced new insights into understanding combustion fluid dynamics [9,10,11]. Significant efforts have focused on predicting flow fields in simplified scenarios, such as cavity flow, bluff-body flow, and flow around cylinders. For instance, convolutional neural networks (CNNs) have been used to predict velocity fields and pressure coefficients around cylinders across varying Reynolds numbers, establishing strong correlations between pressure and the flow field [12,13]. Cheng et al. [14] further enhanced prediction capabilities by combining physics-informed neural networks (PINNs) with ResNet blocks, achieving superior accuracy in predicting cavity flow and flow around cylinders compared to conventional ML methods. In flight aerodynamics, data-driven methods have shown promise in predicting flow fields [15], surface pressure [16], nonlinear unsteady aerodynamics [17,18], and turbulence modeling [19] around airfoils. More recently, deep learning approaches have also been applied to internal flow studies. Kong et al. [20,21] successfully predicted velocity fields in scramjet engines and detected the shock train leading edge using a CNN model, demonstrating the potential of deep learning in internal flow applications. Chen et al. [22] further extended the above deep learning method to the reconstruction of a flow field in a self-ignition supersonic combustion. Additionally, water flow within a heated metal porous tube has been accurately predicted with adaptive network-based fuzzy inference systems (ANFIS) and differential evolution-based fuzzy inference systems (DEFISs) [23,24]. For swirling gas–liquid flow, a self-organizing neural network (SONN) has been used to identify flow regimes, effectively mapping the complex relationships within these regimes [25].

In conclusion, machine learning (ML) applications in flow prediction have been extensively studied, particularly in simplified flows, airfoil aerodynamics, and internal flows in scramjet engines. Swirling flows in swirl-stabilized aero-engine combustors feature complex topologies, including outer recirculation zones (ORZ), inner recirculation zones (IRZ) with vortex breakdown bubbles (VBB), annular swirling jets, and shear layers [2]. The rise of centrally staged Lean Premixed Prevaporized (LPP) combustion has introduced even more complex multi-swirl configurations, as seen in technologies like the twin annular premixing swirler (TAPS) [26], Rolls-Royce lean burn combustor [27], and the low-emissions stirred swirl (TeLESS-II) combustor [28,29]. To our knowledge, few studies have explored ML in predicting flow patterns within swirl-stabilized combustors. Deep learning holds potential for capturing these complex flow dynamics, but the field remains underexplored. This paper aims to propose a rapid, cost-effective ML-based approach for predicting flow fields in swirl-stabilized combustors.

In this work, we propose a data-driven deep neural network (DNN) to predict swirling flow fields and analyze combustion-relevant characteristics from the reconstructed flow data. The structure of this paper is as follows: first, the experimental setup and modeling methods section details the combustor structure, the PIV test system, and the applied deep learning methods. Next, the optimization and predictive performance of the DNN model are presented. The model’s generalization capability is then evaluated using extrapolated flow fields. Finally, combustion-relevant characteristics are analyzed and extracted based on the DNN-predicted flow fields.

2. Experimental Setup and Modeling Method

2.1. Optical Model Combustor

A schematic of the aero engine model combustor used for the flow measurements is shown in Figure 1. The burner features a centrally staged arrangement of swirling injectors consisting of two stages: the central pilot stage and the outer main stage. The two stages are separated by a lip structure, which plays an important role in flame stabilization. A swirl cup design is applied as the pilot stage with dual radial swirlers, generating the inner swirling flow. The main stage adopts an axial swirler containing 20 straight blades with a swirl number of 0.5. The combustor is equipped with quartz windows on the side wall, allowing full optical access.

2.2. PIV Test System

Particle Image Velocimetry (PIV) is an effective diagnostic method for the flow field. Its system is shown in Figure 2, consisting of a double-cavity Nd:YAG laser with 10 Hz repetition (532 nm), a CCD camera (double-shutter, Lavision), a programmable timing unit (PTU), and a commercial PIV software (FlowMaster). The 532 nm laser beam is reflected by multiple mirrors and converted to a laser sheet with a height of 75 mm and thickness of 1 mm via the sheet optics and the collimator optics. This vertical laser sheet is used for the illumination inside the combustor. The time interval between two laser pulses is 15 $\mathsf{\mu }$s. The air is seeded with ${\mathrm{TiO}}_2$ particles, 1 $\mathsf{\mu }$m in diameter, at 1.5 m upstream of the combustor, ensuring the tracers are distributed uniformly. Mie scattering light from the tracers is collected by the CCD camera with the stray light filtered by a filter (532 nm ± 5 nm).

Ambient air at room temperature and atmospheric pressure is supplied to the swirl combustor from a plenum. Inlet total pressure $P_3^{*}$ of the combustor is monitored via a pressure piezometer. Size calibration of the camera pixel is performed before PIV experiments, as shown in Figure 3a. The effective imaging area is 75 mm × 118 mm and contains 156 × 247 sampling points. The air velocity fields are generated from the raw Mie scattering images using the software FlowMaster. The multi-scale cross-correlation is carried out with the interrogation window size of 16 pixels × 16 pixels and an overlap of 50%. A total of 500 instantaneous velocity fields are averaged to obtain the time-averaged flow fields. Flow fields are measured at a total pressure drop ratio ($\sigma$) ranging from 1% to 5%. The $\sigma$ is defined as [30]

$$\begin{array}{cc}\hfill \sigma & ={\displaystyle \frac{P_3^{*}-P_4^{*}}{P_3^{*}}},\hfill \end{array}$$

(1)

2.3. Deep Learning Method

2.3.1. Data Preprocessing

The present study aims to explore the relationship between the inlet condition ($\sigma$), spatial coordinates, and flow velocities using the machine learning method. The PIV experimental data used to train the deep learning model include the working conditions of $\sigma$ = 0.5%, 1%, 2%, and 3%, as shown in Figure 4. The flow field dataset is a matrix of N × 5. The first three columns are the input parameters, and the last two columns are the 2D velocities. The total training dataset contains 154,128 groups of data, of which each condition includes 38,532 groups of data. All parameters were normalized to the range of [0, 1] and shuffled to improve the model accuracy and accelerate the convergence rate. PIV data for the operating condition of $\sigma$ = 4% were selected as the extrapolation dataset to verify the generalization abilities of the deep learning model for unknown datasets.

2.3.2. Neural Network Framework

Figure 5 shows the architecture of the multilayer neural network established in the present work. The input layer consists of three neurons, corresponding to the total air pressure drop $\sigma$, x coordinates, and y coordinates. The output layer includes two neurons, which indicate the axial velocity $V_x$ and the radial velocity $V_y$. The hidden layers are a fully connected multilayer structure with eight neurons in each hidden layer. The number of neurons is chosen based on the balance between the accuracy and the training time. The effect of the number of neurons on prediction accuracy is detailed in the Supplementary Materials. All neurons in the hidden layers are independent of each other. The number of hidden layers is selected according to the comparison of prediction accuracy and training time, as discussed in Section 3.1.

The present study applies optimization algorithms to train the deep learning model from the dataset. The training and prediction process of the DNN model is sketched in Figure 6. Original PIV data are first divided into two groups: input dataset and extrapolation dataset. The input dataset is normalized, shuffled, and further divided randomly into three groups: the training sets (60%) for DNN fitting, the validation sets (20%) for hyper-parameter optimization, and the test sets (20%) for evaluating the generalization ability of untrained data. The above division ratio of the input dataset has been optimized by considering both prediction accuracy and training time, as detailed in the Supplementary Materials. The data are propagated forward through the DNN network to obtain the weight and bias of each neuron. Back-propagation of the data and hyper-parameter optimization continue until the prediction accuracy of the DNN model meets the goal. The fully trained DNN model is capable of performing reconstruction and extrapolation prediction of flow fields.

2.3.3. Training Setup

A loss function is usually selected to evaluate the error between the true and predicted value of the DNN model. It is set as the optimization objective during the training process, which is to minimize the loss function. The root mean squared error (RMSE) is applied as the loss function in the present paper and is expressed as follows [31]:

$$\begin{array}{cc}\hfill RMSE& ={\left[{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(V_{true}-V_{pred}\right)}^2\right]}^{1/2},\hfill \end{array}$$

(2)

where $V_{true}$ and $V_{pred}$ are the true values and predicted values of velocities, respectively.

A fully-connected neural network is essentially a fitting model that includes two processes: forward propagation and backward propagation [32]. The purpose of forward propagation is to generate prediction values, and the expression is as follows [33]:

$$\begin{array}{cc}\hfill y& =f_{act}\left(\sum _{i=1}^m{w}_i{x}_i+b\right),\hfill \end{array}$$

(3)

where m is the number of neurons in the upper adjacent layer, $x_i$ is the input to the ith neuron, $w_i$ represents the weight factor of the i-th neuron, b indicates the bias, and $f_{act}$ is the activation function. In the present study, the sigmoid function [34] is applied as the activation function:

$$\begin{array}{cc}\hfill f_{act}\left(x\right)& ={\displaystyle \frac{1}{1+e^{-x}}},\hfill \end{array}$$

(4)

Backward propagation is an optimization process of the weight coefficient and the bias. The Levenberg–Marquard (L-M) algorithm [35] is applied, which is a nonlinear minimization algorithm that benefits from the advantage of gradient descent and the Gauss–Newton method. The learning rate is set as 0.01 and the weight decay rate is set as 1 × 10⁻⁷ in the training process.

3. Results and Discussion

3.1. Network Architecture Optimization

The fully connected multilayer neural network with three inputs and two outputs is applied in the present work. Each hidden layer contains eight neurons. The number of hidden layers is an important hyper-parameter affecting the prediction accuracy and efficiency of the models. To evaluate the error between predicted values and experimental data, the dimensionless coefficient $R^2$ is selected [36]:

$$\begin{array}{cc}\hfill R^2& =1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{\left(V_{exp}-V_{pred}\right)}^2}{{\displaystyle \sum _{i=1}^n}{\left(V_{exp}-{\overline{V}}_{pred}\right)}^2}},\hfill \end{array}$$

(5)

where the quantities $V_{exp}$ and $V_{pred}$ represent the velocity of the PIV test and the DNN prediction, respectively, and n is the sampling points of the velocity fields. It is seen from Equation (5) that a $R^2$ value close to 1 means a better prediction ability of the model.

Figure 7 shows the $R^2$ of the training set and the extrapolation set as a function of the number of hidden layers. The $R^2$ of the training set is seen to increase with the number of hidden layers. The increase in $R^2$ diminishes when the number of hidden layers exceeds three. For the extrapolation set, however, when the number of hidden layers exceeds four, $R^2$ decreases. This phenomenon is called over-fitting in the area of machine learning and needs to be avoided. Figure 7 also presents the neural network training time for different numbers of hidden layers. The related algorithm is implemented using MATLAB 2022b, running on a personal computer with an 11th Gen Intel(R) Core(TM) i7-11800H @ 2.30 GHz. It is seen that the training time of the model increases with the number of hidden layers.

According to Figure 7, a complex neuron network with a large number of hidden layers and neurons effectively extracts flow field features from the training datasets. The cost is a longer model training time and over-fitting of the extrapolation set. A DNN model with four hidden layers demonstrates high prediction accuracy ($R^2$ = 0.98) and a moderate training time (55 s) without an over-fitting phenomenon. Hence, four hidden layers are implemented for the DNN framework in the present work.

We train the DNN model using the Levenberg–Marquardt method. The model demonstrates excellent predictive performance during the training process, as shown in Figure 8. After 40 epochs, the DNN model reaches good prediction ability. The prediction RMSE of the training set, validation set, and test set is less than 1.05. There is no significant gap between the RMSE of the three datasets, indicating that the trained model achieves good prediction accuracy and generalizes well in the untrained flow field data from new operating conditions. Based on the discussion above, the complex multi-swirling flow fields are predicted and extrapolated via the optimized neuron network.

3.2. Performance of Trained DNN Model

In this section, the reconstruction ability of the flow field is evaluated based on the trained DNN model. Comparisons of axial velocity $V_x$ and radial velocity $V_y$ predictions from the DNN model with PIV experimental results at an air pressure drop of 0.5%, 1%, 2%, and 3% are shown in Figure 9. These comparisons are conducted after randomly dividing the dataset into a training set (60%), a validation set (20%), and a test set (20%). The predicted flow velocities from the DNN model show good agreement with experimental values. The $R^2$ values for the three datasets are high, ranging from 0.97 to 0.99, and the RMSE is low, ranging from 0.73 to 0.74. The results reveal that the trained DNN model is able to capture the complex unknown relationship between the input combustor coordinates/air pressure drops and the output air velocities.

We examine the prediction fidelity of the detailed flow field topologies at an air pressure drop of 0.5% and 1%, as shown in Figure 10. The error of velocity component e is calculated by the difference between the PIV test and the DNN prediction. The flow field features a separated dual swirl structure, consisting of two high-speed swirling jets (SWJ) entering from the inner pilot swirlers and the outer main swirler. A lip recirculation zone (LRZ) is formed near the lip between the inner and outer swirlers, which separates the inner and outer swirling jets. Inner recirculation zones (IRZ) characterized with negative axial velocities are found inside the swirling jets, generated by the negative pressure gradient. An exit positive velocity zone (EPZ) is observed at the center of the combustor outlet. It is seen from Figure 10 that the above complex flow regions have been successfully learned and captured by the deep learning model. The velocity field and vectors reconstructed by the trained neural network meet well with the flow field obtained from PIV experiments under the air pressure drops of 0.5% and 1%.

The flow fields with higher pressure drops of 2% and 3% are predicted by the DNN model, as shown in Figure 11. Swirling jets with high axial velocities up to 40 m/s are observed for the pressure drops of 3%. When increasing the air pressure drops, recirculation zones are characterized by higher reverse flow velocities arising from the increase in the axial pressure gradient. Generally, similar flow topologies and velocity vectors are found, except for small differences in local values. The local errors are seen around the boundary of the prediction domain. This results from a lack of surrounding data fed to the DNN model for the boundary grids, and a similar phenomenon is also seen in the previous study [37].

3.3. Extrapolation of Flow Fields

The above discussion illustrates that the trained deep learning model can reconstruct the detailed flow field well in the operating condition of $\sigma$ = 0.5–3%. In this section, we evaluate the generalization ability of the DNN model to ensure the network will not perform poorly in unknown operating conditions. Figure 12 shows the contour of the axial and radial velocities from the DNN prediction and PIV experiment, respectively, with an extrapolated operating condition of $\sigma$ = 4%. It is interesting to find that the DNN model can still give good predictions of the swirling flow field and velocity vectors, although the $\sigma$ = 4% is not involved in the training dataset. Despite the overall excellent prediction ability, it is admitted that small errors have occurred locally based on the prediction of the DNN model. The difference in axial velocity is located near the inner side of outer swirl jets (x = 40–50 mm, y = 35 mm). The radial velocity distribution reconstructed by the DNN model is smoother, while the flow field from the experiment manifests as discontinuous wrinkled structures. This phenomenon is similar to the results of the training datasets, as discussed in Section 3.2. The issue does not affect the present deep learning model much due to its successful prediction of most samples of the flow field at various air pressure drops and the wide variety of application scenarios presented in the next section.

Figure 13 shows the comparison of axial and radial flow velocities between the deep model prediction and the PIV experiments in the unfamiliar condition of $\sigma$ = 0.04. Generally, the flow velocities predicted by the deep learning model find good agreement with the experiment values. Most of the sampling points lie around the line y = x with a goodness of fit $R^2$ ranging from 0.92 to 0.97. The RMSE of the axial velocity $V_x$ is 1.616, while the RMSE of the radial velocity is 1.3976.

Table 1. Prediction performance of the DNN model.

RMSE	Training Set	Validation Set	Test Set	Extrapolation Set
$V_x$	0.7398	0.7369	0.7385	1.6160
$V_y$	0.7407	0.7415	0.7359	1.3976
$R^2$	Training set	Validation set	Test set	Extrapolation set
$V_x$	0.9873	0.9875	0.9872	0.9675
$V_y$	0.9753	0.9742	0.9812	0.9269

gives the overall prediction performance of the DNN model between different datasets. Considering that the extrapolated condition has never been fed into the DNN model, DNN relies on the inlet conditions and coordinates to reconstruct the flow velocities. Despite some degrees of deviation from the line y = x, the high goodness of fit shows the strong representation abilities of DNN in the swirling flow fields. The good generalization capability indicates that the established DNN model cannot only accurately reconstruct the flow fields of known training datasets but also predict the flow fields of unknown operating conditions, as detailed in the following Section 3.4.1.

Figure 14 presents a comparison of streamline plots between the DNN prediction and experimental values at the extrapolated condition of $\sigma$ = 0.04. The results show that the deep learning model accurately captures the complex multi-swirling flow topologies and gives good streamline predictions. Lip vortex structures are both identified downstream of the lip from the streamline plots of the DNN model and PIV experiment, which is a unique feature of separated multi-swirl flow. The coordinates of two large vortices from the PIV test are (70, 51) and (68, −50), while those vortices from DNN prediction are located at (66, 51) and (67, −50). We can conclude from the flow velocities and vortice structures that the deep learning model is capable of capturing the complex flow behaviors of swirling combustors and is applicable for preliminary combustor designs.

3.4. Combustor Physics Analysis Based on DNN Model

Section 3.2 and Section 3.3 discussed the good prediction performance of the constructed DNN model. In this section, the present study aims to explain what combustion physics can be extracted by DNN and its potential applications in combustion technologies.

3.4.1. Flow Analysis

Several pertinent gradient-based quantities such as strain rate and vorticity provide useful information for the analysis of coupling between flow and heat release/flame development. These quantities are calculated using the 2D flow velocity components from PIV experiments and DNN reconstruction. The out-of-plane vorticity (${\mathit{\omega }}_z$) is computed from the curl of the velocity field as [38]

$$\begin{array}{cc}\hfill {\mathit{\omega }}_z& ={\displaystyle \frac{\partial V_y}{\partial x}}-{\displaystyle \frac{\partial V_x}{\partial y}},\hfill \end{array}$$

(6)

where the quantity $V_i$ denotes the velocity component in the direction i. The strain rate in the x–y plane ($S_{xy}$) is calculated from the expression as follows [39]:

$$\begin{array}{cc}\hfill S_{xy}& ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial V_x}{\partial y}}+{\displaystyle \frac{\partial V_y}{\partial x}}\right),\hfill \end{array}$$

(7)

Figure 15a,b present the strain rate distribution in the x–y plane calculated from PIV data and DNN prediction, respectively. The shear strain rate, also known as the angular deformation rate, represents the amount of angular deformation per unit time of fluid elements. High values of strain rates (up to 1500 s⁻¹) are seen at the interfaces between the outer swirling jet flow and the surrounding recirculation flow. Two opposite orientations of fluid angular deformation are reflected in the positive and negative values of strain rate. For the high strain rate zone at the outlet of the outer swirling channel, the negative values indicate the OSL while the positive values correspond to the ISL. Moderate values of strain rate are also observed between the inner swirling flow and the recirculation zone, which are representative of CSL.

Figure 15c,d display the comparison of the 2D distribution of vorticity between the PIV data and the DNN prediction. As expected, the 2D vorticity field is similar to the distribution of the strain rate, indicating that the viscous fluid shear leads to the rotation of fluid elements and mutual shear deformation at the same time. Positive and negative vorticity represents the counterclockwise and clockwise orientation of flow vortex rotations. The outer SWJ is found with high vorticities of more than 3000 s⁻¹, which also corresponds to the ISL and OSL shown in Figure 15b. These regions of peak vorticities illustrate that the intense mixing and vortex rotations occur between the SWJ and recirculating flow.

A comparison of gradient-based quantities between experimental data and DNN prediction shows that the deep learning method is able to capture the complex vortex structures and strain rate characteristics. Reconstruction of the vorticity field and strain rate distribution based on the DNN model provides guidance for combustion performance design, such as ignition, flame stabilization, fuel dispersion, emissions, and so on.

Figure 16 compares the PIV experimental values and DNN predicted values of shear strain rate at two selected downstream locations (x = 18 mm, x = 50 mm). It is found that the predicted values of strain rate are in good agreement with the PIV test values at the two downstream locations. For the location of x = 18 mm, the strain rates are a bimodal distribution with peak values identified at the radial location of y = −54 mm–−33 mm, which is representative of the outer SWJ, OSL, and ISL. As the axial location moves to 50 mm, intense stain rates are found at the vicinity of the combustor liner wall, deteriorating the performance of ignition and flame kernel generation (detailed in Section 3.4.3).

Figure 17 displays a contour plot of the turbulent kinetic energy (K). In the present study, K is calculated as

$$\begin{array}{cc}\hfill K& ={\displaystyle \frac{1}{2}}\left({V_x^{\prime }}^2+{V_y^{\prime }}^2+{V_z^{\prime }}^2\right),\hfill \end{array}$$

(8)

in which $V_i^{\prime }$ is the fluctuation of the velocity in the direction i. The 2D PIV experiment only gives the axial velocity and radial velocity, and hence, the velocity fluctuations in the z direction are assumed to be the same as the radial velocity fluctuations for the calculation of K [39]. The annular jet flow (SWJ) issued from the outer swirler channel and the inner swirler are characterized by high turbulent kinetic energy. The predicted turbulent kinetic energy field is found to agree well with the turbulent kinetic energy field from the PIV test.

3.4.2. Fuel Droplets Dispersion

Fuel droplet transportation and spatial distribution makes up one of the key factors controlling the ignition process for spray-forced ignition. The spray dispersion is determined by the cold flow field and atomization process. Mainly two models of particle motion have been adopted: the continuum model based on the Euler coordinate system [40], and the discrete phase model in the Lagrange coordinate system [41]. The Lagrange discrete phase model has been widely used for its ability to track particle trajectories. The general process of the particle tracking model is: Step 1, solving the N-S equation numerically to obtain the air velocity field; Step 2, calculating the gas force to the particle based on the previous velocity field; Step 3, integrating the displacement for the particle motion trajectory. However, in practical combustion applications, the combustor velocity field requires some computational resources and time. In this section, the present work explores the potential of tracking fuel droplets based on the flow field reconstructed by the deep learning model.

The motion equation of fuel droplets in a cylindrical coordinate system without considering the evaporation term and gravity is expressed as [42]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\pi d_{Di}^3}{6}}{\rho }_{Di}{\displaystyle \frac{D\overrightarrow{v_{Di}}}{Dt}}& =C_D{\displaystyle \frac{\pi d_{Di}^2}{4}}{\displaystyle \frac{\rho }{2}}\left|\overrightarrow{v_{ai}}-\overrightarrow{v_{Di}}\right|(\overrightarrow{v_{ai}}-\overrightarrow{v_{Di}}),\hfill \end{array}$$

(9)

where $C_D$ is the drag coefficient, ${\rho }_{Di}$ is the density of fuel droplets, $\rho$ is the density of air, $d_D$ is the diameter of droplets, $v_D$ is the velocity of fuel droplets, $v_a$ is the flow velocity, and i represents the particle number. Solving the equation via the discretization method, the spray dispersion is calculated. The air temperature is set as 298 K, and the number of particles for the tracking calculation is set as 50. The initial particle parameters such as the Sauter Mean Diameter (SMD), radial position, and 3D velocity components all follow the normal distribution to simulate realistic conditions. Details of the initial droplet parameters are shown in

Table 2. Initial condition of fuel droplets.

Parameter	SMD (μm)	y (mm)	${\mathit{v}}_{\mathit{Dx}}$ (m/s)	${\mathit{v}}_{\mathit{Dy}}$ (m/s)	${\mathit{v}}_{\mathit{Dz}}$ (m/s)
N:Normal distribution	N(30, 10)	N(9, 1)	N(17.32, 3)	N(10, 3)	N(10, 3)

, and the N(A,B) indicates the normal distribution with a mean value of A and a variance of B.

Figure 18 presents the calculated spatial dispersion of the spray based on PIV test velocity fields and DNN-predicted flow velocities. An experimental spray distribution tested by Mie Scattering is provided for validation. The spray pattern from deep learning model is basically consistent with that from the PIV test. Fuel droplets injected from the central pilot nozzle first move with the pilot swirl flow. Then, the recirculating flow transports the fuel droplets outside to the inner side of the outer SWJ. Subsequently, droplets follow the outer swirling flow and finally reach the combustor liner wall. This stepwise fuel transportation process generally meets the experimental Mie scattering results, which is an important feature in separated multi-swirl flows. It is concluded from the above results that the deep learning model has the potential to predict spray dispersion and is engineering-applicable.

3.4.3. Ignition and Flame Propagation

Swirl spray flames in aero-engine combustors are ignited by multiple sparks at the liner wall. The flow velocity at the vicinity of the ignition plug is important for the early ignition phase, i.e., flame kernel generation. Figure 19 shows the histogram and probability density function (PDF) of velocities at the ignition region with an air pressure drop of 3%. It is found that the statistical velocity distribution of DNN prediction is similar to that of the PIV test with a peak PDF velocity of 19 m/s. The deep learning model provides reliable guidance for evaluating the ignition performance at the preliminary stage of combustor design.

Once the flame kernel is successfully generated, the subsequent flame propagation in turbulent swirl flow also needs evaluation. During the flame propagation process, the local flow regime highly affects the kernel propagation routes and the survival of the flame kernel. Figure 20 analyzes the possible flame propagation routes based on the strain rates from the PIV experiment and the DNN prediction. The regions enclosed by the purple line represent that the local strain rate is lower than 100 s⁻¹. The interface between the ISL and OSL is characterized by low strain rates. However, this region is unfavorable for flame propagation due to the high turbulent velocities (up to 40 m/s). The upstream flame propagation along the inner side of ISL will lead to ignition failure due to the absence of fuel. Two trajectories of the kernel propagating downstream to the combustor exit means the flame kernel has blown out. The successful flame propagation route is represented by the blue line shown in Figure 20. The predicted flame propagation route based on the deep learning model is generally consistent with that using the PIV test and also fits well with the ignition experiments.

3.5. Discussion

The above results show that the DNN model constructed in the present work can predict the swirling flow field for known and unknown conditions with high accuracy, and relevant combustor physics can be analyzed based on the predicted flow velocities. Now the ML-based method of flow field prediction is compared to conventional methods. Experimental flow fields tested from PIV measurements are usually missing in some regions due to the restriction of the imaging area and the window being polluted by the tracer particles. In addition, the PIV test needs optical windows that destroy the structural integrity of the combustor. Compared to PIV, the advantages of the DNN model are two-fold: (1) cost-effective; (2) ability to predict entire combustor regions without missing velocities; (3) fast predictions of flow fields without destroying combustor structures.

Numerical simulations have been widely used by combustor designers to evaluate swirling flow fields. However, the computational domain is often simplified from realistic combustor geometries, as the models are sensitive to the meshing of complex structures. Additionally, CFD simulations remain relatively time-consuming, requiring extensive iterative calculations [11]. In particular, coupling turbulence modeling with chemical kinetics and atomization models is complicated in swirl-stabilized combustors, and these models require high-precision data for calibration. In comparison to CFD, the advantages of the present deep learning model are threefold: (1) cost-effectiveness; (2) reduced operation time, allowing for rapid flow field predictions that facilitate performance optimization during the combustor design stage; and (3) real-time flow field prediction, enabling immediate output of flow field data based on monitoring combustor pressure in practical combustion applications.

4. Conclusions

This work proposes a deep learning model for predicting the swirling flow field in a multi-swirl aero-engine combustor. A deep neural network (DNN) is constructed, with the input consisting of the pressure drop across the combustor and 2D spatial coordinates, while the output comprises the 2D velocity components. An experimental Particle Image Velocimetry (PIV) dataset, covering an air pressure drop range from 0.5% to 4%, is utilized to train and test the model. The main conclusions drawn from this work are as follows.

(1) Increasing the number of hidden layers enhances the prediction accuracy of the training dataset; however, this comes at the cost of longer training times. Over-fitting occurs when the number of hidden layers exceeds five. Therefore, a deep neural network (DNN) model with four hidden layers is constructed for this work.

(2) The optimized DNN model achieves good accuracy after 40 epochs, with a training time of 55 s. The two velocity components in the swirling flow field predicted by the DNN model agree well with the values obtained from the PIV experiments. The flow field exhibits a separated dual swirl structure, consisting of two annular swirling jets (SWJ) and two recirculation zones (LRZ and IRZ), which have been effectively learned and captured by the DNN model.

(3) The trained DNN model has a good generalization ability that accurately reconstructs the flow field of unknown operating conditions.

(4) Flow analysis results indicate that intense viscous fluid shear occurs between the SWJ and the surrounding recirculation flow, leading to rotation and mutual shear deformation of the fluids. The deep learning model effectively learns the intrinsic flow mechanisms within the combustor, accurately predicting the vorticity field, strain rate field, and turbulent kinetic energy field. More importantly, the combustion-related characteristics, including fuel droplet dispersion and flame propagation, have been accurately extracted and analyzed based on the predicted flow fields generated by the DNN model.

(5) This paper provides a new approach for the flow field prediction in a multi-swirl combustor and facilitates the combustor performance evaluation during the design stage. The proposed deep learning model will be further extended and investigated, with a focus on flow field repair and real-time instantaneous predictions.