Multilayer perception (MLP)

MLP is one of the essential parts of PINNs, which is a fully connected neural network. When MLP is being trained, the known input and output would be fed as the labels so that the parameters in neural network would be optimized to minimize the loss functions. Activation functions are added to increase the nonlinearity of the mapping. In this study, the mapping is expressed as:

where uι¯, p, and uι′uȷ′ represent the mean velocity components, pressure, and Reynolds stress, respectively. Previous study (Huang et al., 2023) has provided the in-depths information about the determination of hyperparameter and MLP structure on solving RANS problems. In this study, 10 hidden layers, each containing 300 neurons is prescribed in MLP structure (Figure 2). Swish function is selected as the activation function, expressed as follows:

Figure 2.

The schematic of PINNs structure, where the output is the quantities from three-dimensional RANS equations.

Loss function

The loss functions of PINNs consist of two parts: loss functions of physics constraints and experimental observation. To obtained the mean flow field, the RANS equations are preparedly designed into the loss function of physics constraints:

The derived term uι′uȷ′¯, traditionally, is closed by semiempirical model based on the Boussinesq eddy viscosity hypothesis:

where S_ij is the mean rate of strain tensor, νt is the eddy viscosity, and K is the turbulent kinetic energy (TKE). The assumption of isotropic νt is considered as the causes that lead to the unsatisfactory numerical results. Meanwhile, the application of Eq. 5 in PINNs couldn’t perform well in previous work. Huang et al. (2023) designed the t-EV model in consideration of the anisotropy. This model has been validated its capability of well-predicting jet in crossflow field. Therefore, the modified symmetric 2-rank anisotropic eddy viscosity νt(ij) is imported in this study so that Eq. 5 could be rewritten as

where νt(ij) is symmetrically expressed as:

Thus, another six formulations would be designed into the loss function of physics constraints:

Note that the derivatives in these PDEs are obtained by automatic differentiation (AD) (Baydin et al., 2018). Therefore, the derivatives of these PDEs can be calculated by applying the chain rule under backpropagation. To satisfy these physical equations, the loss functions of physics constraints should approach zeros, which is formulated as

where N_e is the number of physical equations. N_d is the domain points in the selected regions.

For the observation loss functions, the two-dimensional and two-component (2D2C) results from PIV measurements are chosen as the labels, containing 2D2C mean velocities and their corresponding Reynolds stress. Furthermore, the non-slip boundary condition is added as known labels. Therefore, the loss function of the observations is reported as

The total loss then written by summing the weighted loss of above loss functions:

where ωe and ωobs are the weighting coefficients. In this study, both of the weighting coefficients are set as the same because of the equal importance. Tensorflow (Abadi et al., 2016) is adopted as the deep-learning package to construct PINNs model. Adam optimizer (Kingma and Ba, 2014), with default hyper-parameters is chosen to minimize the loss function due to its efficient optimization algorithm. The batch sizes for both loss functions of physical equations and experimental observations are set as 20,000. A single NVIDIA Tesla V100s GPU is chosen for PINNs’ training. The losses during training are depicted in Figure 3, where three samples preliminarily serve as training datasets. When the loss was found to be stable, the PINNs’ regression would be regarded as the convergence. Generally, it’s observed that the losses tend to be stable after 30,000.

Figure 3.

Loss profile of PINNs’ regression with three samples strategies.

Multi-sources strategies

For the swirling flow in the combustor, scalar field is equally important as flow field. As the key variable to evaluate the film cooling performance, film cooling effectiveness was relatively measurable in the scalar field. Therefore, besides for the source from flow observation, the source from scalar field would also be imported to further improve PINNs’ regression. In this study, the scalar (i.e., film cooling effectiveness) observation of effusion plate is supplemented. The scalar distribution, as shown in Figure 4, is experimentally obtained by pressure-sensitive paint (PSP) measurement (Jiang et al., 2022). The PSP molecules could return to the ground state via the oxygen quenching process thus the cooling effectiveness could be acquired according to the calibrated luminosity-pressure correlation. The adopted 1,000 pairs of snapshots were captured for each case with a frequency of 1,800 Hz. The uncertainty of the PSP measurement was calculated to be around 2.7%.

Figure 4.

Scalar (film cooling effectiveness) distribution on the effusion plate.

The scalar equation is thus added into the loss function of physical equations:

where Pr is the Prandtl number, set as 0.71 in this study. uι′c′¯ is the turbulent scalar flux. Similarly, the derived term uι′c′¯ is also unclosed. In this study, the simple gradient diffusion hypothesis (SGDH) is adopted:

where Prt is the turbulent Prandtl number, set as 0.85 in this study. To better characterize the anisotropy, the isotropic νt is substituted as νt(ij). Therefore, other three physical equations are supplemented:

Dataset establishment

The dataset is established employing the PIV measurement (Jiang et al., 2022), where 1,000 pairs of snapshots were captured for each case. The spatial resolution of each snapshot was 0.023 mm/pixel with a frequency of 1 Hz. The uncertainty of the PIV measurement was calculated to be around 3%. As shown in Figure 5, seven observations were accessed during the measurement, containing two planes at XoZ direction (Y = 5 and 70 mm) and 5 planes (Z = 0, ±25.25, and ±50.5 mm) at XoY direction. In subsequent study, the coordinate is normalized, with a measured length L = 25.25 mm. To find out PINNs’ performance based on various number of observations, several training datasets are designed, as listed in Table 1. The two planes at XoZ direction are adopted throughout all datasets while the planes at XoY direction are increasingly sampled.

Figure 5.

Observation sampled by PIV measurement, (a) is the sampling zones, (b) is three sampling strategies to be fed into PINNs.

Effect of sampling strategy on field reconstruction

Number of observations

In this section, we focus on investigating the effect of the number of observations on reconstructing swirling flow, with the streamwise velocity as the main variable of interest. Following the sampling strategy employed in previous work (Cai et al., 2021b), sample 1 consists of four planes surrounding the regions of interest. However, unlike the well-reconstructed fields obtained in previous work, the performance of PINNs in predicting swirling flow fields is poor. As shown in Figure 6, (a) depicts experimental observations from PIV measurement, while (b) represents predictions made by PINNs using sample 1. It’s observed that the test dataset fails to capture the characteristics of the swirling flow, especially at the exit of the swirler, where the swirling jet should have emerged. Owing to the supplement of plane 1, the region where jet hit on the effusion plate could be predicted, but presented with shrunk area. This suggests that the vanilla sampling strategy is insufficient in leveraging the limited information provided by sample 1 to reconstruct complex swirling flow fields. It is therefore suggested that increasing the number of observations in the training dataset could improve the performance of PINNs. Subsequently, sample 2 and 3, which include additional information on the swirling flow field, will be adopted in the subsequent study.

Figure 6.

Comparison of streamwise velocity between (a) experimental observation and (b) prediction conducted by PINNs with sample 1.

Figure 7 presents the predictions of PINNs using sample 2, where planes 4 and 5 serve as the test dataset. Compared to the predictions from sample 1, the overall performance shows improvement. Figure 6(a) shows that the jet at the exit of the swirler is captured for plane 4, although the velocity magnitude is weakened. However, a failed prediction is obtained for plane 5 in Figure 7b. The magnitudes of the recirculation zone and swirling jet at the swirler’s exit are much weaker than those of the experiments, as marked with red lines. In contrast, a better prediction is achieved when PINNs are trained with sample 3, as shown in Figure 8. The predicted streamwise velocity distribution of plane 3 shows satisfactory agreement with that of the experiment, including the recirculation zone, the jet at the swirler’s exit, and near the effusion plate. Furthermore, an improved denoise on the distribution could also be observed. Thus, the increasingly improved predictions suggest that compared to simple turbulent flow, swirling flow requires a higher number of fed observations to better capture the three-dimensional and complex swirling flow characteristics using PINNs.

Figure 7.

Comparison of streamwise velocity between experimental observation (top) and predictions conducted by PINNs with sample 2 (bottom), where (a) is plane 4, (b) is plane 5.

Figure 8.

Comparison of streamwise velocity between experimental observation (top) and prediction conducted by PINNs with sample 3 (bottom), by evaluating plane 3.

Effective sampling strategy

From the results above, it can be concluded that the jet at the swirler’s exit is the most sensitive region to sampling strategies. In addition, Karniadakis et al. (2021) revealed that PINNs models often struggled on penalizing the PDEs residuals when tackling the solution with steep target gradients. Therefore, the regions with steep velocity gradients, such as the swirling jets, brought by the complex vortex structures and intense shearing would similarly increase the difficulty in PINNs’ accurate regression. Therefore, supplementing partial information at these sensitive regions can improve the prediction of PINNs. To improve the performance of PINNs, sample 1, which performed the poorest, was selected for investigation. In the updated sampling strategy, partial information containing the jet at the exit of the swirler was imported for planes 6 and 7 in the XoY direction, as shown in Figure 9a. The updated sample was named as sample 4. The predictions of PINNs with sample 4, shown in Figure 9b, indicate a significant improvement in the prediction of the unknown zones, such as the jet near the effusion plate and recirculation zones, compared with the poorly performing sample 1. However, it is also observed that the prediction of the jet highly close to the effusion plate is still unsatisfactory, where the streamwise velocity impinging onto the effusion plate is weakened. Statistically listed in the Table 1, compared with the poorly performed sample 1, sample 4 had the training volume of 106,308, where the additional observation of sample 4 reduced the numbers by 50% and 75%, respectively, compared with sample 2 and sample 3. To quantitatively analyse the PINNs’ performances in the test dataset of, relative L₂ error is imported as the metrics, reported as:

Figure 9.

Scheme of sample 4, (a) is the sampling strategy, and (b) is the prediction conducted by PINNs with sample 4.

(15)

εf=‖f(xf)−g(xf)‖2‖g(xf)‖2.

Sample 1 and 4 were chosen as the comparison due to the same test dataset. It was found that the relative L₂ error of the test dataset has been effectively dropped from 0.84 to 0.24, with an reduction of 71.4%. Therefore, it can be concluded that the performance of PINNs for the prediction of complex swirling flow can be improved by increasing the number of observations, particularly in sensitive regions, to better capture the three-dimensional and complex characteristics of swirling flows.

Multi-source strategy

The scalar distribution of the effusion plate (shown in Figure 4) highlights the complex interaction between the effusion film and swirling flow. The low FCE zone is attributed to the mixing of swirling jets with the film, resulting in film coverage destruction. Supplementing observations of scalar sources prompts the PINNs to extract latent flow information and improve their performance.

The investigation focuses on samples 2 and 4. In sample 2, the results in Figure 10 reveal improvements in the predicted flow below Y/L = 1. The regions marked by the red lines show more agreement with experimental observations, indicating a better distribution compared to the single-source strategy. For Plane 4 (Figure 10a), the marked zone is corrected as a higher velocity zone. While for plane 5 (Figure 10b), although the high streamwise velocity at the swirler outlet is not predicted accurately, the jet’s majority at the exit of the swirler agrees well with the experiments, indicating a better distribution than the single-source strategy. Note that the improvements propagating vertically are clearly reduced after Y/L > 1, where only limited correction could be observed, and the streamwise velocities of recirculation zone are still ill-matched with that of experimental observations. For sample 4, as shown in Figures 11a and 11b, the improvement is also observed, mainly for the marked regions near the effusion plate. To quantitatively analyse the amelioration provided by the multi-source strategy, the streamwise velocity profiles at the ill-performed marked region are shown in Figure 11c. It’s found that the more upstream zone shows the more noticeable improvement. For plane 5 (Z/L = 1), whose ill-prediction is concentrated in the downstream zone (Figure 9), the mean error of streamwise velocity weakly reduces from 18.7% to 13.5% after adopting multi-source strategy. While for planes 3 (Z/L = 0) and 4 (Z/L = −1), the improvements are obvious, where the mean errors are reduced from 18.1% to 6.9% and 6.8% to 1.6%, respectively. Note that the improvements for planes 5 and 3 near the jet impingement zone (Y/L < 0.5) are limited to some extents due to the insufficient downstream observations, suggesting that a satisfactory reconstruction of swirling flow field requires both designed multi-source and sampling strategies.

Figure 10.

PINNs’ prediction based on sample 2 with single source (top) and multi-source (bottom), validated by (a) plane 4 and (b) plane 5.

Figure 11.

PINNs’ prediction based on sample 2 with single source (a) and multi-source (b). (c) Are the quantitative comparisons of marked region for validation set.

To show the three-dimensional analysability of the reconstructed flow field, PINNs regression based on sample 4 and the multi-source strategy is selected as the investigation. As shown in Figure 12, the vortex structures are extracted from the predicted velocity field based on the Q criterion with a normalized value of 0.113. Note that only half of the vortex can be predicted because the PIV experiments only captured the half of the swirling flow field. The shown vortexes are found as a ring-like pattern similar with the previous literature (Chen and Driscoll, 1989; Oberleithner et al., 2011), named as recirculation vortex or recirculation bubble. The contours of vortexes in Figures 12a and 12b are marked by vertical velocity of Y direction and lateral velocity of Z direction, respectively. The black streamlines indicate the general swirling direction of swirling flow. The adiabatic cooling effectiveness is also shown. It’s found from the vortex structure distributions that the vortexes are mainly captured near the exit of the swirler (around 0 < X/L < 2.5), corresponding to the direct impingement region of the swirling jet in Figure 11b. From the contour in Figure 12a, it’s observed that the lateral velocity near the effusion plate is shown as negative value, indicating that the swirling flow would induce the effusion film deviating towards the opposite direction of the Z-axis. Therefore, the cooling effectiveness distribution presents the swept pattern. The region marked with the dashed red rectangle shows the obvious deviation of the cooling effectiveness distribution because of the strongest sweeping effect brought by the closest interaction of swirling flow. As the swirling flow develops downstream (2.5 < X/L < 4.5), there are still vortexes be recognized and marked as the negative lateral velocity, meaning that the swirling flow could continuously affect the effusion cooling and deviate the cooling effectiveness distribution. It’s then found from Figure 12b that both positive and negative values of vertical velocity are found as balanced distributions. The region with positive and negative value indicates that the swirling flow go through the upwash and downwash motions, respectively. It’s found near the exit of the swirler that the vertical velocity distributed as the negative value, revealing that the swirling flow downwards impacts and damages the surface of the effusion film, reducing the cooling effectiveness. Similarly, the region marked with the dashed red rectangle still shows the most severe deterioration of cooling performance than the surroundings due to the most direct impingement of swirling flow. Due to the occlusion of the vortex structures, it could be observed combining with Figure 4 that the effusion films at Z/L < −1 are less destructed due to the upwash motion of swirling flow. As the swirling flow develops downstream, the recognized vortexes are marked as weakly positive values. It is revealed that the swirling flow is slightly lifted off, meaning that the destructions on the cooling performance are gradually attenuated, thus leading to the recovery in cooling effectiveness.

Figure 12.

The predicted three-dimensional vortex structures using sample 4 and multi-source strategy, the vortexes are extracted based on Q criterion (Q = 0.113).