Flow field reconstruction of trash rack based on generative adversarial networks

Trash racks play crucial roles in ensuring the safe operation of hydroelectric power stations (Nguyen & Naudascher 1991; Naudascher & Wang 1993) and protecting fish ecosystems (Raynal et al. 2013; Lučin et al. 2020). These devices are primarily designed to intercept large debris that might potentially harm the turbine components, while also being strategically designed to prevent fish from inadvertently entering the turbine units. However, the dynamic water flow loads that the trash racks are exposed to make them susceptible to fatigue failure. This is particularly evident when the frequency of the vortex street matches the natural frequency of the trash racks, leading to vortex-induced vibrations (Alazwari et al. 2022; Liu et al. 2022). Such vibrations can compromise the stability of the trash racks or even result in their damage. Therefore, it is essential to have a comprehensive understanding of the flow field characteristics downstream of the trash racks that are helpful in their design and maintenance.

Trash racks are commonly simplified as cylinders when analyzing the flow field downstream of their trailing edges. Currently, high-resolution wake flow fields are primarily obtained using computational fluid dynamics (CFD) methods (Hariri Asli et al. 2023), such as RANS models (Wu et al. 2022), large eddy simulation (LES) (Wu et al. 2020), and direct numerical simulation (DNS) (Trias et al. 2015). While CFD has been successful in many aspects of hydrodynamic analysis and design, obtaining a high-resolution flow field with a large number of grid points requires abundant computational resources. Given the substantial investment required for both experiments and numerical simulations, proposing a rapid and efficient approach to deriving the high-resolution flow fields and the turbulence characteristics is crucial. The promotion of the integration of artificial intelligence and big data methods (Guo et al. 2023) in fluid numerical computation, along with further exploration of the physical laws governing fluid motion, has emerged and developed rapidly (Kutz 2017; Kim et al. 2021).

Deep learning, an important branch of machine learning, has been extensively utilized in recent years in the field of fluid mechanics (Wang et al. 2018; Lee & You 2019; Raissi et al. 2019). Besides flow control (Zhou et al. 2020), deep learning can also be applied to solve the Navier–Stokes equations. Deep learning models utilized in flow control can optimize multiple parameters to achieve desired target characteristics while generating new complex turbulent motions. This advancement aids in enhancing the understanding of turbulent physical processes, which proves to be a challenging task with traditional control techniques. When solving the Navier–Stokes equations, CFD methods predominantly rely on modeling turbulent physics. On the other hand, deep learning decouples the Navier–Stokes equations primarily through the utilization of neural networks and the incorporation of residuals into the original loss function. This approach leads to the development of physics-informed neural networks (PINN) with fast convergence and high accuracy (Pang et al. 2019; Krishnapriyan et al. 2021). Kharazmi et al. (2021) encoded the Navier–Stokes equations using PINN and coupled them with the structural dynamic equations to derive the structural parameters, velocity fields, and dynamic structural motions. Ling et al. (2016) proposed a tensor-based neural network model that incorporated Galilean invariants of the Reynolds stress mean equation. This approach resulted in computational results that were superior to those obtained using nonlinear eddy viscosity models. Kim & Lee (2020) utilized generative adversarial networks and recurrent neural networks to achieve long-term sequence prediction of turbulence development. Their predicted results showed consistent spatiotemporal correlations with direct numerical simulation (DNS).

Besides the above-mentioned research, flow field results can be affected by potential issues such as noise, low resolution due to defects in measuring equipment, instability in observation scenarios in experiments, or insufficient grid resolution in CFD simulations. Recently, the research about deriving high-resolution (HR) flow fields from low-resolution flow fields (LR) is becoming increasingly popular, which is commonly referred to as super-resolution reconstruction (SR). Dong et al. (2014) first used deep learning models to reconstruct HR images with higher efficiency than the conventional bicubic interpolation method. Ribeiro et al. (2020) and Zhang et al. (2023) both utilized U-shaped convolutional networks to implement the reconstruction of time-averaged flow fields behind a cylinder. However, deep learning models face a significant challenge when it comes to reconstructing irregular turbulent flows, which are more complex than the regular time-averaged flow fields. Kim et al. (2021) employed an unsupervised model called Cycle-GAN to generate high-resolution turbulent flows from different filtered scales. Moreover, they validated the coherence of the reconstructed flow by comparing the high-order turbulence indicators and turbulent characteristics. Zhou et al. (2022) introduced a novel SR model that successfully reconstructed three-dimensional high-resolution turbulent fields using low-resolution flow field data.

Many previous studies have successfully utilized deep learning models to reconstruct turbulent flow fields. However, these models often have low input and output resolutions, typically at 64 × 64 or 128 × 128. In a recent study conducted by Kim et al. (2021), the impact of different input resolutions on the reconstruction of flow fields was investigated. Their findings revealed that higher input resolutions resulted in better reconstruction of turbulent flow fields across various models. Zhou et al. (2024) successfully reconstructed the wake flow field structure of a cylinder using LGF-CNN with an output resolution of 55 × 37. Although the lower output resolution can enhance computational efficiency, it also introduces significant interpolation errors. In simulations of the sliced 2D flow fields that adopted LES or DNS, tens even hundreds of thousands of grid nodes will be necessary. Interpolating such a vast number of grids into 128 × 128 slices of turbulent data may lead to the loss of important details, which finally alter the original characteristics of the flow field. Hence, it is essential to employ higher-resolution images in deep-learning models.

Considering the high demand of memory in the derivation of high-resolution flow fields, the current investigation is devoted to incorporating a sub-pixel convolution layer in the computational framework, which is capable of generating detailed feature maps at a resolution of 512 × 512, while not requiring a significant increase in memory. This layer learns filters to map features to higher resolutions, thus avoiding direct convolution and deconvolution computations in HR space, which significantly alleviates the GPU burden. Additionally, the proposed approach utilizes the Wasserstein distance as a loss function and incorporates gradient penalty features. Combining these techniques, the SRWgan-GP neural network framework successfully achieves the reconstruction of the HR(512 × 512) turbulent flow field of the trash rack.

The spacing between each trash rack bar is 0.16 m. Given that the trash rack comprises several bars, the current simulations select three adjacent bars to reflect actual engineering conditions, thereby considering the influence of the outer bars on the central one. Thus, the flow field results obtained are solely for this middle bar. The bar width B = 0.16 m and thickness D = 0.022 m. To reduce the number of grids, the height of the computational domain H is taken as 0.022 m.

As is shown in Figure 1(b), the original flow field is obtained from the LES simulations, then a 2D slice is extracted at the height of Z = 0.5 H. Two probes for validating the accuracy of time series results, namely J1 and J2, are positioned at x/d = 1 and x/d = 4 on the central axis behind the middle trash racks bar. The reconstruction velocity filed region in the x- and y-directions behind the trash rack bars in this section are interpolated to create an HR flow field. This HR flow field is the target for the deep learning model's reconstruction. Additionally, the flow field is interpolated to a LR 16 × 16 domain to eliminate the original flow field characteristics. This LR flow field serves as the input for the deep learning model. The main objective of this paper is to utilize the spatial information from the instantaneous LR flow field to generate the corresponding HR flow field, achieving the mapping from LR data to HR data, denoted as HR(t) = F(LR(t),θ), where θ represents the model parameters.

In the neural network training process, 400 flow field datasets are used for training, 100 for validation, and 500 for testing. Each dataset has a time interval of 0.0001 s, resulting in a total test data duration of 0.5 s.

In the grid independency study, three different grid configurations were selected with total computational node amounts of 2.1, 5.3, and 8 million, respectively. The height of the first layer grid is 4 × 10⁻³, with a y⁺ value under 5, and the time step is set at 1 × 10⁻³.

Table 1 lists the results derived from different grids and taking the results from the finest grid as reference, the deviation of Strouhal number (St) changed by 5.6% from the coarse grid to the fine grid, and only 0.7% from the medium grid to the fine grid.

The drag coefficient (C_d) for both fine and medium grids is consistent, aligning with results obtained using DNS in Trias et al. (2015). However, the calculated fluctuation of the drag coefficient (C_d′) for the fine grid is slightly lower than that of Trias, with an absolute error of 0.03. Regarding the fluctuation of the lift coefficient (C_l′), the results calculated by LES are consistently lower than those obtained by DNS. The fluctuation of the lift coefficient (C_l′) presented in this study falls within the range calculated by Cao & Tamura (2016). To balance computation time and accuracy, a medium-density grid is employed, and further verification of key flow field indicators has been conducted.

As the adoption of the current medium grid and boundary condition settings meets the accuracy requirements. These same settings are applied to calculate the flow field of the trash racks (composed of three 7.23:1 rectangles).

In current literature, the high-resolution flow field used as a model input is predominantly obtained through bicubic interpolation (Dong et al. 2014). Compromising the significant demand of computational resources and time, the 2D flow field is typically interpolated into a 64 × 64 or 128 × 128 flow field. However, flow fields derived from DNS or LES can have millions of grid cells within a 2D slice. Interpolating such a vast amount of grid node data into a 128 × 128 plane will inevitably result in significant numerical errors or even alter the original flow field characteristics. Therefore, it is necessary to interpolate the computed results into higher-resolution flow fields, such as 512 × 512 and 1,024 × 1,024.

To enhance spatial resolution without consuming significant GPU memory, Shi et al. (2016) proposed a network framework based on a sub-pixel convolution layer. The design concept is as follows.

(1) In the encoder: Multiple blocks are utilized to extract features, and the resulting feature maps from the block outputs are saved to be used in the Unet's skip connections. Each block consists of two convolutional layers with a kernel size of 3 and one max pooling layer. While the convolution layers do not alter the size of the feature maps, the pooling layer reduces the size of the feature maps by half. (2) In the latent representation space: two residual blocks are employed to enhance the model's ability to capture high-dimensional information while maintaining the constant feature map size of 64 × 64. (3) In the decoder: deconvolution operations are used to increase the size of the feature maps by a factor of 2. The output result of each block is then fused with the corresponding skip connection from the Encoder using concatenation. Finally, a pixel shuffle (PS) layer is incorporated to map the size of the feature maps to a super-high-resolution flow field of 512 × 512.

The discriminator's framework is relatively simple, with its main function being to determine the authenticity of images. Firstly, the high-resolution flow field (512 × 512) is transformed into higher-dimensional data with a smaller feature size using the PixelUnShuffle operation, effectively reducing network parameters. Subsequently, convolution operations are applied multiple times to convert the authenticity of the image into the similarity of image distributions.

The neural network model in this study is constructed using PyTorch and developed using an NVIDIA GeForce 3060Ti. The Adam optimizer is employed to minimize the loss function. Both the generator and the discriminator adopt a learning rate of 0.0001, and stable convergence is achieved after 2,000 training iterations. A total of 400 flow field datasets are used for training 100 for validation, and 500 for testing. Each dataset has a time interval of 0.0001 s, resulting in a total test data duration of 0.5 s.

In the formula, m,n are the dimensions of the sample, MAX_I represents the maximum possible pixel value of the sample, μ_I, μ_J are the average pixel values, σ_I,σ_J are the variance of the pixel values, σ_IJ is the covariance, and c₁ and c₂ are constants to stabilize the division with a weak denominator.

In the formula, MSE_ideal is 0, PSNR_ideal is 60, and SSIM_ideal is 1.

When water flows through a trash rack, Kármán vortex streets are formed behind it inducing multiple micro vortices. These structures are mainly concentrated around the vortex street, making the overall flow field appear disordered therefore bringing great challenge in the reconstruction of such a complex flow using deep learning methods. The current investigation adopts SRWgan-GP to reconstruct the high-resolution instantaneous flow field. To facilitate appropriate comparisons, the performance of SRWgan-GP was assessed in contrast to Unet and CNN models, which have been widely utilized in reconstructing flow fields. Specifically, for the SRWgan-GP model, the generator component was implemented using the Unet framework. This allowed for the evaluation of reconstruction performance without introducing the discriminator component, which is a crucial element in the original SRWgan-GP architecture.

In this study, the SRWgan-GP network framework was successfully employed to reconstruct high-resolution (512 × 512) turbulent flow structures of trash racks. Most researchers (Liu et al. 2020; Ribeiro et al. 2020; Zhou et al. 2024) use deep learning network frameworks that output flow field resolutions at 128 × 128 or even lower. For simple flow features or laminar flow fields, this resolution might be sufficient, but for the DNS and LES turbulent flow field data, hundreds of thousands to millions of computational cells are required to capture smaller-scale turbulent structures. Additionally, higher resolutions provide a broader reconstructed field, allowing for a more comprehensive analysis of the flow field's dynamic characteristics. Therefore, we propose using a sub-pixel convolution layer in the final layer of the generator to enhance the flow field resolution from 128 × 128 to 512 × 512 which significantly improves the resolution. For fine turbulent fields that rely on higher grid resolutions, this model offers substantial advantages.

Due to the chaotic behavior of turbulence across a wide range of spatiotemporal scales, its inherent irregularity and disorder make it difficult to accurately reconstruct the turbulent field. Kim et al. (2021) reconstructed isotropic turbulence from DNS calculations using a Cyclegan model at a lower scaling factor (r = 4), where the reconstructed flow field was very close to the actual field, with the smallest MSE error of 0.00548. However, at a higher scaling factor (r = 16), the reconstructed flow field differed significantly from the actual field, resulting in a much larger MSE error of 0.087. According to Zhou et al. (2024), their LCF-CNN model was applied to reconstruct the PIV experimental flow field of a cylinder with a Reynolds number of 33,000, where at r = 4(16), the model reconstructed average relative errors in the x- and y-directions of the flow field of 0.0335(0.0872) and 0.1274(0.457), respectively. In our study, using a scaling factor of 32 (512/16 = 32), the proposed method based on Wgan-Gp enhanced the training stability and successfully reconstructed the trash rack flow field with an MSE error of 0.001, accurately capturing the turbulent structures and the Karman vortex street of the trash racks.

In summary, the model proposed in this paper, with its higher flow field resolution, is better suited for engineering applications involving the reconstruction of flow fields with a large number of grids and extensive computational domains. Additionally, the model possesses high accuracy, capable of precisely reconstructing actual flow characteristics, thereby providing high-quality flow field reconstruction results.

This paper presents the SRWgan-GP neural network framework for reconstructing ultra-high-resolution (512 × 512) flow fields, achieving successful reconstruction of 2D flow fields behind the trash racks. Through a comparison with two typical generative models, namely Unet and CNN, this study analyzes and validates the accuracy in reconstructing flow fields in the x- and y-directions, as well as capturing turbulence characteristics and higher-order quantities using deep learning models. The main conclusions of our study are as follows:

• (1) The SRWgan-GP model demonstrates a high accuracy in reconstructing the flow field in the x- and y-directions, evidenced by its MSE of 0.001, SSIM of 0.994, and PSNR of 46.557. These metrics suggest that the model is effective in capturing the nuances of the Kármán vortex street and the turbulent structures present downstream of the trash rack. Unet is observed that efficiently captures the vortex street but exhibits a larger error in reconstructing smaller turbulence structures. The flow field reconstructed by the CNN exhibited significant errors, as indicated by an MSE of 0.006, SSIM of 0.954, and PSNR of 38.234.

• (2) The SRWgan-GP model demonstrates effective reconstruction of the flow velocity time series in both the x- and y-directions. Notably, the reconstruction error for the y-direction time series is more pronounced than in the x-direction, suggesting a higher turbulence intensity in the y-direction. The shedding frequency of the Kármán vortex street downstream of the trash rack is observed to be 10 Hz. Both the SRWgan-GP and Unet models align with this frequency, whereas the CNN model overestimates it, reaching 14 Hz. In the higher frequency domain (>100 Hz), the energy represented in the power spectra of all three models is consistently overestimated, suggesting challenges in accurately capturing flow dynamics at smaller temporal scales.

• (3) The results demonstrate that SRWgan-GP exhibits the smallest error and effectively reconstructs the instantaneous Reynolds shear stress field. However, Unet shows a larger error in the vicinity of the trash racks’ trailing edge.

This research is supported by the National Nature Science Foundation of China (Grant No. 52179060, 52209081, and 51909024).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.