In addition to two-dimensional flow field prediction, 30–32 some researchers also studied three-dimensional problems using deep learning, 33–35 and they have achieved some good prediction results. However, in their test results, the reconstructed flow field exhibited an obvious sawtooth phenomenon. 29 used a multi-scale graph NN model for flow field prediction. The extracted geometric parameters were then used along with the Reynolds number and angle of attack (AOA) as the inputs for a multilayer perceptron (MLP) network to predict the flow field. 28 used CNNs to extract geometric features from airfoil images. They took the geometric parameters and physical coordinates of the airfoil as the inputs of the NN, and the velocity and pressure of the current physical coordinates were the output in this way, they established a flow field prediction model for subsonic airfoils. 27 proposed an efficient deep-NN approach to predict compressible flows over transonic airfoils. Here, point-to-point modeling is defined as the process of constructing nonlinear mapping functions between input parameters and prediction parameters based on black-box models such as neural networks. 26 Previous studies have attempted to use point-to-point modeling based on deep learning to predict the flow field. However, in the near wall region of an airfoil, this pixelation will inevitably lead to a lack of information regarding the flow field, and it may even generate nonphysical solutions. The existing flow-field prediction methods using the SDF and airfoil images as NN inputs are all based on uniform Cartesian grids to carry out the NN training and testing tasks. 19–21 Machine learning methods can use a large amount of CFD-generated data to model, analyze, and predict aerodynamic coefficients of airfoils. 18 Recently, data-driven machine-learning methods have been widely applied as a fourth paradigm for studying aerodynamics. However, the traditional reduced-order model is difficult to apply to multi-scale, transient, and discontinuous processes. Reduced-order methods, such as proper orthogonal decomposition (POD) 16 and dynamic mode decomposition (DMD), 17 greatly reduce the complexity of solving complex systems and improve the efficiency of modeling and solving. Currently, using a CFD solver to solve an airfoil flow field requires large numbers of iterative processes these have high memory demands and they are computationally expensive and time-consuming. ![]() However, in practical engineering applications, as the computational objects become increasingly complex, the number of computational grid points also increases geometrically. In comparison with wind tunnel tests, CFD methods are more flexible. Large-scale computer simulations provide high-precision aerodynamic flow field data for aircraft design. With the rise in computer techniques, CFD approaches have become widely used for flow field simulations of airfoils. Large eddy simulation (LES), 5–7 direct numerical simulation (DNS), 8–12 and Reynolds-averaged NS (RANS) equations 13–15 are also widely used in CFD simulations. ![]() ![]() ![]() For example, both wind tunnel tests and computational fluid dynamics (CFD) simulations are based on the Navier–Stokes (NS) equations. Theoretical science provides support for both experimental and computational science, the latter of which is the third paradigm. The second paradigm is model-based theoretical science. Through analysis of a large number of qualitative and quantitative experimental results, it is concluded that the proposed DAN can improve the interpretability of the model while obtaining good prediction accuracy and generalization capability for different airfoils and flow-field states. The geometric parameters extracted from the transformer encoder, together with the Reynolds number, angle of attack, flow field coordinates, and distance field, are input into a multilayer perceptron to predict the flow field of the airfoil. To extract the geometric representation of the input airfoils, the grayscale image of the airfoil is divided into a set of patches, and these are input into the transformer encoder by embedding. In this article, a novel data-driven deep attention network (DAN) is proposed for reconstruction of incompressible steady flow fields around airfoils. The traditional method for obtaining aerodynamic parameters of airfoils by solving Navier–Stokes equations is a time-consuming computing task.
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