The complex flow modeling based on machine learning is becoming a promising way to describe multiphase fluid systems. This work demonstrates how a physics-informed neural network promotes the combination of traditional governing equations and advanced interface evolution equations without intricate algorithms. We develop physics-informed neural networks for the phase-field method (PF-PINNs) in two-dimensional immiscible incompressible two-phase flow. The Cahn–Hillard equation and Navier–Stokes equations are encoded directly into the residuals of a fully connected neural network. Compared with the traditional interface-capturing method, the phase-field model has a firm physical basis because it is based on the Ginzburg–Landau theory and conserves mass and energy. It also performs well in two-phase flow at the large density ratio. However, the high-order differential nonlinear term of the Cahn–Hilliard equation poses a great challenge for obtaining numerical solutions. Thus, in this work, we adopt neural networks to tackle the challenge by solving high-order derivate terms and capture the interface adaptively. To enhance the accuracy and efficiency of PF-PINNs, we use the time-marching strategy and the forced constraint of the density and viscosity. The PF-PINNs are tested by two cases for presenting the interface-capturing ability of PINNs and evaluating the accuracy of PF-PINNs at the large density ratio (up to 1000). The shape of the interface in both cases coincides well with the reference results, and the dynamic behavior of the second case is precisely captured. We also quantify the variations in the center of mass and increasing velocity over time for validation purposes. The results show that PF-PINNs exploit the automatic differentiation without sacrificing the high accuracy of the phase-field method.

Recent experiments have found that there is unstable vaporous cavitation around a surface-piercing hydrofoil at high Froude numbers and small yaw angles, and it would promote ventilation formation¹, but the cavity shedding dynamics and the mechanism of cavitation-vortex interaction are still open problems. In this paper, the unstable vaporous cavities around a surface-piercing hydrofoil are numerically investigated using the Large Eddy Simulation coupled with Schnerr-Sauer cavitation model. Numerical simulations can predict the cavity features, including an aerated base cavity aft of the hydrofoil trailing edge, vaporous cavitation at the hydrofoil suction surface, and tip-vortex cavitation. A U-shaped vapor cloud shedding together with a horse-shoe vortex is observed during the unsteady cavitation evolution, i.e. the cavity development, cut-off, and collapse. This irregular shedding is related to the three-dimensional re-entrant jet induced by the velocity reflection at the vaporous cavity closure line. Furthermore, the effects of the vaporous cavitation on the vorticity generation are attributed to vortex stretching, baroclinic torque, and vortex dilatation by using the vorticity transport equation. This study could contribute to the novel hydrofoil designs and their flow control.

Proper orthogonal decomposition (POD) enables complex flow fields to be decomposed into linear modes according to their energy, allowing the key features of the flow to be extracted. However, traditional POD requires high-quality inputs, namely, high-resolution spatiotemporal data. To alleviate the dependence of traditional POD on the quality and quantity of data, this paper presents a POD method that is strengthened by a physics-informed neural network (PINN) with an overlapping domain decomposition strategy. The loss function and convergence of modes are considered simultaneously to determine the convergence of the PINN-POD model. The proposed framework is applied to the flow past a two-dimensional circular cylinder at Reynolds numbers ranging from 100 to 10 000 and achieves accurate and robust extraction of flow structures from spatially sparse observation data. The spatial structures and dominant frequency can also be extracted under high-level noise. These results demonstrate that the proposed PINN-POD method is a reliable tool for extracting the key features from sparse observation data of flow fields, potentially shedding light on the data-driven discovery of hidden fluid dynamics.

Unsteady Reynolds-averaged Navier–Stokes (URANS) equations have been widely used in engineering fields to investigate cavitating flow owing to their low computational cost and excellent robustness. However, it is challenging to accurately obtain the unsteady characteristics of flow owing to cavitation-induced phase transitions. In this study, we propose an implicit data-driven URANS (DD-URANS) framework to analyze the unsteady characteristics of cavitating flow. In the DD-URANS framework, a basic computational model is developed by introducing a cavitation-induced phase transition into the equations of Reynolds stress. To improve the computational accuracy and generalization performance of the basic model, the linear and nonlinear parts of the anisotropic Reynolds stress are predicted through implicit and explicit methods, respectively. A data fusion approach, allowing the input and output of characterized parameters at multiple time points, is presented to obtain the unsteady characteristics of the cavitating flow. The DD-URANS model is trained using the numerical results obtained via large-eddy simulation. The training data consist of two parts: (i) the results obtained at cavitation numbers of 2.0, 2.2, and 2.7 for a Venturi flow, and (ii) those obtained at cavitation numbers of 0.8 and 1.5 for a National Advisory Committee for Aeronautics (NACA) 66 hydrofoil. The DD-URANS model is used to predict the cavitating flow at cavitation numbers of 2.5 for a Venturi flow and 0.8 for a Clark-Y hydrofoil. It is found that the DD-URANS model is superior to the baseline URANS model in predicting the instantaneous periodic shedding of a cavity and the mean flow fields.