Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations. We employ PINNs for solving the Reynolds-averaged Navier–Stokes equations for incompressible turbulent flows without any specific model or assumption for turbulence and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier–Stokes equations for laminar flows by solving the Falkner–Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Unsteady fluid systems are nonlinear high-dimensional dynamical systems that may exhibit multiple complex phenomena in both time and space. Reduced Order Modeling (ROM) of fluid flows has been an active research topic in the recent decade with the primary goal to decompose complex flows into a set of features most important for future state prediction and control, typically using a dimensionality reduction technique. In this work, a novel data-driven technique based on the power of deep neural networks for ROM of the unsteady fluid flows is introduced. An autoencoder network is used for nonlinear dimension reduction and feature extraction as an alternative for singular value decomposition (SVD). Then, the extracted features are used as an input for a long short-term memory (LSTM) network to predict the velocity field at future time instances. The proposed autoencoder-LSTM method is compared with non-intrusive reduced order models based on dynamic mode decomposition (DMD) and proper orthogonal decomposition. Moreover, an autoencoder-DMD algorithm is introduced for ROM, which uses the autoencoder network for dimensionality reduction rather than SVD rank truncation. The results show that the autoencoder-LSTM method is considerably capable of predicting fluid flow evolution, where higher values for the coefficient of determination R² are obtained using autoencoder-LSTM compared to other models.

Dynamic Mode Decomposition (DMD) is a data-driven reduced order method, which is known for its power to capture the basic features of dynamical systems. In fluid dynamics, modal analysis of unsteady fluid flows over moving structures is significant in terms of state estimation and control. However, the underlying algorithm of the DMD requires a fixed spatial domain, which is an obstacle for applying the DMD on the numerically investigated problems using dynamic meshes. In this study, a hybrid method called Hybrid Dynamic Mode Decomposition (HDMD) is presented for analysis of unsteady fluid flows over moving structures based on the DMD and machine learning. According to the assessment of several data interpolation methods, the K-nearest neighbor algorithm is employed for the interpolation of the numerical data from dynamic meshes at each time step to a single stationary grid. Three different case studies (rotating cylinder, oscillating airfoil, and Savonius wind turbine) are assessed to ensure the validity of the proposed method. Minimum mean R² equal to 0.92 has been obtained for all of the mentioned cases, indicating the robustness of the HDMD algorithm for a variety of fluid flow simulations.