PREDICTING FLUID PARTICLE TRAJECTORIES WITHOUT FLOW COMPUTATIONS: A DATA-DRIVEN APPROACH

The Lagrangian analysis of a fluid flow entails calculating the trajectories of fluid particles, which are governed by an autonomous or non-autonomous dynamical system, depending on whether the flow is steady or unsteady. In conventional methods, a particle's position is incremented time step by time step using a numerical solver for ordinary differential equations (ODEs), assuming that the fluid velocity field is known analytically or can be acquired through either numerical simulation or experimentation. In this work, we assume instead that the velocity field is unavailable but abundant trajectory data are available. Leveraging the data processing power of deep neural networks, we construct data-driven models for the increment in particles' positions and simulate their trajectories by applying such a model recursively. We develop a novel, more experiment-friendly model for non-autonomous systems and compare it with two existing models: one developed for autonomous systems only and one developed for non-autonomous systems with some knowledge of the time-varying terms. Theoretical analysis is performed for all three that sheds a new light on the existing models. Numerical results obtained for several benchmark problems confirm the validity of these models for advancing fluid particles' positions and reveal how their performance depends on the structure of the neural network and physical features of the flow, such as vortices.