Physics-Constrained Deep Learning of Incompressible Cavity Flows

Abstract

High resolution simulations of incompressible flows have become routine across a range of engineering applications. Despite their routine use, due to the high dimensional parameter space present for most practical applications, a comprehensive exploration of the available parameter space is often impractical. In this work, we demonstrate the ability of physics-constrained deep learning methods to provide an efficient means of exploring high-dimensional parameter spaces with minimal amounts of data from high resolution computational fluid dynamic simulations. As a specific application, we choose the well established problem of a 2D lid driven cavity flow. While giving an extensive treatment of the classic case of a square cavity, we extend the analysis to treat an isosceles trapezoid. In so doing, the number of parameters determining the solution includes not just the Reynolds number, but also two additional parameters characterizing the geometry of the cavity. Thus, together with the $\left( x, y \right)$ variation of the flow and pressure in configuration space, the presence of these three parameters results in the solution varying in a 5D space. It is shown that in the absence of data, physics-constrained methods are able to provide an accurate description of the cavity flow in this 5D space up to intermediate values of the Reynolds number, but fails to train for sufficiently high Reynolds numbers. In contrast, using a small quantity of flow data, a single neural network is able to provide an accurate description for a broad range of Reynolds numbers and cavity geometries. Once trained, such a model provides a rapid surrogate that can be used to efficiently explore the 5D space. This 5D surrogate model is subsequently used to identify critical parameter values for the merger and splitting of vortices as the Reynolds number and cavity geometry are varied.