Updated Lagrangian particle hydrodynamics (ULPH) modeling for free-surface fluid flows

In this work, we develop an accurate and stable Updated Lagrangian particle hydrodynamics (ULPH) modeling to simulate complicated free-surface fluid flows. Leveraging its inherent properties as a Lagrangian particle method, the ULPH has natural advantages in modeling free-surface flows. However, similar to other meshfree methods, ULPH is subject to numerical instabilities and non-physical pressure fluctuations when solving the Navier–Stokes equation in the explicit numerical scheme. Within the framework of the ULPH method, several innovative enhanced treatment techniques have been proposed and combined with other previouly developed methods to establish an ULPH single-phase flow model. The main novelties of these techniques are the derivation of the density diffusive term in the continuum equation inspired by δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-SPH to eliminate pressure oscillations, and the proposal of a new free-surface search algorithm to determine the particles and their normal vectors at the free surface. The ULPH is a nonlocal fluid dynamics model, which is in fact a prototype of Peridynamics in fluid mechanics. Considering the nature of free-surface fluid flows, we strategically implement the diagonalization and renormalization of the shape tensor for particles located in close proximity to the free-surface region to improve the numerical stability of computations. Several complex free-surface flow benchmark examples have been simulated, which confirms that the enhanced treatment techniques can effectively capture the details of surface flow evolution and maintain long-term stability. Moreover, the qualitative and quantitative analyses of the results indicate that the proposed ULPH surface flow model is highly accurate and stable for simulating complex free-surface fluid flows.