Explicit calculation for the pressure Poisson equation to simulate incompressible fluid flows in a mesh‐free method

Abstract

In this study, an equation is derived to explicitly solve the pressure Poisson equation (PPE) in the moving particle semiimplicit method. In the derivation, the PPE is discretized by an improved Laplacian model and the incompressible condition is satisfied by establishing a direct relationship between pressure and the known flow information. An improved gradient model is adopted in the method while a repulsive force is used to handle particle clustering. To validate the method, a hydrostatic problem, the impact of two rectangular fluid patches and dam-breaking flows are simulated. The numerical results are compared with analytical solutions and experimental measurements in terms of free surface, pressure, and velocity. Good agreements in the comparisons are achieved, showing that the method can calculate smooth pressure field and accurate pressure and velocity distributions. K E Y W O R D Sdam-break flow, explicit method, free surface, mesh-free method, pressure field INTRODUCTIONThe mesh-free method is advantageous in handling interface flows, 1,2 such as free surface flows and two-phase flows.To simulate incompressible fluid flows, one significant branch for the mesh-free method is to solve the pressure Poisson equation (PPE) as the projection-based particle method. There are several types of this method with wide application in simulating free-surface flows referred as the moving particle semiimplicit method (MPS) 3-6 and the incompressible smoothed particle hydrodynamics (ISPH). [7][8][9] The MPS method was initially introduced to model complex thermal-hydraulic problems by using particles. The method includes the particle interaction models representing pressure gradient, diffusion, incompressibility, and a free surface boundary condition, in which the particle interaction is localized by using a kernel function. In the method, an implicit pressure calculation procedure is used to model the incompressible fluid flows. 10 Koshizuka et al. 3 used the MPS method to simulate two types of breaking waves on slopes as the plunging and spilling breakers. They obtained the critical value to separate the two breaker types and calculated the repeated process for a float of moving offshore and inshore. Koshizuka 5 reviewed the numerical developments of the MPS method in the calculation of the thermal-hydraulic problems in terms of fluid dynamics and heat transfer. Gotoh 11 presented a pioneering work of the large eddy simulation-based turbulence modeling in the context of particle methods by developing the so-called subparticle scale turbulence model. The model has been applied to both MPS and smoothed particle hydrodynamics (SPH) to capture the turbulence. 12,13 The MPS method is also modified by many researchers to enhance its performance in different research areas. Khayyer and Gotoh 14 proposed a corrected MPS method to track water surface in breaking waves, in which the pressure 3034