Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> B. INCOMPRESSIBLE SPH <s> Abstract An incompressible smoothed particle hydrodynamics (SPH) method is presented to simulate Newtonian and non-Newtonian flows with free surfaces. The basic equations solved are the incompressible mass conservation and Navier–Stokes equations. The method uses prediction–correction fractional steps with the temporal velocity field integrated forward in time without enforcing incompressibility in the prediction step. The resulting deviation of particle density is then implicitly projected onto a divergence-free space to satisfy incompressibility through a pressure Poisson equation derived from an approximate pressure projection. Various SPH formulations are employed in the discretization of the relevant gradient, divergence and Laplacian terms. Free surfaces are identified by the particles whose density is below a set point. Wall boundaries are represented by particles whose positions are fixed. The SPH formulation is also extended to non-Newtonian flows and demonstrated using the Cross rheological model. The incompressible SPH method is tested by typical 2-D dam-break problems in which both water and fluid mud are considered. The computations are in good agreement with available experimental data. The different flow features between Newtonian and non-Newtonian flows after the dam-break are discussed. <s> BIB001 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> B. INCOMPRESSIBLE SPH <s> Grid-based methods have difficulty resolving features on or below the scale of the underlying grid. Although adaptive methods (e.g., RLE, octrees) can alleviate this to some degree, separate techniques are still required for simulating small-scale phenomena such as spray and foam, especially since these more diffuse materials typically behave quite differently than their denser counterparts. In this paper, we propose a two-way coupled simulation framework that uses the particle level set method to efficiently model dense liquid volumes and a smoothed particle hydrodynamics (SPH) method to simulate diffuse regions such as sprays. Our novel SPH method allows us to simulate both dense and diffuse water volumes, fully incorporates the particles that are automatically generated by the particle level set method in under-resolved regions, and allows for two-way mixing between dense SPH volumes and grid-based liquid representations. <s> BIB002 </s> Survey on Smoothed Particle Hydrodynamics and the Particle Systems <s> B. INCOMPRESSIBLE SPH <s> A constant-density approach, which corrects intermediate density errors by adjusting the half-time-step velocity with exact projection, is proposed for the multi-phase SPH method developed in our previous work [X.Y. Hu, N.A. Adams, An incompressible multi-phase SPH method, J. Comput. Phys. 227 (2007) 264-278]. As no prescribed reference pressure is required, the present approach introduces smaller numerical viscosity and allows to simulate flows with unprecedentedly high density ratios by the projection SPH method. Numerical examples for Taylor-Green flow, capillary waves and for Rayleigh-Taylor instability are presented and compared to theoretical solutions or references from the literature. <s> BIB003
In computational fluid dynamics, incompressibility is usually applied by the incompressible SPH (ISPH) BIB001 - BIB003 method based on pressure projection, and pressure is obtained by solving the pressure Poisson equation by using the conjugate gradient method. This method has also been introduced into the field of graphics BIB002 . The ISPH first uses a force other than pressure to predict the intermediate velocity of the particle and then uses the discrete pressure Poisson equation to project the intermediate velocity into the non-dispersion vector space and calculate the pressure and pressure accordingly. The ISPH can allow large time steps and impose strict incompressibility, but it requires considerable time to calculate linear equations in each time step, and the overall computational overhead is high.