A fourth-order kernel for improving numerical accuracy and stability in Eulerian and total Lagrangian SPH

The error of smoothed particle hydrodynamics (SPH) using kernel for particle-based approximation mainly comes from smoothing and integration errors. The choice of kernels has a significant impact on the numerical accuracy, stability and computational efficiency. At present, the most popular kernels such as B-spline, truncated Gaussian (for compact support), Wendland kernels have 2nd-order smoothing error and Wendland kernel becomes mainstream in SPH community as its stability and accuracy. Due to the fact that the particle distribution after relaxation can achieve fast convergence of integration error respected to support radius, it is logical to choose kernels with higher-order smoothing error to improve the numerical accuracy. In this paper, the error of 4th-order Laguerre-Wendland kernel proposed by Litvinov et al. \cite{litvinov2015towards} is revisited and another 4th-order truncated Laguerre-Gauss kernel is further analyzed and considered to replace the widely used Wendland kernel. The proposed kernel has following three properties: One is that it avoids the pair-instability problem during the relaxation process, unlike the original truncated Gaussian kernel, and achieves much less relaxation residue than Wendland and Laguerre-Wendland kernels; One is the truncated compact support size is the same as the non-truncated compact support of Wendland kernel, which leads to both kernels' computational efficiency at the same level; Another is that the truncation error of this kernel is much less than that of Wendland kernel. Furthermore, a comprehensive set of $2D$ and $3D$ benchmark cases on Eulerian SPH for fluid dynamics and total Lagrangian SPH for solid dynamics validate the considerably improved numerical accuracy by using truncated Laguerre-Gauss kernel without introducing extra computational effort.Comment: 37 pages and 12 figure

An explicit multi-time stepping algorithm for multi-time scale coupling problems in SPH

Simulating physical problems involving multi-time scale coupling is challenging due to the need of solving these multi-time scale processes simultaneously. In response to this challenge, this paper proposed an explicit multi-time step algorithm coupled with a solid dynamic relaxation scheme. The explicit scheme simplifies the equation system in contrast to the implicit scheme, while the multi-time step algorithm allows the equations of different physical processes to be solved under different time step sizes. Furthermore, an implicit viscous damping relaxation technique is applied to significantly reduce computational iterations required to achieve equilibrium in the comparatively fast solid response process. To validate the accuracy and efficiency of the proposed algorithm, two distinct scenarios, i.e., a nonlinear hardening bar stretching and a fluid diffusion coupled with Nafion membrane flexure, are simulated. The results show good agreement with experimental data and results from other numerical methods, and the simulation time is reduced firstly by independently addressing different processes with the multi-time step algorithm and secondly decreasing solid dynamic relaxation time through the incorporation of damping techniques.Comment: 37 pages 20 figure

Extended Eulerian SPH and its realization of FVM

Eulerian smoothed particle hydrodynamics (Eulerian SPH) is considered as a potential meshless alternative to a traditional Eulerian mesh-based method, i.e. finite volume method (FVM), in computational fluid dynamics (CFD). While researchers have analyzed the differences between these two methods, a rigorous comparison of their performance and computational efficiency is hindered by the constraint related to the normal direction of interfaces in pairwise particle interactions within Eulerian SPH framework. To address this constraint and improve numerical accuracy, we introduce Eulerian SPH extensions, including particle relaxation to satisfy zero-order consistency, kernel correction matrix to ensure first-order consistency and release the constraint associated with the normal direction of interfaces, as well as dissipation limiters to enhance numerical accuracy and these extensions make Eulerian SPH rigorously equivalent to FVM. Furthermore, we implement mesh-based FVM within SPHinXsys, an open-source SPH library, through developing a parser to extract necessary information from the mesh file which is exported in the MESH format using the commercial software ICEM. Therefore, these comprehensive approaches enable a rigorous comparison between these two methods.Comment: 34 pages and 13 figure

Simulation of a GOx-GCH4 Rocket Combustor and the Effect of the GEKO Turbulence Model Coefficients

In this study, a single injector methane-oxygen rocket combustor is numerically studied. The simulations included in this study are based on the hardware and experimental data from the Technical University of Munich. The focus is on the recently developed generalized k–w turbulence model (GEKO) and the effect of its adjustable coefficients on the pressure and on wall heat flux profiles, which are compared with the experimental data. It was found that the coefficients of ‘jet’, ‘near-wall’, and ‘mixing’ have a major impact, whereas the opposite can be deduced about the ‘separation’ parameter Csep, which highly influences the pressure and wall heat flux distributions due to the changes in the eddy-viscosity field. The simulation results are compared with the standard k–# model, displaying a qualitatively and quantitatively similar behavior to the GEKO model at a Csep equal to unity. The default GEKO model shows a stable performance for three oxidizer-to-fuel ratios, enhancing the reliability of its use. The simulations are conducted using two chemical kinetic mechanisms: Zhukov and Kong and the more detailed RAMEC. The influence of the combustion model is of the same order as the influence of the turbulence model. In general, the numerical results present a good or satisfactory agreement with the experiment, and both GEKO at Csep = 1 or the standard k–# model can be recommended for usage in the CFD simulations of rocket combustion chambers, as well as the Zhukov–Kong mechanism in conjunction with the flamelet approach