High-order consistent SPH with the pressure projection method in 2-D and 3-D

Abstract Mesh-free methods such a smoothed particle hydrodynamics (SPH) have advantages over mesh-based methods for flow in complex domains but attaining consistent high-order accurate solutions with the conventional form of SPH has yet to be resolved. The high-order smoothing error dominates the SPH error until increasingly fine resolutions cause the low-order discretisation error to dominate; this is related to discretising a volume integral into a summation. In this paper, a high-order consistency correction based on modified SPH (MSPH) and the modified finite particle method (FPM) is proposed for improving the order of the limiting discretisation error. The new technique is an arbitrarily high-order extension of these schemes where the complexity of the consistency correction and the required computations are reduced by using simplified versions of the smoothing kernel derivatives. Tested in Eulerian form, the proposed high-order consistent SPH technique (HOCSPH) is combined with new high-order SPH kernel functions, designed to improve the order of the SPH smoothing error, and the resulting hybrid technique is shown to converge according to the smoothing error initially before converging according to the HOCPSH error once the latter becomes dominant. The initial high-order convergence lowers the computational effort required to achieve higher accuracy with hybrid HOCSPH in comparison to HOCSPH with second-order smoothing accuracy kernel functions. However, for highly irregular distributions it is found that the use of kernels with second-order smoothing accuracy provides more consistent convergence properties. A number of flows are simulated in 2-D and 3-D using the new HOCSPH technique in combination with the pressure projection method, and the results show that the method is accurate and able to model highly complex flow patterns. Some issues with stability of the projection method related to pressure–velocity collocation are identified, and several remedies are proposed. While effective for the test cases herein, these remedies are new in this context and require further attention for generalisation in future studies.