On zero-order consistency residue and background pressure for the conservative SPH fluid dynamics

TL;DR

This paper analyzes the zero-order consistency residue in conservative SPH methods, revealing its impact on flow damping and energy dissipation, and evaluates correction techniques in various flow scenarios including complex geometries.

Contribution

It provides a comprehensive analysis of the zero-order consistency residue's effects and limitations of correction schemes in SPH, especially under high background pressure conditions.

Findings

Residue causes non-physical energy dissipation in SPH flows.

Reverse kernel gradient correction reduces but cannot eliminate residue effects.

High background pressure exacerbates the zero-order consistency issues.

Abstract

As one of the major challenges for the conservative smoothed particle hydrodynamics (SPH) method, the zero-order consistency issue, although thought to be mitigated by the particle regularization scheme, such as the transport velocity formulation, significantly damps the flow in a long channel for both laminar and turbulent simulations. Building on this finding, this paper not only thoroughly analyzes the damping reason in this pressure-driven channel flow, but also relates this problem with the excessive numerical dissipation in the gravity-driven free-surface flow. The common root cause of the non-physical numerical damping in the two typical flow scenarios, the zero-order gradient consistency residue, is exposed. The adverse influence of the background pressure on the residue for the two scenarios is revealed and discussed. To comprehensively understand the behavior of the residue and mitigate its potential adverse effects, we conduct both theoretical analysis and numerical experiments focusing on the key sensitive factors. For studying the residue-induced non-physical energy dissipation in the gravity-driven free-surface flow, the water depth and input dynamic pressure in the inviscid standing wave case are tested. To investigate the velocity loss in the pressure-driven channel flow, we examine the effects of the channel length, resolution, and outlet pressure. The state-of-the-art reverse kernel gradient correction technique is introduced for the two typical flows, and proved to be effective in reducing the residue effect, but we find its correction capability is fundamentally limited. Finally, the FDA nozzle, an engineering benchmark, is tested to demonstrate the residue influence in a complex geometry, highlighting the necessity of correction schemes in scenarios with unavoidable high background pressure.