Comparative study on volume conservation among various SPH models for flows of different levels of violence

The weakly-compressible smoothed particle hydrodynamics (WCSPH) models offer distinct advantages in simulating free-surface flow. Within the rapidly evolving WCSPH field, Classical WCSPH model, based on the artificial viscous term, and its derived models, which additionally introduce numerical diffusive terms into the continuity equations - including the δ-SPH, δ-SPH^C, δ-SPH^C+, δ-LES-SPH, and δ-SPH^F models have gained prominence recently. The numerical diffusive term is found to alter the particle distributions in the free-surface region, leading to a change in the mean water level that reflects the volume conservation, which might result in inaccuracies in some simulations. This study evaluates the volume conservations of the aforementioned WCSPH models for flows at quasi-hydrostatic states evolved from varying levels of flow violence under prolonged simulations, through a series of numerical tests involving hydrostatic, standing-wave, and dam-break cases. For the hydrostatic and standing-wave cases, the performance of different types of WCSPH models remains consistent overall. The Classical WCSPH and δ-LES-SPH models exhibit minimal changes in volume, with almost no decrease in the mean water level occurring. The δ-SPH, δ-SPH^C, δ-SPH^C + models demonstrate decreases in the mean water level converging to around 0.2 times the SPH particle diameter (Δx), indicating favorable volume conservation. Conversely, the δ-SPH^F model consistently exhibits a decrease in the mean water level exceeding 1 Δx at t = 2000s in a linear fashion, resulting in a noticeable reduction in volume. Under the dam-break case, most of the mean water levels simulated by different models experience a small increase, which is close to the decrease observed under the hydrostatic and standing-wave cases with corresponding models. Only the δ-SPH^F model continues to show a continuous decreasing tendency. In summary, considering volume conservation, the δ-LES-SPH model demonstrates the best performance (excluding the Classical WCSPH model as it cannot simulate violent flow), followed by the δ-SPH^C, δ-SPH^C+, δ-SPH and δ-SPH^F models.