Analysis of free-surface flows through energy considerations: Single-phase versus two-phase modeling

The study of energetic free-surface flows is challenging because of the large range of interface scales involved due to multiple fragmentations and reconnections of the air-water interface with the formation of drops and bubbles. Because of their complexity the investigation of such phenomena through numerical simulation largely increased during recent years. Actually, in the last decades different numerical models have been developed to study these flows, especially in the context of particle methods. In the latter a single-phase approximation is usually adopted to reduce the computational costs and the model complexity. While it is well known that the role of air largely affects the local flow evolution, it is still not clear whether this single-phase approximation is able to predict global flow features like the evolution of the global mechanical energy dissipation. The present work is dedicated to this topic through the study of a selected problem simulated with both single-phase and two-phase models. It is shown that, interestingly, even though flow evolutions are different, energy evolutions can be similar when including or not the presence of air. This is remarkable since, in the problem considered, with the two-phase model about half of the energy is lost in the air phase while in the one-phase model the energy is mainly dissipated by cavity collapses.

Figure 1

Sketch of the confined dam-break flow problem. Contours are representative of the pressure field in the initial configuration.

Figure 2

Confined dam-break flow: comparison between SPH (left column) and LS-FVM (right column) solvers for the time instants $t\sqrt{g/H}=2.0$, 5.8, 6.2, 6.8. Contours are representative of the pressure field (single-phase model).

Figure 3

Confined dam-break flow: comparison between SPH (left column) and LS-FVM (right column) solvers for the time instants $t\sqrt{g/H}=8.0$, 8.4, 8.8, 9.2. Contours are representative of the pressure field (single-phase model).

Figure 4

Time histories of the mechanical energy evaluated by SPH and LS-FVM solvers (single-phase model).

Figure 5

Time histories of the SPH mechanical energy with varying the spatial resolution (single-phase model).

Figure 6

Contour of the volume-specific dissipation ${\epsilon }_V$ [see Eq. (15)] for four time instants at $\text{Re}=5000$ using a spatial resolution $H/\mathrm{\Delta }x=800$ (single-phase model).

Figure 7

Confined dam-break flow: SPH time evolution of the two-phase simulation. The plots refer to time instants $t\sqrt{g/H}=6.11$, 8.02, 8.43, 9.35 (two-phase model).

Figure 8

Contour of the volume-specific dissipation ${\epsilon }_V$ [see Eq. (15)] for four time instants using a spatial resolution $H/\mathrm{\Delta }x=400$ (two-phase model).

Figure 9

Comparison of the SPH viscous dissipation terms ${\mathcal{Q}}_{V_{\mathrm{water}}}$ and ${\mathcal{Q}}_{V_{\mathrm{air}}}$ during the confined dam-break flow evolution (two-phase model).

Figure 10

Comparison of the mechanical energy decay between SPH single-phase (dashed) and SPH two-phase model (solid) during the confined dam-break flow evolution (two-phase model).

Figure 11

Time histories of the SPH mechanical energy of the water phase varying the spatial resolution (two-phase model).

Figure 12

Time histories of the SPH energy terms $\mathrm{\Delta }{\mathcal{E}}_{C_{\mathrm{water}}},\mathrm{\Delta }{\mathcal{E}}_{C_{\mathrm{air}}}$, and $\mathrm{\Delta }{\mathcal{E}}_{M_{\mathrm{air}}}$ (two-phase model).