Numerical study on the dissipation mechanisms in sloshing flows induced by violent and high-frequency accelerations. II. Comparison against experimental data

In Part I of this series [Marrone et al., Numerical study on the dissipation mechanisms in sloshing flows induced by violent and high-frequency accelerations. I. Theoretical formulation and numerical investigation, Phys. Rev. Fluids 6, 114801 (2021)], a theoretical formulation and the numerical model were developed in order to obtain a complete perspective of the energy balance of a violently accelerated flow confined inside a rectangular tank. The tank-fluid system was periodically excited with a predetermined law of motion and the force between the wall and the fluid and the global energy balance were computed. In this second part, the experimental validation of the previous formulation is presented. In order to make a comparison with a previous experimental campaign, where the tank moves along a single degree of freedom mechanical guide, two numerical problems have been studied: in the first, the decaying movement of the tank is prescribed according to the experimental measurements, and in the second the tank is coupled to a mass-spring-damper equation, and the sloshing force produced by the confined fluid acts as an external force. Both problems have been studied for two different fluids, water and oil, which implies a difference of two orders of magnitude in terms of Reynolds number. A complete description of the energy balance inside the fluid tank is performed and the complexity of the fluid dynamic behavior that takes place inside the tank is explained. The results are compared to the experimental measurements in terms of fluid-wall interaction and energy dissipation.

Figure 1

Experimental setup and outline. (1) Load cell, (2) metallic plate with mass $m_p=0.06$ kg, (3) upper set of springs with stiffness constant $k_1=2154.17$ N/m, (4) lower set of springs with stiffness constant $k_2=2154.17$ N/m, (5) laser sensor, (6) mechanical guide, (7) accelerometer, (8) methacrylate tank and C-shaped wooden structure with mass $m_s=2.403$ kg, (9) pair of solenoids acting as release mechanism.

Figure 2

Experimental snapshots of the SDOF vertical sloshing water experiments carried out in [2] for 10 g and 50% filling level.

Figure 3

Sketch of the problem geometry and initial conditions (see Part I for more details).

Figure 4

Tank motion recorded in the experiment of [2] plotted in terms of elevation (dashed line) and acceleration of the tank (solid line).

Figure 5

Time history of the energy decay obtained by the SPH simulation at $N=400$ and by the 1D analytical model as described in Part I (solid lines). The dashed line represents the tank elevation. Labels (a) to (f) correspond to the same time instants of Figs. 6 and 7.

Figure 6

Four representative instants of the flow evolution obtained by the SPH simulation at $N=400$ for the $\text{Re}=233000$ case. Contours refer to turbulent viscosity ratio ${\mu }_T/\mu$. See the Supplemental Material [19].

Figure 7

SPH simulation at $N=400$ for the $\text{Re}=233000$ case: vorticity contour at $t/T=10.56$ (left plot) and $t/T=24.64$ (right plot). See the Supplemental Material [19].

Figure 8

Left: time history of the energy components defined in Sec. 3: resolved viscous dissipation ${\mathcal{E}}_V$, modeled turbulent viscous dissipation ${\mathcal{E}}_V^{\mathrm{turb}}$, numerical diffusion ${\mathcal{E}}_{\mathcal{N}}$, total energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$ for the case $N=400$ and $\text{Re}=233000$. Right: ensemble average of of the energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$ time history for resolutions $N=50$, $N=100$, $N=200$, and $N=400$; error bars refer to the computed standard deviation.

Figure 9

Four representative instants of the flow evolution obtained by the SPH simulation at $N=400$ for the $\text{Re}=4660$ case. Contours refer to turbulent viscosity ratio ${\mu }_T/\mu$. See the Supplemental Material [19].

Figure 10

SPH simulation at $N=400$ for the $\text{Re}=4660$ case: vorticity contour at the same time instants of Fig. 7. See the Supplemental Material [19].

Figure 11

Left: Time history of the energy components defined in Sec. 3: resolved viscous dissipation ${\mathcal{E}}_V$, modeled turbulent viscous dissipation ${\mathcal{E}}_V^{\mathrm{turb}}$, numerical diffusion ${\mathcal{E}}_{\mathcal{N}}$, total energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$ for the case $N=400$ and $\text{Re}=4660$. Right: Ensemble average of of the energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$ time history for resolutions $N=50$, $N=100$, $N=200$; error bars refer to the computed standard deviation.

Figure 12

Ensemble average of the total vertical force for resolutions $N=50$, $N=100$, $N=200$ for $\text{Re}=233000$. The $F_y^{\mathrm{stat}}$ force component is also reported.

Figure 13

Force comparison between the experimental data and the SPH simulation at $N=400$ and $\text{Re}=233000$: total force (top) and $F_y^{\mathrm{dyn}}$ component (bottom). On the bottom plot the initial part of the experimental $F_y^{\mathrm{dyn}}$ is colored in red where incorrect overestimation of the force is linked to the release mechanism.

Figure 14

SPH simulation at $N=400$ and $\text{Re}=233000$. Top plot: total force against the tank velocity; bottom plot: time history of the power $-{\mathcal{P}}_{\mathrm{ext}}^{\mathrm{dyn}}$ (see definition in the text).

Figure 15

SPH simulation at $N=400$ and $\text{Re}=233000$. Time history of the powers $-{\mathcal{P}}_{\mathrm{ext}}^{\mathrm{dyn}}$, $-{\dot{\mathcal{E}}}_M^{\mathrm{dyn}}$, and ${\mathcal{P}}_V$ (see definitions in the text).

Figure 16

SPH simulation at $N=400$ and $\text{Re}=233000$. Time history of the mechanical energy ${\mathcal{E}}_M^{\mathrm{dyn}}$, the dissipated energy ${\mathcal{E}}_{\mathrm{diss}}$, the external work ${\mathcal{W}}_{\mathrm{ext}}^{\mathrm{dyn}}$. The solid line with $\mathrm{\Delta }$ symbols represents ${\mathcal{W}}_{\mathrm{ext}}^{\mathrm{dyn}}$ as computed through the experimental data.

Figure 17

SPH simulation at $N=400$ and $\text{Re}=4660$. Time history of the mechanical energy ${\mathcal{E}}_M^{\mathrm{dyn}}$, the dissipated energy ${\mathcal{E}}_{\mathrm{diss}}$, the external work ${\mathcal{W}}_{\mathrm{ext}}^{\mathrm{dyn}}$. The solid line with $\mathrm{\Delta }$ symbols represents ${\mathcal{W}}_{\mathrm{ext}}^{\mathrm{dyn}}$ as computed through the experimental data.

Figure 18

Evolution of the vertical position of the tank over time obtained from the FSI-SPH simulation at $N=200$. Top figure corresponds to $\text{Re}=233000$ for water and bottom figure to $\text{Re}=4660$ for oil. The experimental signal obtained from the accelerometer is plotted for comparison. Variability of the numerical outcome, represented in terms of standard deviation, is plotted in lighter color (light blue on the top plot and pink on the bottom one).

Figure 19

Vorticity fields of the simulations with free-slip (left plot) and no-slip (right plot) conditions at time $t/T=2.43$, $N=200$.

Figure 20

Time history of the ensemble average of the energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$ for resolutions $N=200$ using free-slip and no-slip conditions on the solid boundaries. Error bars refer to the computed standard deviation. Left: $\text{Re}=233000$ for water case. Right: $\text{Re}=4660$ for oil case.