Numerical study on the dissipation mechanisms in sloshing flows induced by violent and high-frequency accelerations. I. Theoretical formulation and numerical investigation

The sloshing motion of a confined liquid inside a vertically moving tank is analyzed in the present series of paper. The main objective of the study is to understand the multiple resulting energy dissipation mechanisms, namely wall-liquid impacts and free surface phenomena, among others. This analysis is connected to the damping effects on the aircraft wings caused by the liquid action inside the fuel tanks. Due to the complexity and nonlinearity of the flow generated inside the tank, only experiments or efficient numerical solvers are suitable for studying the problem. In this paper, the tank-fluid system is periodically excited with a prescribed law of motion and the nonlinear features are observed, the force between the wall and the fluid and the global energy balance are computed. A weakly compressible smoothed particle hydrodynamics model has been reformulated and adapted to this kind of violent and turbulent flow. The evolution of the different terms that appear in the energy conservation law are computed, and the formulation is compared to other alternatives where the advantages of the present formulation are indicated. The comparison with the experimental results and the fluid-structure interaction case is carried out in Part II [Marrone et al., Numerical study on the dissipation mechanisms in sloshing flows induced by violent and high-frequency accelerations. II. Comparison against experimental data, Phys. Rev. Fluids 6, 114802 (2021)] of this work.

Figure 1

Left: Sketch of the problem geometry. Right: Example of the tank motion and consequent flow evolution.

Figure 2

Flow evolution in the forced harmonic heave motion obtained by $\delta$-LES-SPH. Colors refer to the ratio ${\mu }_T/\mu$. Plot sequence is from top to bottom and from left to right. Spatial resolution $N=200$.

Figure 3

Dissipated energy ${\mathcal{E}}_{\mathrm{diss}}$ (solid line) and corresponding tank elevation (dashed line). Labels (a) to (f) are related to plots of Fig. 2.

Figure 4

Left: Vorticity contour at $t/T=0.98$, corresponding to plot (e) of Fig. 2. Right: Contour plot of the vertical component of the velocity field at $t/T=0.56$ corresponding to plot (c) of Fig. 2.

Figure 5

Flow evolution obtained by LES-SPH. Colors refer to the ratio ${\mu }_T/\mu$. Plot sequence is from top to bottom and from left to right. Spatial resolution $N=200$.

Figure 6

Comparison of the energy dissipated in the LES-SPH scheme (dash-dotted line) and in the present $\delta$-LES-SPH scheme (solid line). The dashed line indicates the tank elevation. Labels (b′) to (e′) are related to plots of Fig. 5.

Figure 7

Pressure measured at the top-left corner of the tank. The red solid line represents the “weakly compressible limit” given by Eq. (4). Top: Comparison between $\delta$-LES-SPH scheme (solid line) and LES-SPH scheme (dashed line) for the time range [0:4] $t/T$. Bottom: Pressure recorded throughout the entire $\delta$-LES-SPH simulation.

Figure 8

Comparison of the energy dissipated in the LES-SPH scheme (dashed line), in the present $\delta$-LES-SPH scheme (solid line) and in the 1D analytical model (dash-dotted line).

Figure 9

Time history of the nondimensional time derivative of the dissipated energy, ${\mathcal{P}}_{\mathrm{diss}}/\mathrm{\Delta }\mathcal{P}$ (solid line) and of the absolute value of the tank displacement (dashed line).

Figure 10

Time history of the kinetic energy ${\mathcal{E}}_K$ (solid line) and of the potential energy ${\mathcal{E}}_P$ (dashed line) in the Ni-FoR.

Figure 11

SPH vorticity field at $t/T=1.25$ using a spatial resolution $N=800$. On the right plot an enlargement of the field is depicted comparing the vortices size with the evaluated Taylor microscale.

Figure 12

SPH turbulent viscosity ${\mu }_T$ field at $t/T=0.98$ using a spatial resolution $N=400$ (left) and $N=800$ (right).

Figure 13

Time history of the energy components defined in Sec. 6: resolved viscous dissipation ${\mathcal{E}}_V$, modeled turbulent viscous dissipation ${\mathcal{E}}_V^{\mathrm{turb}}$, numerical diffusion ${\mathcal{E}}_N$, total energy dissipation ${\mathcal{E}}_{\mathrm{diss}}$. Spatial resolution is $N=200$.

Figure 14

Energy dissipation over the simulation time duration. that is, 140 periods of oscillation. The solid line represents the raw simulation data, whereas the related regression line is drawn with the dashed line.

Figure 15

History of the energy dissipation without of its linear component $\overline{\mathcal{P}}t=-2.906t$.

Figure 16

Sketch of different levels of random noise applied on the initial particle lattice. The right plot shows the 1% noise configuration adopted for the simulations.

Figure 17

History of the energy dissipation without its linear component $\overline{\mathcal{P}}t$ for simulations N.2 (blue line), N.9 (red line), and the ensemble average of the series (black line). See Table 1 for label references.

Figure 18

Ensemble average of the energy dissipation time history for the three resolutions $N=25,50,100$. Error bars refer to the computed standard deviation. For the $N=200$ simulation only a single realization is reported.

Figure 19

Average dissipation rates $\overline{\mathcal{P}}$ for resolutions $N=25,50,100,200,400,800$. The error bars refer to the computed standard deviation for $\overline{\mathcal{P}}$ evaluated at each semiperiod of the tank oscillation. For the higher resolution $N=400$ and 800, because of the too high CPU costs, only 20 oscillations are simulated.