Fully coupled model for simulating highly nonlinear dynamic behaviors of a bubble near an elastic-plastic thin-walled plate

Existing studies of bubble-structure interaction mainly focus on the interaction between a bubble and a movable rigid body or deformable linear elastic structure. The strong nonlinear interaction of a bubble with an elastic-plastic plate has rarely been studied while the inherent dynamic behavior is not clear. In this paper, we develop a three-dimensional fully coupled model to investigate the interaction between a bubble and an elastic-plastic thin-walled plate, which can consider the fluid disturbance on both sides of the thin-walled plate. In this developed model, the dynamic behaviors of the bubble are obtained by the boundary integral method based on the potential flow theory, and the nonlinear elastic-plastic responses of the structure are resolved by the explicit finite element method on the basis of the Mindlin-Reissner plate theory. The structural nonlinear responses are incorporated into the fluid boundary integral equation (BIE), and the extra relation between the ${\varphi }_t$ (time derivative of velocity potential) jump across the two sides of the thin-walled plate and its normal derivative ${\varphi }_{tn}$ is derived, which can describe the hydrodynamic balance on both sides of the submerged plate. The derived relation is added to the BIE about ${\varphi }_t$, so that the bubble loading acting on the plate can be accurately calculated. The established coupled model is validated by comparing with experimental results. Using this numerical model, the influence of the standoff distance and plate thickness on the bubble-plate interaction is discussed. Subjected to the violent loading from the bubble, the oscillation characteristics and elastic-plastic deformation of the plate are analyzed. Due to the elastic-plastic effects of the plate, the bubble can display different interesting featured patterns, including attractive motion, repulsive motion, or splitting.

Figure 1

Sketch of interaction between a bubble and a plate.

Figure 2

Comparison of the bubble–thin membrane interaction between the experimental images [40] and numerical results at typical dimensionless moments. The parameters in the simulation are $R_m=4.1\mathrm{mm}, R_0=0.419\mathrm{mm}, P_{\infty }=7.844\times 10^4\mathrm{Pa}, P_v=0.23\times {10}^5\mathrm{Pa}, {\rho }_s=1250\mathrm{kg}/{\mathrm{m}}^3, E=3.16\mathrm{MPa}, \upsilon =0.5, h=0.299\mathrm{mm}, {\gamma }_w=0.55, \varepsilon =400$, and $\lambda =1.4$.

Figure 3

Dimensionless velocity potential contour of bubble–thin membrane interaction results using Eq. (36) at the nondimensional times $t=0$, 0.384, 0.837, 1.144, and 1.262. All the parameters are the same as in Fig. 2.

Figure 4

Dimensionless velocity potential contour of bubble dynamics near a thin-walled plate with ${\gamma }_w=1.75$ at typical nondimensional times. The other parameters are $R_m=0.112\mathrm{m}, R_0=8.904\mathrm{mm}, P_{\infty }=1.098\times 10^5\mathrm{Pa}, P_v=0, {\rho }_s=7850\mathrm{kg}/{\mathrm{m}}^3, E=203\mathrm{GPa}, \upsilon =0.3, {\sigma }_0=235\mathrm{MPa}, E_T=400\mathrm{MPa}, h=20\mathrm{mm}, \varepsilon =588.76$, and $\lambda =1.25$. Top: The interaction between the bubble and a thin-walled plate. Below: An enlarged view of the evolution of the bubble.

Figure 5

Dimensionless displacement contour of plate in the $y$ direction at ${\gamma }_w=1.75$. All the parameters are the same as in Fig. 4.

Figure 6

Time history of the dimensionless bubble volume and plate center displacement for the case shown in Fig. 4.

Figure 7

Dimensionless velocity potential contour of bubble dynamics near a submerged plate for various distance parameters ${\gamma }_w=0.80$, 1.00, 1.25, and 1.50 with the remaining parameters being the same as in Fig. 4. Top: The interaction between the bubble and a thin-walled plate. Below: An enlarged view of the evolution of the bubble.

Figure 8

Time evolution of bubble dynamics near a 20-mm-thick plate with different distance parameters for the case shown in Fig. 7. (a) Dimensionless bubble volume, (b) dimensionless bubble center, and (c) dimensionless Kelvin's impulse.

Figure 9

Time evolution of the structural dynamics at the center of 20-mm-thick plate with different distance parameters for the case shown in Fig. 7. (a) Dimensionless hydrodynamic load, and (b) dimensionless displacement.

Figure 10

Evolution of von Mises stress distribution of the plate at the location $\widehat{z}=–9\mathrm{mm}$ in thickness direction at ${\gamma }_w=0.80$. All the parameters are the same as in Fig. 4.

Figure 11

Dimensionless normal velocity contour of the interaction between the bubble and the thin plate. (a) ${\gamma }_w=0.80$; (b) ${\gamma }_w=1.75$. The other parameters are $R_m=0.112\mathrm{m}, R_0=8.904\mathrm{mm}, P_{\infty }=1.098\times 10^5\mathrm{Pa}, P_v=0, {\rho }_s=7850\mathrm{kg}/{\mathrm{m}}^3, E=203\mathrm{GPa}, \upsilon =0.3, {\sigma }_0=235\mathrm{MPa}, E_T=400\mathrm{MPa}, h=2\mathrm{mm}, \varepsilon =588.76$, and $\lambda =1.25$.

Figure 12

Comparison of structural dynamics at the plate center between different plate thicknesses at ${\gamma }_w=1.75$ for the case shown in Figs. 4 and 11. (a) Dimensionless hydrodynamic load, and (b) dimensionless displacement.

Figure 13

Dimensionless normal velocity contour of bubble jet patterns near a 2-mm-thick plate for various distance parameters ${\gamma }_w=0.80$, 1.25, 1.50, and 1.75 with the remaining parameters being the same as in Fig. 11.

Figure 14

Dimensionless pressure contour of the fluid field around the bubble for ${\gamma }_w=1.50$ at typical dimensionless times. The other parameters are the same as in Fig. 11. The plate surface is not shown in this figure.

Figure 15

Time evolution of bubble dynamics near a 2-mm-thick plate for various distance parameters ${\gamma }_w=0.80$, 1.0, 1.25, 1.50, and 1.75 with the remaining parameters being the same as in Fig. 11. (a) Dimensionless bubble center, and (b) dimensionless bubble volume.

Figure 16

Time evolution of the structural dynamics at the center of 2-mm-thick plate for various distance parameters ${\gamma }_w=0.80$, 1.0, 1.25, 1.50, and 1.75 with the remaining parameters being the same as in Fig. 11. (a) Dimensionless hydrodynamic load, and (b) dimensionless displacement.