A lattice Boltzmann study of the thermodynamics of an interaction between two cavitation bubbles

Xiaolong He; Xiang Song; Haonan Peng; Hao Yuan

doi:10.3934/dcdss.2023156

Article Contents

2024, Volume 17, Issue 11: 3252-3277. Doi: 10.3934/dcdss.2023156

This issue Previous Article Key ingredients for wall-modeled LES with the Lattice Boltzmann method: Systematic comparison of collision schemes, SGS models, and wall functions on simulation accuracy and efficiency for turbulent channel flow Next Article Binary fluid flow simulations with free energy lattice Boltzmann methods

A lattice Boltzmann study of the thermodynamics of an interaction between two cavitation bubbles

1.
Southwest Research Institute for Hydraulic and Water Transport Engineering, Chongqing Jiaotong University, Chongqing 400074, China
2.
School of River and Ocean Engineering, Chongqing Jiaotong University, Chongqing 400074, China
3.
Laboratory for Waste Management, Paul Scherrer Institute, CH, 5232, Villigen PSI, Switzerland

^*Corresponding author: Haonan Peng and Hao Yuan
^*Corresponding author: Haonan Peng and Hao Yuan

Received: November 2022

Revised: July 2023

Early access: August 24, 2023

Published online: August 24, 2023

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Abstract

A two-dimensional double-distribution-function thermal pseudo-potential lattice Boltzmann model is applied to study the inception and evolution processes of cavitation bubbles. The inception process is realized by a high-temperature spot, while the growth and collapse processes are realized through the variation of the boundary pressure. Interactions between two equal-sized bubbles and two unequal-sized bubbles are systematically investigated. A modified Rayleigh–Plesset equation for the two cavitation bubbles that includes a weak interaction and thermal effects is proposed. For the weak interaction mode, the highest collapse temperature is lower than the initial input temperature due to energy dissipation. However, for the strong interaction mode, the maximum temperature is ultimately higher than the initial input temperature due to the superposition of an exothermic process. For the weak interaction mode between two unequal-sized bubbles, when the distance between the two bubbles is small, the re-entrant jets and pressure wave generated by the small bubble cause the inner wall of the large bubble to flatten. For the strong interaction film-thinning mode, the microjets and high pressure generated by the small bubble change the collapse direction of the large bubble.

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Mathematics Subject Classification: Primary: 35Q20; Secondary: 76B10.

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Figure 1. The relationship between the pressure difference $ \Delta p $ and the bubble radius with $ \rho_l/\rho_g = 720 $ and $ 290 $. The inner figure represents the density and velocity distribution after the bubble reaches the equilibrium state

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Figure 2. A comparison of the thermodynamic curves obtained with the present model and the theoretical solution

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Figure 3. The relationship between the square of the dimensionless diameter $ (D(t)/D_0)^2 $ and time $ t $

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Figure 4. The variation of the pressure on the boundary $ p_{\infty} $ versus time $ t $

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Figure 5. A qualitative comparison between the simulation results and the experimental results for the single cavitation bubble evolution process in an infinite quiescent liquid. (a) Temperature field (right) obtained using the present method, with contour line $ \rho = 0.23 $. (b) The experimental results obtained by Zhang et al. [55]

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Figure 6. A comparison between the radius obtained using the present model and the R–P prediction

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Figure 7. The evolution processes of (a) the pressure inside the bubble during the initial growth stage, and (b) the bubble radius with different $ T_{ini} $

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Figure 8. A comparison of the (a) experimental results reported by Bremond et al. [7] and (b) the SCMP pseudopotential LBM simulation proposed by Peng et al. [39], (c) the macroscopic numerical simulation obtained using the PhantomCloud model [8], (d) the macroscopic numerical simulation obtained using the 3DYNAFS-BEM model [8], and (e) the results obtained with the present LB model

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Figure 9. A schematic diagram of the two-bubble interaction

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Figure 10. Snapshots of the bubble morphology and the temperature distribution and comparisons with the experimental results provided by Bremond et al. [6] for (a) the W mode, (b) the SF mode, (c) the SCC mode, and (d) the SGC mode of two equal-sized bubbles interacting with $ T_{ini} = 1.2T_c $. Both the white line and the black line are density contours with $ \rho = 0.25 $. The arrows in parts (c) and (d) represent the collapse direction in the final collapse stage, and the number in the upper right corner of each subfigure represents the dimensionless time

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Figure 11. The maximum dimensionless temperature evolution processes of different interaction modes with $ T_{ini} = 1.2T_c $

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Figure 12. Snapshots of the bubble morphology and the temperature distribution for (a) the W mode, (b) the SF mode, (c) the SCC mode, and (d) the SGC mode of interaction between two unequal-sized bubbles. Both the white line and the black line are density contours with $ \rho = 0.25 $. The arrows in the subfigures represent the collapse direction in the final collapse stage, and the number in the upper right corner of each subfigure represents the dimensionless time

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Figure 13. The relationship between the lifetime and different dimensionless interval distances $ d^* $ with $ T_{ini} = 1.2T_c $

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Figure 14. The lifetime of (a) the larger bubble, and (b) the smaller bubble with different initial distances between the two bubbles

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Figure 15. The range of $ d^* $ and the corresponding interaction modes for two unequal-sized bubbles with different $ T_{ini} $

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Figure 16. A quantitative comparison of the bubble radius evolution processes in the LBM simulation and the R–P prediction with $ d^* = 1.723 $ and $ d^* = 3.333 $

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Figure 17. A quantitative comparison of the bubble radius evolution processes from the LBM simulation and the R–P prediction for an interaction between two unequal-sized bubbles with $ d^* = 3.301 $. $ r_s $ denotes the radius of the small bubble and $ r_l $ denotes the radius of the large bubble

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Table 1. The comparison of the dimensionless bubble radius from LBM simulation and experimental study [55]

$ R/R_{max} $
LBM	$ 400 \Delta t $	$ 1200 \Delta t $	$ 2000 \Delta t $	$ 2530 \Delta t $	$ 2800 \Delta t $
LBM	0.65	1.00	0.88	0.56	0.14
Exp	$ 25 \mu s $	$ 75 \mu s $	$ 125 \mu s $	$ 158.33 \mu s $	$ 175 \mu s $
Exp	0.73	1.00	0.96	0.62	0.17

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Table 2. A comparison between the different cavitation models

Model type	Present	Peng et al. [39]	3DY [8]	PhC [8]
Shield effect	✔	✔	✔	✔
Wall effect	✔	✔	✔	×
Toroidal shape	✔	×	✔	×
Smooth interface	✔	✔	×	✔

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Table 3. The range of $ d^* $ and the corresponding interaction modes for two equal-sized bubbles with $ T_{ini} = 1.2T_c $

$ d^* $	$ 0.28 \leq {d^*} \leq 0.38 $	$ 0.47 \leq {d^*} \leq 0.64 $	$ 0.73 \leq {d^*} \leq 1.23 $	$ 1.31 \leq {d^*} \leq 3.33 $
Mode	SGC	SCC	SF	W

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