A new 3-D multi-fluid model with the application in bubble dynamics using the adaptive mesh refinement

Highlights

• A compressible multi-fluid model is presented for bubble dynamics.

• The toroidal bubble motion beside a vertical wall motion is investigated.

• The crescent-shape bubble and cushion effect are observed and analyzed.

Abstract

Violent pulsating bubbles behave diversely in different circumstances. It is a multi-scale problem in both space and time. In 3-D problems, the numerical simulation is usually too expensive to implement in practice with a fixed grid. In this paper, a 3-D multi-fluid model is established based on the Eulerian finite element method and the adaptive mesh refinement technique to investigate the bubble evolution and its toroidal motion near a solid vertical wall. The mixture formula for compressible multi-fluid flow is adopted to ensure conservativeness. By means of the block-based adaptive mesh refinement, the accuracy and the efficiency of the simulation are well balanced. The present model is validated by comparing the results with an underwater explosion experiment and the existing numerical results. The results agree well and a fast convergence is observed. Then, several cases with different buoyancy parameters are simulated, and the toroidal bubble motion and their pressure load on the solid wall are analyzed. The bubble’s motion exhibits complex physics, such as the formation of the crescent-shaped bubble, the air cushion effect during the jet penetration, and the nonlinear relationship between the jet impact pressure and the angle between the jet and the opposite bubble surface.

Introduction

The bubble dynamics has always been a significant aspect in the research of the fluid dynamics because of its various applications, such as in underwater explosion (Cole, 1948, Klaseboer et al., 2005, Wang, 1998, Wang et al., 2018, Barras et al., 2012, Daramizadeh and Ansari, 2015), sea resources exploration (Li et al., 2020, Chelminski et al., 2019, Zhang et al., 2019), biology (Dollet et al., 2019), cavitation erosion and surface cleaning (Blake et al., 1986, Brennen, 1995, Chahine et al., 2016). It is a typical multi-scale, transient, and nonlinear problem with the large deformation of the multi-fluid interface. The complexity of the bubble motion will be further increased if a nearby structure is presented. The boundary of the structure is usually assumed to be solid if it is rigid enough, such as the copper propeller blade to a cavitation bubble or the concrete dam to an underwater explosion bubble.

As one of the pioneers in the field of bubble dynamics research, Rayleigh (1917) proposed the first practical equation describing the collapse of a spherical cavitation bubble in a free and perfect fluid field. Subsequently, a lot of past work (Plesset and Chapman, 1971, Prosperetti and Lezzi, 1986) contributed to improving the equation by including the other factors, such as the compressibility of the surrounding fluid, heat conduction, viscosity, and surface tension. However, researchers soon realized that the collapsing bubble did not maintain spherical due to the surface instability. Kornfeld and Suvorov (1944) suggested that the non-spherical motion of a collapsing bubble might produce a much higher pressure than a spherical one on a nearby solid boundary. Naude (1960) theoretically and experimentally validated this suggestion and found that the solid wall would contract the collapsing bubble to develop a high-speed jet. The bubble dynamics near a wall with different geometric shapes and material properties have renewed interest in the subject. Brujan et al. (2018) and Cui et al. (2020) also implemented the experiments to study the interaction between a pulsating bubble and two solid joint walls. Their results show that the two walls compete to dominate the jet development, and complex evolution features are observed. Chahine et al. (2019) employed both experimental and numerical fluid–structure interaction methods to investigate the cavitation bubble collapse near a solid wall with polymetric coating and found that the material property of the coating layer had a significant influence on the jet development and the impact pressure.

The Boundary Element Method (BEM) (Blake et al., 1986, Wang, 1998, Klaseboer et al., 2006, Klaseboer et al., 2005, Zhang and Liu, 2015) is one of the numerical methods that are first successfully used in the non-spherical bubble dynamics simulation because it reduces the spatial dimensions by 1. This was a significant advantage over the other domain-mesh methods in the late 20th century when the computational resources were minimal. It is the boundary that needs to be discretized instead of the whole flow domain. Besides, the far-field condition is automatically satisfied such that the truncation error of the computational domain is avoided. Many researchers contributed to this field and made significant progress in the non-spherical bubble dynamics simulation. Blake et al. (1986) established an axisymmetrical model based on the BEM to investigate a cavitation bubble collapsing nearby a solid wall. Wang et al. (2005) extended the simulation to the toroidal bubble generated by the jet penetration by introducing a vortex ring inside the bubble. This approach resolves the conflict between the potential theory used in the BEM and the velocity circulation around the toroidal bubble. When the solid wall’s normal does not align with the gravity, the problem is no longer axisymmetrical. Thus, Wang (1998) employed a 3-D numerical model based on the BEM to investigate the jet development subject to the combined effects of an inclined wall and the buoyancy. However, the toroidal stage of the bubble is not included in this paper. Thus, Klaseboer et al. (2005) combined the 3-D BEM and the vortex ring model and simulated an underwater explosion bubble interacting with a vertical wall. Their results of the impact pressure were compared and agreed with the experimental results well. Furthermore, Zhang and Liu (2015) improved the 3-D bubble dynamics model based on the BEM to capture more deformation details of the toroidal bubble, and it was adopted in Liu et al. (2016) where the influence of the free surface was included in the interaction between a bubble and the nearby solid wall. It was found that the toroidal bubble might further break into two toroidal or a crescent-shape bubble (Zhang et al., 2015, Liu et al., 2016). Because of the requirement of complex mesh manipulations for bubble split and merge, introducing and removing vortex rings, it is non-trivial to implement in the 3-D BEMs.

The domain-mesh methods for computational fluid dynamics have developed rapidly along with the growth of computational power. However, the interface treatments require special attention because the interface is usually not coincident with the cell boundaries. In the cut-cell method (Ye et al., 2001), the interface cell cut by the moving interface and the irregular subcells are solved as normal cells. However, explicit tracking of the interface is required, and special manipulations should be included to avoid small and narrow subcells. With a proper implicit interface capture method, e.g., the level-set method (Fedkiw et al., 1999, Liu et al., 2003, Wang and Shu, 2010a), the volume of fluid (VOF) method (So et al., 2012, Youngs, 1982, Hirt and Nichols, 1981), or the $\gamma$-based method (Cheng et al., 2020), the complex topology change of the collapsing bubble requires no special treatments and is no longer a difficulty. Besides, suppose the Euler equation describes the fluid flow and discretized in the numerical model. In that case, the fluid’s compressibility is naturally considered while it is non-trivial in the BEMs. One significant challenge in these implicit interface capture methods is the conservativeness issue. The level-set method coupled with the ghost fluid method proposed by Fedkiw et al. (1999) solves two single-medium problems with properly defined ghost cell states in which the conservativeness is not enforced. A lot of efforts have been made to improve the conservativeness within the framework of the level-set method and the ghost fluid method (Wang and Shu, 2010b, Liu et al., 2021). By contrast, the conservation in the mixture formula (So et al., 2012, Murrone and Guillard, 2005) is trivial although a transition layer of several cells is required to represent the interface. Johnsen and Colonius (2009) employed the quasi-conservative interface capturing formula to investigate the non-spherical bubble collapse beside a solid wall induced by a strong shock. Tian et al. (2020b) established an axisymmetric model based on the Eulerian finite element method (Eulerian FEM) (Benson, 1992a) and investigated the jet development of an underwater explosion bubble and its impact on a solid horizontal wall. Another difficulty in the pulsating bubble simulation is that the computation is expensive if the compressibility of the surrounding fluid is considered. A large number of time increments are required due to the Courant–Friedrichs–Lewy (CFL) condition.

When it comes to a fully 3-D simulation, the unknown degrees of freedom (DOF) in the domain-mesh methods dramatically increases to an impractical level, especially in the multi-scale bubble pulsating problem. Thus, an Adaptive Mesh Refinement (AMR) technique is essential to reduce the DOFs by coarsening the local mesh where the solution is not important and has little influence on the global solution. The patch-based AMR first introduced by Berger and Colella (1989) proposed have been widely used in CFD codes, such as the BoxLib, AMROC, and Chombo. In 3-D block-based AMR, a set of non-overlapping boxes is find to cover the region requiring refinement. These boxes have meshed with higher resolution than the base mesh. The results calculated with the coarser mesh is mapping to the finer one to provide it a ghost boundary, while the results obtained in the finer mesh with this boundary is projected back to the coarse one. Another approach to the AMR is the block-based AMR (MacNeice et al., 2000, Olson and MacNeice, 2005), in which each block is a box with the same data structure, and they are refined based on the oct-tree algorithm. One or more layers of guardcells are attached to each block’s outside boundary, and their data is copied from the adjacent blocks for communication.

In the present paper, the Eulerian FEM is adopted to resolve the motion of a pulsating bubble. We use the block-based AMR implemented with the open-source package Paramesh (MacNeice et al., 2000, Olson and MacNeice, 2005) such that the efficiency and accuracy can be balanced in simulating the bubble pulsation and toroidal motion. The rest of the present paper is organized as follows. In Section 2, the underlying theory and numerical techniques are presented relating to the Eulerian FEM and the AMR implementation. Then, the numerical model is validated by comparing it with an underwater explosion experiment in Section 3. In Section 4, several cases of a high-pressure bubble moving beside a vertical wall are simulated and analyzed with the present model. The results are also compared with the BEM to verify the accuracy. At last, some conclusions are drawn in Section 5.