A compressible Lagrangian framework for the simulation of the underwater implosion of large air bubbles

Abstract

A fully Lagrangian compressible numerical framework for the simulation of underwater implosion of a large air bubble is presented. Both air and water are considered compressible and the equations for the Lagrangian shock hydrodynamics are stabilized via a variationally consistent multiscale method. A nodally perfect matched definition of the interface is used and then the kinetic variables, pressure and density, are duplicated at the interface level. An adaptive mesh generation procedure, which respects the interface connectivities, is applied to provide enough refinement at the interface level. This framework is verified by several benchmarks which evaluate the behavior of the numerical scheme for severe compression and expansion cases. This model is then used to simulate the underwater implosion of a large cylindrical bubble, with a size in the order of cm. We observe that the conditions within the bubble are nearly uniform until the converging pressure wave is strong enough to create very large pressures near the center of the bubble. These bubble dynamics occur on very small spatial (0.3 mm), and time (0.1 ms) scales. During the final stage of the collapse Rayleigh–Taylor instabilities appear at the interface and then disappear when the rebounce starts. At the end of the rebounce phase the bubble radius reaches 50% of its initial value and the bubble recover its circular shape. It is when the second collapse starts, with higher mode shape instabilities excited at the bubble interface, that leads to the rupture of the bubble. Several graphs are presented and the pressure pulse detected in the water is compared by experiment.

Introduction

The underwater implosion of air-filled bubbles has been studied by many authors in the last century due to its main role in a number of phenomena in science, like sonoluminescence, sonochemistry and sonofusion and a series of applications in engineering like cavitation damages, seabed detection and structure safety in the vicinity of the imploded volumes.

Historically, the first work on the cavitation and bubble dynamic was done by Rayleigh [1], who considered the collapse of both empty and gas filled cavities in an inviscid incompressible liquid. Plesset extended his work by adding surface tension and viscous effects that resulted in the famous Rayleigh–Plesset equation [2]. The presence of high pressures in liquid near the interface in addition to the damping oscillations of the bubble lead many authors to take into account for the compressibility of the surrounding liquid in their analysis [3], [4], [5]. Depending on the initial radius of the bubble and external driving pressure two types of behavior are observed in the bubble motion, namely weakly oscillating and strongly collapsing.

In physics, violent collapse of μm size bubbles excited by the sound waves may lead to UV-light emission of picoseconds duration that is known as sonoluminescence, SL [6]. Super compression of the internal air and high velocities obtained during the final stage of the violent collapse put in doubt the stability of the bubble. Later studies on the shape stability of the bubble [7], [8] revealed that the spherical symmetry assumption cannot be rigorously correct especially in the final stage of the violent collapse. Bogoyavlenskiy [8] demonstrated that time derivatives of shape perturbations grow significantly as the bubble radius vanishes. In general two different types of instabilities are prone to be excited during this stage, (i) interfacial instabilities, i.e. Rayleigh–Taylor (RT) instabilities, which occur when a gas is strongly accelerated into the liquid, and (ii) shape instabilities due the excitation of non-spherical modes causing the bubble to take on a non spherical shape. The shape instability is well shown in the DNS results provided by Nagrat et al. [9] for a micron-size bubble implosion. Although they predict the RT instabilities during the very short time interval that the bubble radius is near its minimum, no numerical evidence is presented in their results.

In engineering applications, on the other hand, larger size bubbles, in the order of cm, are of interest. Acoustic waves emanated from broken glass spheres are used to indicate the contact of the equipment with the seabed. Orr et al. [10] have reported the pressure signatures and energy–density spectra for a series of preweakend hollow glass spheres imploded at ocean depths of approximately 3 km. The signature of all implosions have many features in common. Basically each consists of a low flat negative-pulse followed by a sharp positive-pressure spike of roughly 0.2 ms duration. Different pressure signature is reported for shallow depth implosion, less than 300 m. Here linear oscillation, resembling a strongly damped sinusoid with damping factor of e per oscillation cycle occurs. McDonald et al. [11] propose turbulent instabilities excited by the shape oscillation of the bubble as the decay mechanism in shallow depth. Recently, Turner [12] studied the influence of the structure failure on the pressure pulse. Four glass spheres of diameter $7.62$ cm were imploded in a pressure vessel at a hydrostatic pressure of $6.996$;MPa and the pressure–time histories were compared with numerical results obtained from different failure rates. He reported an error of 44% for models that do not account for structure failure.

The development of efficient algorithms to understand rapid bubble dynamics presents a number of challenges. The foremost challenge is to efficiently represent the coupled compressible fluid dynamics of internal air and surrounding water. Secondly, the method must allow one to accurately detect or follow the interface between the phases. Finally, it must be capable of resolving any shock waves which may be created in air or water during the final stage of the collapse. Regarding the Eulerian approach and for small bubbles, μm-size, Nagrat et al. [9] proposed a DNS solution of the full hydrodynamics set of equations stabilized by the SUPG method. The interface is tracked by a modified level-set method and a DC operator is added to provide smooth transition in shock zones. Surface tension is also included to see its influence on the final shape of the bubble. Concerning large bubbles, in the order of cm, Farhat et al. [13] solved the Euler equation for the multi-fluid problem using a ghost fluid method for the poor (GFMP) generalized for an arbitrary equation of state (EOS). Viscous effect and surface tension are neglected due to the size of the bubbles and an exact Riemann solver is used to resolve the shock at the interface.

Lagrangian frameworks to solve the Euler equations in the presence of the shock waves and with large mesh movement have been developed by different authors. Efforts have been dedicated to improve the robustness of the simulation with respect to mesh distortion, while maintaining second order accuracy in smooth regions of the flow [14], [15], [16]. Scovazzi et al. [17], [18], [14], [19], [20] developed a robust second-order FEM method with continuous linear approximation of kinematic and kinetic variables that is stabilized by operators driven from the variational multiscale paradigm. This work is recently equipped with a conservative synchronized ALE remap approach to reposition the mesh in distorted zones without changing connectivities [20], [21]. In moving Lagrangian curvilinear coordinates, traditional staggered grid hydro (SGH) methods, that use continuous linear representation for kinematic variables and discontinuous constant field for thermodynamic variables, have been extended for higher order elements [22].

All of the above mentioned Lagrangian methods are able to treat air or water phase as well as to represent shock waves in them. None of them, however, is designed to capture possible large distortions of the interface that appear in multi-fluid flow. The recent developments in the Particle Finite Element Method (PFEM) [23], [24], [25], [26] to deal with multi-fluid flows provide a good basis to capture interface instabilities in the final stage of the collapse. In particular, Idelsohn et al. [25], [26] successfully track the interface in an incompressible heterogeneous flows in the presence of large density and viscosity jumps. Interface is forced to match the nodes and due to the jump in pressure gradient across the interface, a discontinuous pressure gradient projection is necessary to stabilize the flow near the interface.

In this work we present a fully Lagrangian shock hydrodynamics framework to solve two-phase flow problems with large distortions at the interface and big pressure and density jumps. To solve the hydrodynamic set of equations in each phase the stabilized variational multiscale method presented by Scovazzi et al. [17], [18], [14] is adopted. Later we improve the interface detecting technique proposed by Idelsohn et al. [25], [26], by conserving the interface connectivities. This method is then extended to compressible multi-fluid flows by considering a discontinuous representation of the kinetic variables, i.e. pressure and density, at the interface level. The simulation of the large-bubble implosion using the proposed framework allows to identify, to our best knowledge for the first time, the appearance of the RT instabilities in these bubbles, at the final stage of the collapse. The possibility of the appearance of such instabilities has been reported by many authors [2], [6], [8], [9], [27]. We continue the simulation during the rebounce phase till instabilities disappear and the second collapse occurs. The second collapse ends up with the rupture of the air bubble.

The outline of the paper is as follows: In the next section a review of the stabilized variational multiscale method developed by Scovazzi et al. [17], [18], [14] to solve the Euler hydrodynamic set of equations is presented. Then, the interface-following technique proposed by Idelsohn et al. [25], [26] is presented and extended for compressible flow, emphasizing on the use of a discontinuous pressure along the interface. In the final section we present some numerical examples that verify the method and show its potential for simulating the implosion of a large size cylindrical bubble.