Bubble models and real bubbles: Rayleigh and energy-deposit cases in a Tait-compressible liquid

• Published in: IMA journal of applied mathematics Vol. 83; no. 4; pp. 556 - 589
• Main Authors: Lauterborn, Werner, Lechner, Christiane, Koch, Max, Mettin, Robert
• Format: Journal Article
• Language: English
• Published: Oxford University Press 25.07.2018
• Subjects: volume of fluid method; jet formation; Euler equations; bubble dynamics
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxy015

Abstract In analytical and numerical studies on bubbles in liquids, often the Rayleigh initial condition of a spherical bubble at maximum radius is used: the Rayleigh case. This condition cannot be realized in practice, instead the bubbles need first to be generated and expanded. The energy-deposit case with its initial condition of a small, spherical bubble of high internal pressure that expands into water at atmospheric pressure is studied for comparison with the Rayleigh case. From the many possible configurations, a single bubble near a flat solid boundary is chosen as this is a basic configuration to study erosion and cleaning phenomena. The bubble contains a small amount of non-condensable gas obeying an adiabatic law. The water is compressible according to the Tait equation. The Euler equations in axial symmetry are solved with the help of the open source software package OpenFOAM, based on the finite volume method. The volume of fluid method is used for interface capturing. Rayleigh bubbles of $R_\mathrm{max} = 500\,\mu $m and energy-deposit bubbles that reach $R_\mathrm{max} = 500\,\mu $m after expansion in an unbounded liquid are compared with respect to microjet velocity, microjet impact pressure and microjet impact times, when placed or being generated near a flat solid boundary. Velocity and pressure fields from the impact zone are given to demonstrate the sequence of phenomena from axial liquid microjet impact via annular gas-jet and annular liquid-nanojet formation to the Blake splash and the first torus-bubble splitting. Normalized distances $D^{\ast } = D/R_\mathrm{max}$ (D = initial distance of the bubble centre from the boundary) between 1.02 and 1.5 are studied. Rayleigh bubbles show a stronger collapse with about 50% higher microjet impact velocities and also significantly higher microjet impact pressures.