Acceleration of Gas Bubble-Free Surface Interaction Computation Using Basis Preconditioners

The computation of gas bubble-free surface interaction entails a time-stepping algorithm whereby a linear system is solved at each time-iteration. In our investigation, the linear systems are derived from a desingularized boundary integral formulation and are poorly conditioned. This leads to poor convergence rates when Krylov subspace methods are used to solve these systems. The convergence rates may however be improved with proper preconditioning. We limit our investigation to gas bubbles initiated at depths sufficiently small such that a spike forms on the free surface during the later stages of evolution. Bubble dynamics dictate that for gas bubbles initiated at such depths, the stages through which the gas bubble and free surface evolve are similar. Based on this fact, we propose to perform one computation run for a gas bubble initiated at one particular depth, obtain a judicious set of a priori basis preconditioners from this run and thereafter, use this set of preconditioners on computation runs for gas bubble initiated at different depths. The computation time taken by the proposed method is, in general, 50% and 20% of the time taken by the present method (without preconditioning) with terminating criteria of 1.0e-5 and 1.0e-7 in the infinity-norm respectively using the Bi-conjugate Gradient Stabilized solver. The present method further enables computation to an infinity-norm terminating criterion of 1.0e-10 in a shorter time compared to the present method with a criterion of 1.0e-5. K. L. Tan is a student with the Singapore-MIT Alliance (SMA), National University of Singapore, 10 Kent Ridge Cresent, Singapore 119260. E-mail: [email protected]. B. C. Khoo and J. K. White are SMA fellows. INTRODUCTION Bubbles surrounded by a liquid can be found in many applications. Examples include bubble jet printers, boiling water and underwater explosions. The chemical industry uses bubble columns to enhance chemical reactions and mixing. The study of bubbles can be divided into two main areas. The first area, bubbles that do not undergo much change in volume, is studied extensively. A seemingly simple problem of this kind, the terminal rise velocity of a single bubble in water, is still not yet fully understood. The second area entails bubble dynamics pertaining to oscillating bubbles. Examples of these include cavitation bubbles and underwater explosion bubbles. Underwater explosion bubbles is an active area of defense research. An explosion or gas bubble is formed when an underwater explosion takes place. Due to inertia, this bubble will overexpand and thereafter collapse. If this collapse takes place near a solid surface, a high-speed jet will be formed, directed towards the solid surface. On the other hand, if it takes place near a free water surface, the highspeed jet will then be directed away from the free surface and a water plume or spike will be formed on the surface. All these problems have been studied using the boundary integral method. This method necessitates the meshing of the surface only and not the entire domain thereby greatly reducing the size of matrices to be solved. In this paper, we seek to reduce the computation time taken to evolve a single gas bubble interacting with a free surface through use of preconditioning. Limiting our investigation to gas bubbles initiated at depths such that a spike forms on the free surface, the preconditioners are chosen and computed based on some a priori knowledge of evolution of such bubbles initiated at different depths. MATHEMATICAL FORMULATION Mathematical Formulation Assuming the fluid domain bounded by the gas bubble and free surface to be inviscid, incompressible and irrotational, the flow within the domain Ω may then be described by a potential field ( ) x, y, z φ , where the fluid velocity is given by φ ∇ , that satisfies Laplace’s equation