Entropy-Stable Positivity-Preserving Schemes for Multiphase Flows

Abstract

High-intensity focused ultrasound is a promising non-invasive medical technology that has been successfully used to ablate tumors, as well as in the treatment of other conditions. Researchers believe high-intensity focused ultrasound could see clinical application in other areas such as disruption of the blood brain barrier and sonoporation. However, such advances in medical technology requires fundamental insight into the physics associated with high-intensity focused ultrasound, such as the phenomena known as acoustic cavitation and the collapse of the ensuing bubble cavity. The multiphase description of flow phenomena is an attractive option for modelling such problems as all fluids in the domain are modelled using a single set of governing equations, as opposed to separate systems of equations for each phase and therefore, separate meshes for each fluid. In this thesis, we are interested in studying the bubble collapse problem numerically, to elucidate the physics behind the collapse of acoustically driven bubbles. We seek to develop high-order numerical methods to solve this problem, due to their potential to increase computational efficiency. However, high-order methods typically have stability issues, especially when considering complex physics. For this reason, high-order entropy-stable summation-by-parts schemes are a popular method used to simulate compressible flow equations. These methods offer provable stability through satisfying a discrete entropy inequality, which is used to prove discrete L2 stability. Such stability proofs rely on the fundamental assumption that the densities and volume (or void) fractions of both phases remain positive. However, we seek numerical schemes that can simulate flows where the densities and volume fractions get arbitrarily close to zero and, as such, could become negative as the simulation progresses. To address this problem, we present a novel high-order entropy-stable positivity-preserving scheme to solve the 1-D isentropic Baer-Nunziato model. The key to our proposed scheme is a novel artificial dissipation operator, which has tuneable dissipation coefficients that allow the scheme to have provable nodewise positivity of the densities and volume fractions. This new scheme is constructed by mixing a high-order entropy-conservative scheme with a first-order entropy-stable positivity-preserving scheme to create a high-order entropy-stable positivity-preserving scheme. Numerical results which demonstrate the convergence, positivity, and shock capturing capabilities of the scheme are presented.