Abstract

Smooth exterior calculus (EC) is a powerful mathematical tool in modern differential geometry, which was originally developed by the famous mathematician Cartan. And its discrete counterpart, discrete exterior calculus (DEC), possesses superior numerical properties, i.e. it is coordinate independent, works in arbitrary dimensions; has superior conservation properties. Previously, the application of DEC for fluid dynamics is limited in simulating single phase flow, while two-phase flow is significant both in industry and nature, in which the discontinuous physical properties and singular surface tension across the interface make it a big challenge. In the present work, the physically-compatible exterior calculus discretization of single-phase flow is extended to simulate the two-phase flow, in which the two-phase Navier-Stokes equations and conservative phase field equation are transformed into DEC framework. The boundedness of the phase field has been proved in the DEC framework. Benefiting from the coordinate independent property of DEC, two-phase flow simulations can be implemented on Riemannian manifolds. Moreover, the governing equations of axisymmetric two-phase flow are introduced into DEC framework and the boundedness of its corresponding phase field is also proved. Furthermore, optimally time dependent (OTD) basis evolution equations for the two-phase flow are transformed into DEC framework to capture transition and asymptotic instability information of the unstable two-phase flow, Rayleigh-Taylor instability, on Riemannian manifolds.

Biography

Minmiao Wang is currently a Ph.D. candidate in the Mechanical Engineering Program under the supervision of Prof. Peter Schmid. Formerly, he was also supervised by Prof. Ravi Samtaney. During his PhD period, his research focuses on extending discrete exterior calculus (DEC) discretization method for single phase flow to two-phase flow on Riemannian manifolds, i.e. solving two-phase Navier-Stokes equations and optimally time-dependent (OTD) modes evolution equations for two-phase flow on Riemannian manifolds.