Development of singular value decomposition based solution scheme for a numerical simulation of multiphase flow

Abstract

This paper presents an innovative solution scheme that may resolve the numerical challenges of simulating the phase appearance and disappearance phenomena in multiphase flow. In this study, a least square problem for the mass and internal energy conservation equations is reviewed, and its inherent ill-posedness, originating from the rank deficiency of the coefficient matrix, in the inverse problem is introduced. Since building mathematically well-conditioned matrices is essential to avoiding the numerical instability as well as the singularity of matrices, first of all, a new solution scheme for the mass and internal energy conservation equations for the simulation of multiphase flow in which phase can appear and disappear under various flow conditions is derived based on a truncated Singular Value Decomposition (SVD)-based low rank approximation. The truncated SVD-based low rank approximation is then used to solve the mass and internal energy conservation equations by computing the pseudo inverse of the low rank approximation matrix. The field equations and closure relations currently being developed by introducing arbitrary values, which replace an extremely low volume fraction of fluids, are redefined in this study. As such, the physical phenomena themselves are directly considered without any modifications to the mathematical equations to compute phase transitions. The characteristics of phase transition are classified into convection, vaporization, and condensation over a wide range of fluid conditions, and the verification of the proposed solution method is completed through simulations of the 3 phase transition mechanisms. The SVD-based low rank approximation method improves computations of phase transition as simulation results indicate that the proposed solution scheme describes the physics better than the conventional one.