A novel decoupled second-order time marching scheme for the two-phase incompressible Navier–Stokes/Darcy coupled nonlocal Allen–Cahn model

Abstract

We construct a novel second-order time marching scheme with the full decoupling structure to solve a highly coupled nonlinear two-phase fluid flow system consisting of the nonlocal mass-conserved Allen–Cahn equation where two types of flow regimes are considered (Navier–Stokes and Darcy). We achieve the full decoupled structure by introducing a nonlocal variable and designing an additional ordinary differential equation for it which plays the key role to maintain the unconditional energy stability. The whole scheme is built upon the pressure correction/ quadratization approach for the fluid equation and nonlinear double-well potential, respectively. At each time step, one only needs to solve several independent elliptic equations with constant coefficients illustrating the high practical efficiency. We strictly prove that the scheme satisfies the unconditional energy stability, and carry out various numerical simulations to prove its stability and accuracy numerically, such as spinodal decomposition and fingering instability due to the continuous injection flow, etc.