Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier-Stokes equation

Abstract

We present a numerical method for two-phase incompressible Navier-Stokes equation with jump discontinuity in the normal component of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuity sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure. Numerical results support the inference that the proposed method converges in L ∞ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.

Solving incompressible Navier--Stokes equations on irregular domains and quadtrees by monolithic approach

“…Although this method creates a larger linear system than projection methods do, the boundary conditions for the velocities are no longer coupled with other variables. Similarly, the jump conditions of two-phase incompressible flow can be treated more easily than they can by projection methods, and recent works [8,10,39] have reported success in obtaining convergence in L ∞ norms. For a single-phase flow problem on irregular domains, a very recent study by Coco [12] solved the equations using the monolithic approach.…”

“…These monolithic methods exhibit convergence with nontrivial analytical solutions; note, however, that discretized linear systems are rank-deficient (the pressure is unique up to a constant). In [10,39], the right-hand side was projected onto the range space of the linear system by formulating the kernel of its transpose. By contrast, Coco [12] augmented the approach with an additional scalar unknown.…”