On the conservative phase-field method with the N-component incompressible flows

This paper presents a conservative Allen-Cahn model coupled with the incompressible Navier-Stokes equation for tracking the interface with the N-component immiscible fluids system. The proposed conservative phase-field model can track the interface with large deformation in divergence-free velocity fields. The erroneous estimation of the normal vector is a challenging numerical issue for the interface capturing due to the appearance of spurious oscillations. The improved phase-field-based method combines the nonlinear preprocessing operation guided by the level-set method with local artificial viscosity stabilization to improve the computation of the discrete normal vector. The interfaces between different immiscible components are replaced by the transition region with finite thickness in the continuous phase field. The surface tension effects are represented with the continuous surface tension force in the system, which is not limited by the number of components. The third-order Runge-Kutta time discretization and second-order spatial discretization are applied for the multi-component system. To eliminate the spurious oscillations caused by discontinuous and steep gradient for capturing the shocks and sharp interfaces, we apply the third-order weighted essentially non-oscillatory method for the advection term. Several quantitative results of numerical tests, such as error estimation with the proposed method, comparative tests with different methods, and convergence rate for classical benchmark test, have been performed to illustrate that our method works well for the interface tracing issue with high numerical accuracy. In addition, various representative qualitative tests have been presented to demonstrate the applicability of our method.