A spectral element-based phase field method for incompressible two-phase flows

In this paper, we propose a spectral element-based phase field method by solving the Navier-Stokes/Cahn-Hilliard equations for incompressible two-phase flows. The high-order differential nonlinear term of the Cahn-Hilliard equation poses a great challenge for obtaining numerical solutions, so the Newton-Raphson method is adopted to tackle this conundrum. Furthermore, we employ the time-stepping scheme to decouple the Navier-Stokes equations to favor the computations with large density and viscosity contrast, in terms of both stability and convergence efficiency. In addition, the continuum surface tension model is used to account for the tangential component of the interfacial force, and thus our numerical method has the ability to simulate thermocapillary flows. We present four examples to demonstrate the interface capture accuracy of the proposed method. The shape of the rotating Zalesak's disk is accurately preserved by the present method even for two periods, which implies less dissipation and higher accuracy at long time numerical simulation. It is also noted that the present method with fourth-order element can achieve similar accuracy with the consistent scheme by evaluating the convective fluxes with the fifth-order weighted essentially non-oscillatory scheme. Moreover, the proposed method appears to comply well with mass conservation. And the results of thermocapillary flow test show good agreement with theoretical prediction. Finally, the rising bubble and Rayleigh-Taylor instability are considered to verify the proposed method for complex changes in interfacial topology, as well as its performance under large density and viscosity contrasts and high Reynolds number conditions