Improved phase field model for two-phase incompressible flows: Sharp interface limit, universal mobility and surface tension calculation

In this paper, we propose an improved phase field model for interface capturing in simulating two-phase incompressible flows. The model incorporates a second-order diffusion term, which utilizes a nonlinear coefficient to assess the degree of deviation of interface profile from its equilibrium state. In particular, we analyze the scale of the mobility in the model, to ensure that the model asymptotically approaches the sharp interface limit as the interface thickness approaches zero. For accurate calculations of surface tension, we introduce a generalized form of smoothed Dirac delta functions that can adjust the thickness of the tension layer, while strictly maintaining that its integral equals one, even when the interface profile is not in equilibrium. Furthermore, we theoretically demonstrate that the spontaneous shrinkage of under-resolved interface structures encountered in the Cahn-Hilliard phase field method does not occur in the improved phase field model. Through various numerical experiments, we determine the range of the optimal mobility, confirm the theoretical analysis of the improved phase field model, verify its convergence, and examine the performance of different surface tension models. The numerical experiments include Rayleigh-Taylor instability, axisymmetric rising bubbles, droplet migration due to the Marangoni effect, partial coalescence of a droplet into a pool, and deformation of three-dimensional droplet in shear flow. In all these cases, numerical results are validated against experimental data and/or theoretical predictions. Moreover, the recommended range of dimensionless mobility has been shown to be universal, as it can be effectively applied to the simulations of a wide range of two-phase flows and exhibits excellent performance.