A hybrid phase field - volume of fluid method for the simulation of three-dimensional binary solidification in the presence of gas bubble

We present a hybrid sharp-diffusive methodology to model three-dimensional binary solidification in the presence of gas bubbles. The solidification is dominated by both temperature and solute concentration, specifically addressing the interactions among growing dendrites and gas phases. Central to this numerical algorithm is the integration of a phase-field method, adept at tracking solidification fronts, and a volume-of-fluid method, optimized for capturing gas-liquid interfaces. The former operates as a diffuse interface scheme, enabling robust simulation of complex porous flows arising in binary solidification scenarios, such as the crystallization of binary alloys or the freezing of salty water. The latter adopts a sharp interface approach, reconstructing gas interfaces through a geometrical scheme, thereby preserving the sharpness of gas interfaces and the conservation of phase-concentration fields within the liquid-solid domain. Moreover, a novel technique is devised for precisely imposing arbitrary contact angles at the trijunction, enhancing the fidelity of interface modeling. Concurrently, an implicit strategy is developed to discretize the governing phase-temperature-concentration equations, facilitating implicit and synchronous coupling of their solutions, which significantly enhances time-step tolerance compared to explicit schemes. The accuracy, efficiency, and robustness of our hybrid method are validated through various benchmark tests, including two more intricate scenarios of freezing water droplets on supercooled solid substrates and interactions between dendrite arms and rising gas bubbles during large-scale dendrite growth. Our results exhibit excellent agreement with experimental observations and theoretical solutions, underscoring the effectiveness and superiority of our proposed hybrid method.