Interfacial evolution such as solidification/melting and precipitation/dissolution is governed mathematically by Stefan-type problems, free boundary problems which relate the interface evolution to the transport of temperature and/or concentration and their gradients at the interface. Typically, the classical Stefan problem considers the diffusive transport of the temperature and/or species in each phase. However, coupling the Stefan problem with fluid flow allows us to consider convective transport effects, which lead to a complex coupling between the interface morphology and corresponding flow dynamics. This type of phenomena is found in a wide variety of settings, from formation of natural landscapes, to engineering systems, to manufacturing processes. The presented dissertation covers numerical approaches for the simulation of such Stefan-type problems coupled with incompressible fluid flow, which are shown to achieve reasonable numerical convergence, reproduce quantitative experimental results, and capture previously observed qualitative phenomena. The first chapter of the dissertation covers a foundational numerical approach to solve this type of problem in the context of solidification/melting of a pure substance. The numerical method utilizes sharp interface techniques that accurately capture discontinuities and gradients at the interface, and adaptive grids which provide a computationally efficient solution. Then, the subsequent chapters will discuss the extension and application of the foundational approach to two different problems of interest. First, Chapter 2 will discuss the extension of the foundational method to the application of mineral precipitation/dissolution in porous media, and will introduce an augmentation to the numerical method in order to accommodate large disparities in the time scales governing the fluid flow and interface evolution. Then, Chapter 3 will discuss the simulation of solidification of multi-component alloys with convective effects, which is achieved by combining the strategies developed in this dissertation with an existing technique for simulating purely diffusion-driven multi-component alloy solidification.