Liquid-gas flows are ubiquitous in both natural and engineering environments, and the processes that occur near the phase interface play a crucial role to the overall flow dynamics. When simulating such flows, the phase interface must be accurately tracked to allow for the handling of discontinuities in fluid properties and pressure. This work presents a level set-based interface capture technique that provides good mass conservation through the use of an auxiliary function. This function must be re-initialized to maintain mass conservation, and an approach for doing this in a way that minimizes re-initialization errors is presented. The resulting method is validated with a milk crown simulation, and it is shown that re-initialization errors in the form of spurious corrugations on the interface are removed from the simulation, leading to improved radial growth of the splashing lamella wall. The new method requires that numerical data be extrapolated from the interface to a band of cells surrounding it. This leads to the development of a framework for performing rapid multidimensional extrapolations that utilize a numerical procedure known as the fast marching method. Results demonstrate that when using this new framework, extrapolations with the same level of error can be obtained orders of magnitude faster than the current state-of-the-art. Applications of advanced numerical methods relevant to the energy sector are presented, including annular flows within horizontal pipes relevant to transport systems and air-assisted primary atomization of a three-dimensional planar liquid layer. For the former, simulations of fully turbulent liquid-gas pipe flows under various conditions are conducted, and the effect of the Froude number on the transition between the annular and stratified regimes is investigated. For the latter, a probability density function of the axial Kelvin-Helmholtz instability wavelength that arises due to shear between the initially parallel flows is extracted from the spatiotemporal data, and results are shown to agree well with experiments. Lastly, a canonical simulation configuration is developed that isolates the interaction between surface tension and surrounding turbulence. The Kolmogorov critical radius/Hinze scale is shown to be a controlling parameter for the resulting interface topology, as well as a source of backscatter of liquid volume fraction variance and kinetic energy from small to large scales.