A numerical approach is developed to gain fundamental insight in liquid-liquid dispersion formation under well-controlled turbulent conditions. The approach is based on a free energy lattice Boltzmann equation method, and relies on detailed resolution of the interaction of the dispersed and continuous phase at the microscopic level, including drop breakup and coalescence. The capability of the numerical technique to perform direct numerical simulations of turbulently agitated liquid-liquid dispersions is assessed. Three-dimensional simulations are carried out in fully periodic cubic domains with grids of size 1003to10003. The liquids are of equal density. Viscosity ratios (dispersed phase over continuous phase) are in the range 0.3-1.0. The dispersed phase volume fraction varies from 0.001 to 0.2. The process of dispersion formation is followed and visualized. The size of each drop in the dispersion is measured in-line with no disturbance of the flow. However, the numerical method is plagued by numerical dissolution of drops that are smaller than 10 times the lattice spacing. It is shown that to mitigate this effect it is necessary to increase the resolution of the Kolmogorov scales, such as to have a minimum drop size in the range 20-30 lattice units [lu]. Four levels of Kolmogorov length scale resolution have been considered K=1, 2.5, 5, and 10 [lu]. In addition, the numerical dissolution reduces if the concentration of the dispersed phase is increased. (c) 2015 American Institute of Chemical Engineers AIChE J, 61: 2618-2633, 2015