Liquid Composite Molding is a composite manufacturing process in which fiber preforms consisting of stitched, woven or braided bundles of fibers, known as fiber tows, are stacked in a closed mold and polymeric resin is injected to impregnate all the empty spaces between the fibers. An important step is to ensure wetting and saturation of all the fiber tows and regions in between them. Mesolevel analysis refers to study at the order of a fiber tow. Numerical simulations at the mesolevel are conducted by incorporating the governing equations for free boundary flows around and inside a fiber tow into a set of subroutines. The simulation can track the advancement of the resin front promoted by both hydrodynamic pressure gradient and capillary action. Efficiency and appropriateness of several stabilization techniques is explored. Numerical results are in good agreement with reported experimental studies. The results clearly show that wicking flow plays an important role at the mesolevel and cannot be omitted.