Ceramic matrix composite (CMC) parts create the opportunity for increased turbine entry temperatures within gas turbines. To achieve the highest temperatures possible, film cooling will play an important role in allowing turbine entry temperatures to exceed acceptable surface temperatures for CMC components, just as it does for the current generation of gas turbine components. Film cooling over a CMC surface introduces new challenges including roughness features downstream of the cooling holes and changes to the hole exit due to uneven surface topography. To better understand these impacts, this study presents flowfield and adiabatic effectiveness CFD for a 7-7-7 shaped film cooling hole at two CMC weave orientations. The CMC surface selected is a 5 Harness Satin weave pattern that is examined at two different orientations. To understand the ability of steady RANS to predict flow and convective heat transfer over a CMC surface, the weave surface is initially simulated without film and compared to previous experimental results. The simulation of the weave orientation of 0°, with fewer features projecting into the flow, matches fairly well to the experiment, and demonstrates a minimal impact on film cooling leading to only slightly lower adiabatic effectiveness compared to a smooth surface. However, the simulation of the 90° orientation with a large number of protruding features does not match the experimentally observed surface heat transfer. The additional protruding surface produces degraded film cooling performance at low blowing ratios but is less sensitive to blowing ratio, leading to improved relative performance at higher blowing ratios, particularly in regions far downstream of the hole.