System and method for cooling the wall of a gas turbine combustor

In one embodiment, a system includes an engine wall (58). The engine wall (58) has a cold side (72) and a hot side (74). The engine wall (58) is formed with one or more dilution holes (70), to each of the dilution holes (70) is one of: a first opening (81) on the cold side (72), a second mouth (83) on a hot side (74 ), and a portion of a thermal barrier coating (TBC) (78) which is mounted on the cold side (72) and having an opening surrounding the first opening (81) substantially.