A method and apparatus for cooling a wall within a gas turbine engine is provided which comprises the steps of: (1) providing a wall having an internal surface and an external surface; (2) providing a cooling microcircuit within the wall that has a passage for cooling air that extends between the internal surface and the external surface; and (3) increasing heat transfer from the wall to a fluid flow within the passage by increasing the average heat transfer coefficient per unit flow within the microcircuit. According to one aspect, the present invention method and apparatus can be tuned to substantially match the thermal profile of the wall at hand.