APPARATUS FOR A TURBINE ENGINE

An apparatus is provided for a turbine engine (20). This turbine engine apparatus includes a turbine engine component (60) that includes a sidewall (62) and a cooling aperture (64). The cooling aperture (64) includes an inlet (98), an outlet (100), a meter section (102), a diffuser section (104) and a transition section (106) between and fluidly coupled with the meter section (102) and the diffuser section (104). The cooling aperture (64) extends through the sidewall (62) from the inlet (98) to the outlet (100). The meter section (102) is at the inlet (98). The diffuser section (104) is at the outlet (100). The transition section (106) is configured to accommodate lateral misalignment between the meter section (102) and the diffuser section (104).