An assembly for a turbine engine is provided. This turbine engine assembly includes a shell and a heat shield with a cooling cavity between the shell and the heat shield. The heat shield defines a plurality of cooling apertures and an indentation in a side of the heat shield opposite the cooling cavity. The cooling apertures are fluidly coupled with the cooling cavity. The indentation is configured such that cooling air, directed from a first of the cooling apertures, at least partially circulates against the side of the heat shield.