A structure is provided for a turbine engine. The structure includes a shell with a first surface, and a heat shield with a textured second surface and a textured third surface. The texture of a portion of the second surface is different than the texture of a portion of the third surface. The first surface and the second surface define a first cooling cavity between the shell and the heat shield. The first surface and the third surface define a second cooling cavity between the shell and the heat shield.