An assembly is provided for a turbine engine. This turbine engine assembly includes a combustor wall including a shell and a heat shield. The combustor wall defines first and second cavities between the shell and the heat shield. The heat shield defines a first outlet and an elongated second outlet. The first outlet is fluidly coupled with the first cavity. The second outlet is fluidly coupled with the second cavity. The combustor wall defines one of the cavities with a tapered geometry.