A gas turbine engine has a combustor supported in a gas generator case. A baffle apparatus is attached to the gas generator case, surrounding an upstream section of the combustor to create at least one passage for directing an air flow discharged from a compressor diffuser to pass therethrough for cooling the combustor. The at least one passage extends from an upstream end of the combustor and has a passage exit immediately upstream of a dilution hole in the combustor. The baffle apparatus is configured to increase a velocity of the air flow entering the at least one passage and passing over the combustor.