A method for operating a gas turbine engine includes channeling fluid from a cooling fluid source to a combustor that includes at least one deflector and flare cone. The deflector and flare cone are coupled together and are configured to define a cooling fluid channel therebetween. The flare cone has a plurality of cooling injectors extending therethrough. The plurality of injectors are spaced circumferentially about a centerline axis of the flare cone and are coupled in flow communication with the fluid source. The plurality of injectors has a plurality of first injectors and a plurality of second injectors. The method also includes directing a portion of the fluid through the plurality of first injectors. The method further includes directing a portion of the fluid through the plurality of second injectors, wherein the first plurality of injectors facilitates cooling a portion of the deflector more than the second plurality of injectors.