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A combustion system includes a first combustion chamber and a second combustion chamber. The second combustion chamber is positioned downstream of the first combustion chamber. The combustion system also includes a pre-mixed, direct-injection secondary fuel nozzle. The pre-mixed, direct-injection se

A combustion system includes a first combustion chamber and a second combustion chamber. The second combustion chamber is positioned downstream of the first combustion chamber. The combustion system also includes a pre-mixed, direct-injection secondary fuel nozzle. The pre-mixed, direct-injection secondary fuel nozzle extends through the first combustion chamber into the second combustion chamber.

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1. A combustion system comprising: a first combustion chamber;a second combustion chamber positioned downstream of the first combustion chamber; anda pre-mixed, direct-injection secondary fuel nozzle extending through the first combustion chamber into the second combustion chamber, the pre-mixed, di

1. A combustion system comprising: a first combustion chamber;a second combustion chamber positioned downstream of the first combustion chamber; anda pre-mixed, direct-injection secondary fuel nozzle extending through the first combustion chamber into the second combustion chamber, the pre-mixed, direct-injection secondary fuel nozzle, comprising: a mixing head comprising a plurality of mixing tubes;a first wall defining a fuel passage into the mixing tubes;a second wall positioned outward of the first wall wherein the first wall and the second wall define a combustion air passage into the mixing tubes, wherein the second wall comprises an upstream portion having a first diameter, a downstream portion having a second diameter that is larger than the first diameter, and a tapered portion having an increasing diameter from the upstream portion to the downstream portion, and wherein the downstream portion of the second wall surrounds the mixing head and is spaced apart from the mixing head to form a gap therebetween;a third wall positioned outward of the second wall, wherein the second wall and the third wall define a cooling air passage, wherein the third wall comprises a plurality of louvers and a plurality of apertures, and wherein the third wall is positioned about the upstream portion of the second wall and at least partially about the tapered portion of the second wall upstream of the mixing head; anda plurality of swirling vanes positioned in the gap between the mixing head and the second wall;a joint located at a downstream end of the third wall that joins the third wall to the tapered portion of the second wall upstream of the mixing head; anda plurality of cooling holes formed through the joint. 2. The combustion system 1, wherein each mixing tube comprises at least one fuel injection hole. 3. The combustion system of claim 2, wherein the at least one fuel injection hole is recessed from an outlet of the mixing tube by a recession distance. 4. The combustion system of claim 1, wherein the pre-mixed, direct-injection secondary fuel nozzle comprises: each mixing tube comprising an inlet and at least one fuel injection hole;the combustion air passage being in communication with the inlets; andthe fuel passage being in communication with the fuel injection holes. 5. The combustion system of claim 4, wherein each mixing tube further comprises an outlet in communication with the second combustion chamber. 6. The combustion system of claim 4, further comprising a fuel plenum positioned about the mixing tubes, the fuel plenum communicating with the fuel passage and the fuel injection holes. 7. The combustion system of claim 4, wherein the combustion air passage extends about the mixing head to cool the mixing head. 8. The combustion system of claim 7, further comprising a plurality of cooling holes formed in the combustion air passage about the mixing head. 9. A pre-mixed direct-injection secondary fuel nozzle, comprising: a mixing head comprising a plurality of mixing tubes;a first wall defining a fuel passage into the mixing tubes;a second wall positioned outward of the first wall, wherein the first wall and the second wall define a combustion air passage into the mixing tubes, wherein the second wall comprises an upstream portion having a first diameter, a downstream portion having a second diameter that is larger than the first diameter and a tapered portion having an increasing diameter from the upstream portion to the downstream portion, and wherein the downstream portion of the second wall surrounds the mixing head and is spaced apart from the mixing head to form a gap therebetween;a third wall positioned outward of the second wall, wherein the second wall and the third wall define a cooling air passage, wherein the third wall comprises a plurality of louvers and a plurality of apertures, and wherein the third wall is positioned about the upstream portion of the second wall and at least partially about the tapered portion of the second wall upstream of the mixing head; anda plurality of swirling vanes are positioned in the gap between the mixing head and the second wall;a joint located at a downstream end of the third wall that joins the third wall to the tapered portion of the second wall upstream of the mixing head; anda plurality of cooling holes formed through the joint. 10. The pre-mixed direct-injection secondary fuel nozzle of claim 9, wherein a plurality of cooling holes are formed through the second wall about the mixing head. 11. The pre-mixed direct-injection secondary fuel nozzle of claim 9, wherein each of the mixing tubes comprises: an inlet in communication with the air passage;an outlet;a plurality of fuel injection holes in communication with the fuel passage, each fuel injection hole being recessed from the outlet. 12. The pre-mixed direct-injection secondary fuel nozzle of claim 9, wherein: the third wall is concentrically disposed about the second wall; andthe second wall is concentrically disposed about the first wall. 13. A combustion system comprising: a first combustion chamber;a second combustion chamber positioned downstream of the first combustion chamber; anda pre-mixed, direct-injection secondary fuel nozzle extending through the first combustion chamber into the second combustion chamber, the pre-mixed, direct-injection secondary fuel nozzle, comprising: a mixing head comprising a plurality of mixing tubes;a first wall defining a fuel passage into the mixing tubes;a second wall positioned outward of the first wall, wherein the first wall and the second wall define a combustion air passage into the mixing tubes,wherein the second wall has an upstream portion having a first diameter, a downstream portion having a second diameter that is larger than the first diameter, and a tapered portion having an increasing diameter from the upstream portion to the downstream portion, wherein the downstream portion of the second wall surrounds the mixing head and is spaced apart from the mixing head to form a gap therebetween; a third wall positioned outward of the second wall, wherein the second wall and the third wall define a cooling air passage, wherein the third wall comprises a plurality of louvers and a plurality of apertures, wherein the third wall is positioned about the upstream portion of the second wall and ends about the tapered portion of the second wall;a joint positioned at a downstream end of the third wall on the tapered portion of the second wall upstream of the mixing head, wherein the joint is configured to join the third wall to the tapered portion of the second wall upstream of the mixing head;a plurality of cooling holes formed through the joint; anda plurality of swirling vanes positioned in the gap between the mixing head and the second wall.