Burner cooling structures

A turbomachine, comprising:

• a compressor section;
• a turbine section;
• and
• a combustor section comprising a plurality of burners, each burner defining an axial direction, a radial direction, and a circumferential direction, each burner comprising:
• a main body comprising an outer surface and defining an interior, the main body having an upstream end axially spaced from a downstream end with respect to an axial centerline of the burner, the outer surface comprising in an axially sequential order:
• a forward portion, a middle portion generally parallel to the axial direction, and a burner front face at the downstream end of the main body, wherein the forward portion diverges radially outwardly from an upstream portion of the main body to the middle portion, wherein the burner front face converges radially inwardly from the middle portion to the downstream end;
• an annular air plenum defined in the main body radially inwardly from the outer surface;
• a cooling air passage extending from the outer surface to the annular air plenum;
• and
• a plurality of cooling channels circumferentially spaced apart from one another along the burner front face, each cooling channel of the plurality of cooling channels extending from a respective inlet configured to receive air from the annular air plenum to a respective outlet defined in the middle portion of the outer surface upstream from the burner front face, wherein the inlet is axially aft of the outlet such that each cooling channel of the plurality of cooling channels extends towards the upstream end of the main body, and wherein the plurality of cooling channels is oriented generally parallel to the burner front face in an axial-radial plane and disposed between the burner front face and the annular air plenum.