A combustor for a turbine comprising: a combustor liner; a first flow sleeve surrounding said combustor liner with a first flow annulus therebetween, said first flow sleeve having a first plurality of cooling apertures formed about a circumference thereof for directing compressor discharge air as cooling air into said first flow annulus; a transition piece body connected to said combustor liner, said transition piece body being adapted to carry hot combustion gases to the turbine; a second flow sleeve surrounding said transition piece body, said second flow sleeve having a second plurality of cooling apertures for directing compressor discharge air as cooling air into a second flow annulus between the second flow sleeve and the transition piece body, said first flow annulus connecting to said second flow annulus; an arch shaped resilient seal structure disposed radially between an aft end portion of said combustor liner and a forward end portion of said transition piece body, a cover sleeve disposed between said aft end portion of said combustor liner and said resilient seal structure, an air flow passage being defined between said cover sleeve and said aft end portion of said combustor liner; said cover sleeve having at a forward end thereof a plurality of air inlet apertures for directing cooling air from said first or second flow annulus into said air flow passage, a radially outer surface of said combustor liner aft end portion defining said air flow passage including a plurality of turbulators projecting towards but spaced from said cover sleeve, characterized in that a center portion of the arch shaped resilient seal structure faces the combustor liner and ends of the arch shaped resilient seal structure bear against an inner surface of the transition piece body; only one or two circumferentially extending rows of supports extend to and engage said cover sleeve to space said cover sleeve from said turbulators to define said air flow passage, wherein said plurality of circumferentially extending rows of supports are disposed at a position substantially aligned with the center portion of the arch shaped resilient seal structure. a combustor liner; a first flow sleeve surrounding said combustor liner with a first flow annulus therebetween, said first flow sleeve having a first plurality of cooling apertures formed about a circumference thereof for directing compressor discharge air as cooling air into said first flow annulus; a transition piece body connected to said combustor liner, said transition piece body being adapted to carry hot combustion gases to the turbine; a second flow sleeve surrounding said transition piece body, said second flow sleeve having a second plurality of cooling apertures for directing compressor discharge air as cooling air into a second flow annulus between the second flow sleeve and the transition piece body, said first flow annulus connecting to said second flow annulus; an arch shaped resilient seal structure disposed radially between an aft end portion of said combustor liner and a forward end portion of said transition piece body, a cover sleeve disposed between said aft end portion of said combustor liner and said resilient seal structure, an air flow passage being defined between said cover sleeve and said aft end portion of said combustor liner; said cover sleeve having at a forward end thereof a plurality of air inlet apertures for directing cooling air from said first or second flow annulus into said air flow passage, a radially outer surface of said combustor liner aft end portion defining said air flow passage including a plurality of turbulators projecting towards but spaced from said cover sleeve, characterized in that a center portion of the arch shaped resilient seal structure faces the combustor liner and ends of the arch shaped resilient seal structure bear against an inner surface of the transition piece body; only one or two circumferentially extending rows of supports extend to and engage said cover sleeve to space said cover sleeve from said turbulators to define said air flow passage, wherein said plurality of circumferentially extending rows of supports are disposed at a position substantially aligned with the center portion of the arch shaped resilient seal structure.