A cooling system for a turbomachine, comprising: a vane, including: an outer diameter platform at one end including outer diameter attachment structure operative to mount the vane to an outer diameter stationary structure, the outer diameter attachment structure having leakage metering outlets formed therein; an outer diameter cavity (257, 357) defined by the outer diameter platform and an outer diameter component; an inner diameter platform at another end of the vane opposite the outer diameter platform and operative to be disposed against an inner diameter component; an inner diameter cavity defined by the inner diameter platform and the inner diameter component; and an airfoil disposed between the outer diameter platform and the inner diameter platform and defining a first cross-passage, wherein the first cross-passage defines a first outer diameter opening in the outer diameter platform, wherein the first cross-passage defines a first inner diameter opening in the inner diameter platform such that a first leakage flow can pass through the vane from the inner diameter platform, through the airfoil, to the outer diameter platform, and into an outer diameter leakage path (225b, 249) of the vane, wherein the outer diameter leakage path (225b, 249) is defined, in part, through the leakage metering outlets of the outer diameter attachment structure. a vane, including: an outer diameter platform at one end including outer diameter attachment structure operative to mount the vane to an outer diameter stationary structure, the outer diameter attachment structure having leakage metering outlets formed therein; an outer diameter cavity (257, 357) defined by the outer diameter platform and an outer diameter component; an inner diameter platform at another end of the vane opposite the outer diameter platform and operative to be disposed against an inner diameter component; an inner diameter cavity defined by the inner diameter platform and the inner diameter component; and an airfoil disposed between the outer diameter platform and the inner diameter platform and defining a first cross-passage, wherein the first cross-passage defines a first outer diameter opening in the outer diameter platform, wherein the first cross-passage defines a first inner diameter opening in the inner diameter platform such that a first leakage flow can pass through the vane from the inner diameter platform, through the airfoil, to the outer diameter platform, and into an outer diameter leakage path (225b, 249) of the vane, wherein the outer diameter leakage path (225b, 249) is defined, in part, through the leakage metering outlets of the outer diameter attachment structure. an outer diameter platform at one end including outer diameter attachment structure operative to mount the vane to an outer diameter stationary structure, the outer diameter attachment structure having leakage metering outlets formed therein; an outer diameter cavity (257, 357) defined by the outer diameter platform and an outer diameter component; an inner diameter platform at another end of the vane opposite the outer diameter platform and operative to be disposed against an inner diameter component; an inner diameter cavity defined by the inner diameter platform and the inner diameter component; and an airfoil disposed between the outer diameter platform and the inner diameter platform and defining a first cross-passage, wherein the first cross-passage defines a first outer diameter opening in the outer diameter platform, wherein the first cross-passage defines a first inner diameter opening in the inner diameter platform such that a first leakage flow can pass through the vane from the inner diameter platform, through the airfoil, to the outer diameter platform, and into an outer diameter leakage path (225b, 249) of the vane, wherein the outer diameter leakage path (225b, 249) is defined, in part, through the leakage metering outlets of the outer diameter attachment structure.