A turbine vane includes a perforated liner defining an annular cavity between an outside wall of the liner and an inside wall of the vane, an air admission opening for feeding the inside of the liner with cooling air, and an air exhaust opening for exhausting a fraction of the cooling air from the vane. The liner is secured to the vane at one end and free at the other end to slide along an inside edge of the vane under the effects of relative thermal expansion between the liner and the inside wall of the vane. An annular gap between the free end of the liner and the inside edge of the vane defines a leakage zone for cooling air and the inside edge includes a recess for generating head loss in this leakage zone so as to reduce the flow rate of cooling air passing through the leakage zone.
