A turbine engine pipe having a longitudinal axis, which, at an upstream end thereof, includes an outer wall and a core, only an upstream portion of which is surrounded by the outer wall. The boundary between the upstream portion and a downstream portion is defined by the intersection between the core and the surface formed by bisectors of acute angles formed by straight lines perpendicular to the outer wall and straight lines perpendicular to an outer surface of the core passing through the downstream end of the outer wall. The outer surface of the downstream portion of the core has a sharply curved convex region close to the boundary, the region having a radius of curvature in a radial plane of less than 30 times the radius of the core at the boundary, and a first slightly curved region between the boundary and the sharply curved convex region.