Solutions for the periodic fully developed regime in an annulus of axially varying cross section were obtained numerically. The annular duct considered here has stream wise-periodic variations of the cross-sectional area, and its dimensions were those of an actual double-pipe heat exchanger (a reflux condenser). The calculations were performed for laminar flow in the Reynolds number range from SO to 1000. In the heat transfer analysis, two boundary conditions were investigated-uniform wall temperature and uniform wall heat flux. The Prandtl number of the fluid was varied from 2 to 10, with liquid water in mind. On the basis of the computed heat transfer coefficients and pressure drops, the periodic annulus appears to be an attractive enhancement configuration relative to the annulus of axially unchanging cross section. For example, at the largest Reynolds and Prandtl numbers investigated (Re ≃. 1000, Pr = 10), the Nusselt number values for the periodic annulus exceed those for the uniform annulus by about a factor of 4, with an increase in the pressure drop of only about a factor of 2. The differences between the Nusselt numbers for the two boundary conditions studied were moderate, typically on the order of 10%, with localized extreme deviations of 15% occurring at the largest investigated Re and Pr.