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The flow in an annular channel of variable cross-section is studied numerically. The flow is assumed to be continuous, so only small angles of inclination of the channel walls are considered. For the Reynolds numbers in the range 3000-10000, the velocity profiles, turbulence energies, and Reynolds stresses and the characteristics of turbulent heat transfer determined by them are calculated. Calculations show a strong influence of the opening angle of the annular channel on the turbulent flow characteristics. An increase in the intensity of turbulence in the case of heat supply to the wall leads to an increase in the heat transfer characteristics. When using such a channel in a heat exchanger, it is possible to increase the Nusselt number and the Reynolds analogy factor without using artificial flow turbulization in comparison with a channel of constant cross-section at the same Reynolds numbers, that is, to increase the heat exchange without significantly increasing the channel resistance. The calculations are carried out using a three-parameter differential turbulence model supplemented with equations for turbulent heat transfer.