In general, flow in an annulus will be strongly asymmetric because of unequal curvatures and, possibly, dissimilar texture of the two boundaries. As a result, the turbulent flow structure in such geometries possesses a number of features which can not satisfactorily be accounted for with a simple model of turbulence. The present paper provides predictions based on a more elaborate model, developed by the author, in which differential transport equations are solved for the turbulent shear stress, the turbulent kinetic energy and the rate of the turbulence energy dissipation, simultaneously with the mean momentum equation. The complete model thus consists of a closed set of four coupled non-linear partial differential equations. Predictions of flow in annuli with both smooth and rough cores with various radius ratios display in all cases very good agreement with experimental results of various authors.