The Reynolds time‐averaged equations are adopted for fully turbulent two‐dimensional flow of an incompressible fluid through a channel with plane smooth walls. Closure is effected by means of so‐called second‐order methods based on the conservation of mean turbulent kinetic energy. In particular, generalizations of the turbulent shear flow models due to Bradshaw and co‐workers and to Spalding and co‐workers, are adopted as representative of recent differential closure models. The models are treated in the limit of large turbulent Reynolds number by means of limit‐process expansions. The principal division of the flow is into a relatively thick defect layer near the center of the channel (in which the channel half‐width characterizes the layer thickness, the velocity may be linearized about its centerline speed, and the Reynolds stress dominates the Newtonian stress to lowest order), and the relatively thin viscous sublayer near the wall (in which a viscous length scale is appropriate, the velocity is small relative to the centerline speed, and the Reynolds and Newtonian stresses are of comparable magnitude). Certain previously unnoticed constraints on the closure models are revealed by the analysis.