Summary: The averaged layered turbulent wall flows of an incompressible fluid are considered for an arbitrary wall roughness and a friction coefficient tending to zero. It is assumed that in the asymptotic limit the flow may be divided into two regions: an outer, only slightly disturbed region, and a thin wall region. A similarity law for the velocity disturbances in the outer region is established for cylindrical tubes with an arbitrary cross section. Using the asymptotically matched solutions in the inner and outer regions, the existence of a logarithmic velocity profile in the matching zone is proved. On the basis of a unique asymptotic approach, the existing semi-empirical solutions for tube flows are brought into consistency. A number of semi-empirical formulas for the velocity profiles, which refine and expand the range of applicability of existing relations, are proposed.