A higher-order asymptotic theory for fully developed turbulent flow in smooth pipes

A complete second-order asymptotic theory for fully developed turbulent flow in smooth pipes at high turbulent Reynolds numbers is presented. The theory is based on Prandtl's mixing-length hypothesis involving a fourth-order polynomial representation for the mixing length and taking into account its dependence on the Reynolds number. Two main contributions with respect to the existing literature have been achieved: (a) the friction law is obtained by asymptotic evaluation of an integral, completely independently of the velocity field, and (b) an axis layer (in addition to the wall layer and the outer layer) has to be included in the analysis in order to remove a nonuniformity appearing in the second-order solution for the velocity field. Closed-form analytic expressions for all constants and wake functions appearing up to the second-order solution in both the friction law and the velocity field are obtained. The results are in a very good agreement with experiments.