Friction factor power law with equivalent log law, of a turbulent fully developed flow, in a fully smooth pipe

Over the past century, Blasius (Forschungsheft 131:1-41, 1913) gave empirical power law friction factors ? = 0.3164 Re-1/4 for a turbulent pipe flow and later work published other empirical values of power law indexes. The present work deals with open Reynolds momentum equations by matched asymptotic expansions for large Reynolds number. In the overlap region, a rational dual solutions have power law and log law velocities and friction factor. If outer layer flow is neglected, the power-law friction factor becomes ? = mRe(-n) in a pipe flow, with power law index n(Re) and prefactor m(Re). Further, tangent at a point on power law envelop gives log law at that point. Thus for each value of power index n with prefactor m, the power law theory holds at a point. As an engineering practice, power law at one point is often used in a limited domain of Reynolds number, which compares in that range with experimental and DNS data reported in the literature. The friction factor log law to higher order has been proposed and the second-order effect compares well for lower Reynolds numbers in an entire range of Reynolds numbers for a turbulent pipe flow.