Asymptotic properties of mixing length closures for turbulent pipe flow

Turbulence is investigated as the Reynolds number approaches infinity for the flow of an incompressible fluid through straight pipes with a circular cross section under the assumptions that the continuum hypothesis holds, the pipe wall is smooth, and the mixing length closure constructed by Cantwell ["A universal velocity profile for smooth wall pipe flow," J. Fluid Mech. 878, 834-874 (2019)] is sufficiently accurate to allow the extrapolation to Reynolds numbers beyond the range of the Princeton superpipe data used as a foundation for the closure model. Two sets of scales are introduced to set up two sets of dimensionless equations and two Reynolds numbers for the near wall region (Re, inner scaling) and the center part of the pipe flow (R-tau, outer scaling). It is shown analytically that the turbulent flow asymptotically approaches a two-layer structure: The core of the pipe flow becomes uniform with constant mean velocity and zero shear stress in the outer scaling and the near wall region (inner scaling) with the mean velocity satisfying the law of the wall and non-zero shear stress. The core part of the flow pushes, as R-tau -> infinity, the near wall layer to the boundary restricting it to a cylindrical subdomain with zero volume.