The Asymptotic Structure of Canonical Wall-Bounded Turbulent Flows

Abstract

Our ability to reliably and efficiently predict complex high-Reynolds-number (Re) turbulent flows is essential for dealing with a large variety of problems of practical relevance. However, experiments as well as computational methods such as direct numerical simulation (DNS) and large eddy simulation (LES) face serious questions regarding their applicability to high Re turbulent flows. The most promising option to create reliable guidelines for experimental and computational studies is the use of analytical conclusions. An essential criterion for the reliability of such analytical conclusions is the inclusion of a physically plausible explanation of the asymptotic turbulence regime at infinite Re in consistency with observed physical requirements. Corresponding analytical results are reported here for three canonical wall-bounded turbulent flows: channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer. The asymptotic structure of the mean velocity and characteristic turbulence velocity, length, and time scales is analytically determined. In outer scaling, a stable asymptotic mean velocity distribution is found corresponding to a linear probability density function of mean velocities along the wall-normal direction, which is modified through wake effects. Turbulence tends to decay in this regime. In inner scaling, the mean velocity is governed by a universal log-law. Turbulence does survive in an infinitesimally thin layer very close to the wall.

The Law of the Wall and von Kármán Constant: An Ongoing Controversial Debate

“…B4. A theory involving a universal law of the wall can essentially contribute to our understanding of the structure of turbulent flows at high Re [13]. It can explain Re requirements to observe the log-law, the structure of self-similar turbulence characteristics, and convergence toward these structures.…”