On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

ABSTRACT

Understanding of the structure of turbulent flows at extreme Reynolds numbers (Re) is relevant because of several reasons: almost all turbulence theories are only valid in the high Re limit, and most turbulent flows of practical relevance are characterized by very high Re. Specific questions about wall-bounded turbulent flows at extreme Re concern the asymptotic laws of the mean velocity and turbulence statistics, their universality, the convergence of statistics towards their asymptotic profiles, and the overall physical flow organization. In extension of recent studies focusing on the mean flow at moderate and relatively high Re, the latter questions are addressed with respect to three canonical wall-bounded flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Main results reported here are the asymptotic logarithmic law for the mean velocity and corresponding scale-separation laws for bulk flow properties, the Reynolds shear stress, the turbulence production and turbulent viscosity. A scaling analysis indicates that the establishment of a self-similar turbulence state is the condition for the development of a strict logarithmic velocity profile. The resulting overall physical flow structure at extreme Re is discussed.

KEYWORDS:

• Wall-bounded turbulent flows
• extreme $R{e}_{\tau }$
• mean flow and turbulence structure

Acknowledgments

The author is very thankful to the referees for their helpful suggestions for improvements.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author would like to acknowledge partial support through Langley Research Center (NASA's NRA) research opportunities in aeronautics program (grant number NNX12AJ71A with Dr P. Balakumar as technical officer) and the National Science Foundation (Directorate for Mathematical and Physical Sciences (DMS - CDS&E-MSS), grant number 1622488 with Dr Y. Zeng as Technical Officer). Substantial support from the Hanse-Wissenschaftskolleg, Institute for Advanced Study, Delmenhorst, Germany (Hanse-Wissenschaftskolleg) (Delmenhorst, Germany, Technical Monitor: W. Stenzel) is gratefully acknowledged.