Quantifying wall turbulence via a symmetry approach. Part I. A Lie group theory

Abstract

First principle based prediction of mean flow quantities of wall-bounded turbulent flows (channel, pipe, and turbulent boundary layer - TBL) is of great importance from both physics and engineering standpoints. Here (Part I), we present a symmetry-based approach which derives analytic expressions governing the mean velocity profile (MVP) from an innovative Lie-group analysis. The new approach begins by identifying a set of order functions (e.g. stress length, shear-induced eddy length), in analogy with the order parameter in Landau's mean-field theory, which aims at capturing symmetry aspects of the fluctuations (e.g. Reynolds stress, dissipation). The order functions are assumed to satisfy a dilation group invariance - representing the effects of the wall on fluctuations - which allows us to postulate three new kinds of invariant solutions of the mean momentum equation (MME), focusing on group invariants of the order functions (rather than those of the mean velocity as done in previous studies). The first - a power law solution - gives functional forms for the viscous sublayer, the buffer layer, the log-layer, and a newly identified central `core' (for channel and pipe, but non-existent for TBL). The second - a defect power law of form $1-r^m$ ($r$ being the distance from the center line) - describes the `bulk zone' (the region of balance between production and dissipation). The third - a relation between the group invariants of the stress length function and its first derivative - describes scaling transition between adjacent layers. A combination of these three expressions yields a multi-layer formula covering the entire flow domain, identifying three important parameters: scaling exponent, layer thickness, and transition sharpness. All three kinds of invariant solutions are validated, individually and in combination, by data from direct numerical simulations (DNS).

25 pages, 3 figures

References in corpus (2)

• More is the Same; Phase Transitions and Mean Field Theories
• Universal Model of Finite-Reynolds Number Turbulent Flow in Channels and Pipes

Cited by in corpus (5)

• A critical examination of the statistical symmetries admitted by the Lundgren-Monin-Novikov hierarchy of unconfined turbulence
• On the physical inconsistency of a new statistical scaling symmetry in incompressible Navier-Stokes turbulence
• Comment on 'Lie symmetry analysis of the Lundgren-Monin-Novikov equations for multi-point probability density functions of turbulent flow'
• Karman constant and accurate mean flow prediction in a turbulent pipe
• A general Reynolds analogy theory for the compressible wall-bounded turbulence