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A simple analytical solution for turbulent plane Couette flow is obtained from a subset of the Navier-Stokes equations. This approach analyses the effect of the unsteady state Lagrangian diffusion of viscous momentum on the smoothed phase velocity. Three initial velocity profiles at the start of a turbulent wall cycle are considered. The predictions adequate identify the existence of three layers usually observed in turbulent flow fields, in particular a viscosity dominated wall layer and a log-law layer. Experimental measurements of time-averaged velocity are also well correlated. Key words: Couette flow, turbulent, attractors, analytical solution, Lagrangian partial derivative, time scale, initial conditions, Comment: 30 pages, 13 figures

Correlations of friction factors are presented for the general case of purely viscous non-Newtonian fluids without requiring a priori the adoption of a rheological model. They are based on an empirical estimate of the shift in the wall layer edge and the Kolmogorov point. The predictions of friction factors have the same level of accuracy as those of the Dodge-Metzner correlation but the visualisation is more compatible with measured velocity profiles. The general correlations obtained can be used to easily retrieve correlations for specific rheological models., Comment: 17 pages, 5 figures; Key words: Turbulent pipe flow, friction factor, rheological models, power law, Bingham plastic, Ellis models

The Blasius empirical correlation for turbulent pipe friction factors is derived from first principles and extended to non-Newtonian power law fluids. Two alternative formulations are obtained that both correlate well with the experimental measurements of Dodge, Bogue and Yoo. Key words: Blasius, turbulent friction factor, power law fluids, Comment: 11 pages, 4 figures

A zonal similarity analysis of friction factors in pipe flow of power law fluids is presented. It uses the critical Reynolds number and friction factor at transition as estimates of the normalised velocity and distance between the wall and log-law layers and successfully collapses both Newtonian and non-Newtonian data for laminar, transition and turbulent flows. Key words: zonal similarity analysis, friction factor, Reynolds number, transition, Comment: 11 pages, 5 figures

This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity relations in turbulence are based on phase-locked parameters linked with the development of secondary flows rather than on time-averaged wall parameters. This similarity analysis collapses the local critical instantaneous friction factor data of both Newtonian and non-Newtonian fluids at the point of bursting into a single curve. The method greatly simplifies the analysis of turbulent transport phenomena in non-Newtonian fluids. Keywords: Turbulence, time-independent non-Newtonian, Power law, pipe flow, wall layer, Comment: 28 pages, 4 figures

A four component decomposition of the local instantaneous velocity is proposed. It brings out more readily the terms in the Navier-Stokes equations associated with different events and fluid structures of turbulent flows than the classic two component decomposition of Reynolds. In particular the new composition highlights the existence of two types of Reynolds stresses: fast and slow. The fast Reynolds stresses can be linked to a streaming process that describes the ejection of wall fluid in the bursting process. It also provides a simple method for modelling the wall layer. The four components are a long time average, a slow fluctuating component based on the difference between the long term average and the smoothed phase velocity developed by passing coherent structures, a fast fluctuating component which is periodic in nature and a streaming component created by the interaction between the fast fluctuations and the fluid viscous effects. Key words: Velocity components, bursting, streaming flow, smoothed phase velocity, fast and slow Reynolds stresses, wall layer, Comment: 31 pages, 9 figures