The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

We review wall-bounded turbulent flows, particularly high–Reynolds number, zero–pressure gradient boundary layers, and fully developed pipe and channel flows. It is apparent that the approach to an asymptotically high–Reynolds number state is slow, but at a sufficiently high Reynolds number the log law remains a fundamental part of the mean flow description. With regard to the coherent motions, very-large-scale motions or superstructures exist at all Reynolds numbers, but they become increasingly important with Reynolds number in terms of their energy content and their interaction with the smaller scales near the wall. There is accumulating evidence that certain features are flow specific, such as the constants in the log law and the behavior of the very large scales and their interaction with the large scales (consisting of vortex packets). Moreover, the refined attached-eddy hypothesis continues to provide an important theoretical framework for the structure of wall-bounded turbulent flows.

/pdf/high-reynolds-number-wall-turbulence-1s7ld2qfbi.pdf

High–Reynolds Number Wall Turbulence

Despite extensive study, there remain significant questions about the Reynolds-number scaling of the zero-pressure-gradient flat-plate turbulent boundary layer. While the mean flow is generally accepted to follow the law of the wall, there is little consensus about the scaling of the Reynolds normal stresses, except that there are Reynolds-number effects even very close to the wall. Using a low-speed, high-Reynolds-number facility and a high-resolution laser-Doppler anemometer, we have measured Reynolds stresses for a flat-plate turbulent boundary layer from Reθ = 1430 to 31 000. Profiles of u′2, v′2, and u′v′ show reasonably good collapse with Reynolds number: u′2 in a new scaling, and v′2 and u′v′ in classic inner scaling. The log law provides a reasonably accurate universal profile for the mean velocity in the inner region.

Reynolds-number scaling of the flat-plate turbulent boundary layer

The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Re at very large ones.

/pdf/scaling-of-the-energy-spectra-of-turbulent-channels-yaaftn9z8z.pdf

Scaling of the energy spectra of turbulent channels

Wall-bounded turbulent flows at high Reynolds numbers have become an increasingly active area of research in recent years. Many challenges remain in theory, scaling, physical understanding, experimental techniques, and numerical simulations. In this paper we distill the salient advances of recent origin, particularly those that challenge textbook orthodoxy. Some of the outstanding questions, such as the extent of the logarithmic overlap layer, the universality or otherwise of the principal model parameters such as the von Karman “constant,” the parametrization of roughness effects, and the scaling of mean flow and Reynolds stresses, are highlighted. Research avenues that may provide answers to these questions, notably the improvement of measuring techniques and the construction of new facilities, are identified. We also highlight aspects where differences of opinion persist, with the expectation that this discussion might mark the beginning of their resolution.

/pdf/wall-bounded-turbulent-flows-at-high-reynolds-numbers-recent-ogu7wzlf0n.pdf

Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×103 to 35×106. Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60 9×103). Von Karman's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600 5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×103 and 35×106, and includes a term to account for the near-wall velocity profile.

Mean-flow scaling of turbulent pipe flow

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Karman constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

/pdf/a-new-friction-factor-relationship-for-fully-developed-pipe-eb05qlazwl.pdf

A new friction factor relationship for fully developed pipe flow

Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

/pdf/friction-factors-for-smooth-pipe-flow-4ujoy32gzd.pdf

Friction factors for smooth pipe flow

The scaling in the overlap region of turbulent wall-bounded flows has long been the source of controversy, and until recently this controversy could not be addressed because measurements did not span a sufficient range of Reynolds number. Mean velocity surveys performed in a new pipe flow experiment span a very large range of Reynolds numbers, 31×103 to 35×106 (based on average velocity and diameter). Here, these experimental data are used to evaluate theories on the scaling in the overlap region. At sufficiently high Reynolds numbers, the mean velocity profile in the overlap region is found to be better represented by a log law than a power law. These results suggest a theory of complete similarity instead of incomplete similarity, contradicting the theories recently developed by Barenblatt et al.

/pdf/log-laws-or-power-laws-the-scaling-in-the-overlap-region-jzlimjou3q.pdf

Log laws or power laws: The scaling in the overlap region

An experimental investigation was conducted to determine the scaling of the mean velocity profile for a fully developed, smooth pipe flow. Measurements of the mean velocity profiles and static pressure gradients were performed at 26 different Reynolds numbers between $31\ifmmode\times\else\texttimes\fi{}{10}^{3}$ and $35\ifmmode\times\else\texttimes\fi{}{10}^{6}$. The profiles indicate two overlap regions: one which scales as a power law and one which scales as a log law, where the log law is only evident when the Reynolds number exceeds approximately $300\ifmmode\times\else\texttimes\fi{}{10}^{3}$. It is proposed that the velocity scales for the inner and outer regions are different, which is contrary to commonly accepted beliefs.

M. V. Zagarola

Papers

Mean-flow scaling of turbulent pipe flow

A new friction factor relationship for fully developed pipe flow

Friction factors for smooth pipe flow

Log laws or power laws: The scaling in the overlap region

Scaling of the mean velocity profile for turbulent pipe flow