Considerable discussion over the past few years has been devoted to the question of whether the logarithmic region in wall turbulence is indeed universal. Here, we analyse recent experimental data in the Reynolds number range of nominally $2\times 1{0}^{4} \lt {\mathit{Re}}_{\tau } \lt 6\times 1{0}^{5} $ for boundary layers, pipe flow and the atmospheric surface layer, and show that, within experimental uncertainty, the data support the existence of a universal logarithmic region. The results support the theory of Townsend (The Structure of Turbulent Shear Flow, Vol. 2, 1976) where, in the interior part of the inertial region, both the mean velocities and streamwise turbulence intensities follow logarithmic functions of distance from the wall.

Modeling wall turbulence remains a major challenge, as a sufficient physical understanding of these flows is still lacking. In an effort to move toward a physics-based model, A.A. Townsend introduced the hypothesis that the dominant energy-containing motions in wall turbulence are due to large eddies attached to the wall. From this simple hypothesis, the attached eddy model evolved, which has proven to be highly effective in predicting velocity statistics and providing a framework for interpreting the energy-containing flow physics at high Reynolds numbers. This review summarizes the hypothesis itself and the modeling attempts made thereafter, with a focus on the validity of the model's assumptions and its limitations. Here, we review studies on this topic, which have markedly increased in recent years, highlighting refinements, extensions, and promising future directions for attached eddy modeling.

The extent or existence of similarities between fully developed turbulent pipes and channels, and in zero-pressure-gradient turbulent boundary layers has come into question in recent years. This is in contrast to the traditionally accepted view that, upon appropriate normalization, all three flows can be regarded as the same in the near-wall region. In this paper, the authors aim to provide clarification of this issue through streamwise velocity measurements in these three flows with carefully matched Reynolds number and measurement resolution. Results show that mean statistics in the near-wall region collapse well. However, the premultiplied energy spectra of streamwise velocity fluctuations show marked structural differences that cannot be explained by scaling arguments. It is concluded that, while similarities exist at these Reynolds numbers, one should exercise caution when drawing comparisons between the three shear flows, even near the wall.

In recent years there has been significant progress made towards understanding the large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers, leaving scope for further work in pipes and channels. In this article the structure of fully developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. Results reveal long meandering structures of length up to 25 pipe radii or channel half-heights. These appear to be qualitatively similar to those reported in the log region of a turbulent boundary layer. However, for the channel case, large-scale coherence persists further from the wall than in boundary layers. This is expected since these large-scale features are a property of the logarithmic region of the mean velocity profile in boundary layers and it is well-known that the mean velocity in a channel remains very close to the log law much further from the wall. Further comparison of the three turbulent flows shows that the characteristic structure width in the logarithmic region of a boundary layer is at least 1.6 times smaller than that in a pipe or channel.

The spring constant of an atomic force microscope cantilever is often needed for quantitative measurements. The calibration method of Sader et al. [Rev. Sci. Instrum. 70, 3967 (1999)]10.1063/1.1150021 for a rectangular cantilever requires measurement of the resonant frequency and quality factor in fluid (typically air), and knowledge of its plan view dimensions. This intrinsically uses the hydrodynamic function for a cantilever of rectangular plan view geometry. Here, we present hydrodynamic functions for a series of irregular and non-rectangular atomic force microscope cantilevers that are commonly used in practice. Cantilever geometries of arrow shape, small aspect ratio rectangular, quasi-rectangular, irregular rectangular, non-ideal trapezoidal cross sections, and V-shape are all studied. This enables the spring constants of all these cantilevers to be accurately and routinely determined through measurement of their resonant frequency and quality factor in fluid (such as air). An approximate formulation of the hydrodynamic function for microcantilevers of arbitrary geometry is also proposed. Implementation of the method and its performance in the presence of uncertainties and non-idealities is discussed, together with conversion factors for the static and dynamic spring constants of these cantilevers. These results are expected to be of particular value to the design and application of micro- and nanomechanical systems in general.