This paper reviews the state-of-the art in vibration energy harvesting for wireless, self-powered microsystems. Vibration-powered generators are typically, although not exclusively, inertial spring and mass systems. The characteristic equations for inertial-based generators are presented, along with the specific damping equations that relate to the three main transduction mechanisms employed to extract energy from the system. These transduction mechanisms are: piezoelectric, electromagnetic and electrostatic. Piezoelectric generators employ active materials that generate a charge when mechanically stressed. A comprehensive review of existing piezoelectric generators is presented, including impact coupled, resonant and human-based devices. Electromagnetic generators employ electromagnetic induction arising from the relative motion between a magnetic flux gradient and a conductor. Electromagnetic generators presented in the literature are reviewed including large scale discrete devices and wafer-scale integrated versions. Electrostatic generators utilize the relative movement between electrically isolated charged capacitor plates to generate energy. The work done against the electrostatic force between the plates provides the harvested energy. Electrostatic-based generators are reviewed under the classifications of in-plane overlap varying, in-plane gap closing and out-of-plane gap closing; the Coulomb force parametric generator and electret-based generators are also covered. The coupling factor of each transduction mechanism is discussed and all the devices presented in the literature are summarized in tables classified by transduction type; conclusions are drawn as to the suitability of the various techniques.

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Energy harvesting vibration sources for microsystems applications

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

The role of coherent structures in the production and dissipation of turbulence in a boundary layer is characterized, summarizing the results of recent investigations. Coherent motion is defined as a three-dimensional region of flow where at least one fundamental variable exhibits significant correlation with itself or with another variable over a space or time range significantly larger than the smallest local scales of the flow. Sections are then devoted to flow-visualization experiments, statistical analyses, numerical simulation techniques, the history of coherent-structure studies, vortices and vortical structures, conceptual models, and predictive models. Diagrams and graphs are provided.

Coherent Motions in the Turbulent Boundary Layer

The turbulent boundary layer on a flat plate, with zero pressure gradient, is simulated numerically at four stations between R sub theta = 225 and R sub theta = 1410. The three-dimensional time-dependent Navier-Stokes equations are solved using a spectra method with up to about 10 to the 7th power grid points. Periodic spanwise and stream-wise conditions are applied, and a multiple-scale procedure is applied to approximate the slow streamwise growth of the boundary layer. The flow is studied, primarily, from a statistical point of view. The solutions are compared with experimental results. The scaling of the mean and turbulent quantities with Reynolds number is examined and compared with accepted laws, and the significant deviations are documented. The turbulence at the highest Reynolds number is studied in detail. The spectra are compared with various theoretical models. Reynolds-stress budget data are provided for turbulence-model testing.

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Direct simulation of a turbulent boundary layer up to R sub theta = 1410

The structure of energy-containing turbulence in the outer region of a zero-pressure-
 gradient boundary layer has been studied using particle image velocimetry (PIV) to measure the instantaneous velocity fields in a streamwise-wall-normal plane. Experiments performed at three Reynolds numbers in the range 930 0) that occur on a locus inclined at 30–60° to the wall.In the outer layer, hairpin vortices occur in streamwise-aligned packets that propagate with small velocity dispersion. Packets that begin in or slightly above the buffer layer are very similar to the packets created by the autogeneration mechanism (Zhou, Adrian & Balachandar 1996). Individual packets grow upwards in the streamwise direction at a mean angle of approximately 12°, and the hairpins in packets are typically spaced several hundred viscous lengthscales apart in the streamwise direction. Within the interior of the envelope the spatial coherence between the velocity fields induced by the individual vortices leads to strongly retarded streamwise momentum, explaining the zones of uniform momentum observed by Meinhart & Adrian (1995). The packets are an important type of organized structure in the wall layer in which relatively small structural units in the form of three-dimensional vortical structures are arranged coherently, i.e. with correlated spatial relationships, to form much longer structures. The formation of packets explains the occurrence of multiple VITA events in turbulent ‘bursts’, and the creation of Townsend's (1958) large-scale inactive motions. These packets share many features of the hairpin models proposed by Smith (1984) and co-workers for the near-wall layer, and by Bandyopadhyay (1980), but they are shown to occur in a hierarchy of scales across most of the boundary layer.In the logarithmic layer, the coherent vortex packets that originate close to the wall frequently occur within larger, faster moving zones of uniform momentum, which may extend up to the middle of the boundary layer. These larger zones are the induced interior flow of older packets of coherent hairpin vortices that originate upstream and over-run the younger, more recently generated packets. The occurence of small hairpin packets in the environment of larger hairpin packets is a prominent feature of the logarithmic layer. With increasing Reynolds number, the number of hairpins in a packet increases.

Vortex organization in the outer region of the turbulent boundary layer

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.

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High–Reynolds Number Wall Turbulence

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

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.

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

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 × 104 < Reτ < 6 × 105 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. © 2013 Cambridge University Press.

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On the logarithmic region in wall turbulence

Contents: 1. Introduction. 2. The Equations of Motion. 3. The Equations for Turbulent Flow. 4. Fundamental Concepts. 5. Boundary Layer Mean Flow Behavior. 6. Boundary Layer Turbulence Behavior. 7. Mixing Layers. 8. Perturbed Boundary Layers. 9. Two-Dimensional Interactions. 10. Three-Dimensional Interactions.

Alexander Smits

Papers

High–Reynolds Number Wall Turbulence

Mean-flow scaling of turbulent pipe flow

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

On the logarithmic region in wall turbulence

Turbulent Shear Layers in Supersonic Flow