Post-critical Regimes Research Articles

Stochastic performance of a torsional-flutter harvester in non-stationary, turbulent thunderstorm outflows

Efficiency of a torsional-flutter energy harvester in thunderstorm-like wind flows is studied. The apparatus has been proposed in recent years as an alternative to other flapping windmills of medium size due to its simple installation and operational mechanism. The paper investigates a rapid change of wind load intensity, induced by a thunderstorm, which may either reduce the output energy conversion or damage the apparatus. Aeroelastic load is replicated through unsteady Wagner’s indicial theory. The study explores the influence of both slowly-varying wind velocity variations, non-stationary turbulence of variable intensity and load perturbations. A stochastic model is derived to investigate both pre-critical and post-critical regimes. Various configurations are considered.

The influence of surface roughness on postcritical flow over circular cylinders revisited

This work investigates the effect of surface roughness on cylinder flows in the postcritical regime and reexamines whether the roughness Reynolds number ( $Re_{k_s}$ ) primarily governs the aerodynamic behaviour. It has been motivated by limitations of many previous investigations, containing occasionally contradictory findings. In particular, many past studies were conducted with relatively high blockage ratios and low cylinder aspect ratios. Both of these factors appear to have non-negligible effects on flow behaviour, and particularly fluctuating quantities such as the standard deviation of the lift coefficient. This study employs a 5 % blockage ratio and a span-to-diameter ratio of 10. Cylinders of different relative surface roughness ratios ( $k_s/D$ ), ranging from $1.1\times 10^{-3}$ to $3\times 10^{-3}$ , were investigated at Reynolds numbers up to $6.8 \times 10^5$ and $Re_{k_s}$ up to 2200. It is found that the base pressure coefficient, drag coefficient, Strouhal number, spanwise correlation length of lift and the standard deviation of the lift coefficient are well described by $Re_{k_s}$ in postcritical flows. However, roughness does have an effect on the minimum surface pressure coefficient (near separation) that does not collapse with $Re_{k_s}$ . The universal Strouhal number proposed by Bearman (Annu. Rev. Fluid Mech., vol. 16, 1984, pp. 195–222) appears to be nearly constant over the range of $Re_{k_s}$ studied, spanning the subcritical through postcritical regimes. Frequencies in the separating shear layers are found to be an order of magnitude lower than the power law predictions for separating shear layers of smooth cylinders.

Identification of distinct flow behaviours around twin rough cylinders at low wind incidence

Wind tunnel experiments are performed on twin rough cylinders with a centre-to-centre spacing ratio L/D=1.2. The unsteady flow field is measured at two different Reynolds numbers, Re = 45 k and 275 k, corresponding to the sub- and post-critical flow regimes, respectively. The wind incidence α is varied from 0°to 10°. The detailed analysis of the lift coefficients as a function of the wind incidence reveals the existence of three flow behaviours in both flow regimes. Bi-stabilities are observed and occur within different ranges of wind incidence depending on the flow regime. A methodology is proposed to decouple the modes which intermittently occur in case of bi-stability, allowing to identify the flow behaviour of each mode. In the sub-critical flow regime, the flow behaviour at very low wind incidences (α=0°–2°) corresponds to an alternate re-attachment on both sides of the rear cylinder. In the post-critical flow regime, the flow behaviour is bi-stable because of the intermittent re-attachment of the shear layers. For larger wind incidences (α= 4°–6°), the lower shear layer intermittently re-attaches steadily or alternately onto the rear cylinder in the sub-critical flow regime, evidencing a bi-stability. On the other hand, it always re-attaches steadily onto the rear cylinder in the post-critical flow regime. A gap flow is observed for the largest wind incidences (α=8°–10°). It is found to be stronger in the sub- than in the post-critical flow regime.

Generalized lift force model under vortex shedding

This paper proposes a generalized model for vortex shedding around a static cylinder. This model is a first step to bridge the gap between two existing families of Vortex Induced Vibration (VIV) models , namely (1) stochastic spectral models and (2) wake-oscillator models with coupled fluid and structural equations. The first family is experimentally-based while the models of the second family typically rely on theoretical and phenomenological concepts. These models use a Van der Pol, Rayleigh or a combination of both oscillators for the lift equation. The proposed model generalizes these models by combining all third degree terms in the case of a static model, i.e. by considering the equation of the fluid oscillator only. It is possible to enrich existing models with a more complex parametric model because its parameters are identified from experimental data. A second specificity of the proposed model stems from the observation that, for a static cylinder in low turbulence flow, the envelope of the measured lift force is not perfectly mono-harmonic, nor deterministic. Turbulence in the wake is expected to create fluctuations in the lift envelope. In the proposed model, stochasticity is consequently introduced to reproduce these fluctuations. It is based as an additive exogenous noise as an input to the generalized shedding model, using a Von Karman spectrum. Its coefficients are adjusted so that the model reproduces the probability density function of the measured lift envelope (and its power spectral density). The methodology is applied to Wind Tunnel data of a static circular cylinder in subcritical and postcritical regimes. This set of data is used to identify the coefficients of non-linear terms and Von Karman spectrum. The lift fluctuation coefficient generated with the model matches results available in the literature for the considered regimes. This work focuses on a methodology to obtain a generalized vortex shedding model from experimental data measured on a static body. It can be applied to other cylinder cross-sections or even extended to cylinders arranged in tandem.

Non-holonomic dynamics of a ball moving inside a spherical cavity

The ball type vibration absorbers are popular to be used at slender structures like masts or towers when a vertical space available and maintenance possibilities are limited. In general, the absorber consists of a cavity in which a heavy ball is rolling. In the paper, the response of a heavy ball rolling inside a semi-spherical cavity under horizontal kinematic excitation is investigated. The system with six degrees of freedom with three non-holonomic constraints is considered. The contact between the ball and the cavity surface is supposed to be perfect without any sliding. The system is strongly non-linear and, consequently, the full 3D mathematical model should be carried out and discussed. With respect to previous experiences the basic strategy declines the Lagrangian procedure including conventional constraints via Lagrangian multipliers. Instead of that the Gibbs-Appell formulation is used. The system has an auto-parametric character. The most important post-critical regimes are outlined and qualitatively evaluated in resonance domain. Numerical experiments have been performed when excitation frequency is slowly swept up and down. Some applications in civil engineering as a tuned mass damper used on slender structures is outlined and compared with a conventional pendulum damper. Strengths and weaknesses of both absorbers types are discussed.

Flapping dynamics of a piezoelectric membrane behind a circular cylinder

The flapping dynamics of a piezoelectric membrane placed behind a circular cylinder, which are closely related to its energy harvesting performance, were extensively studied near the critical regime by varying the distance between the cylinder and the membrane. A total of four configurations were used for the comparative study: the baseline configuration in the absence of the upstream circular cylinder, and three configurations with different distances (S) between the cylinder and the membrane (S/D=0, 1, and 2). A polyvinylidene fluoride (PVDF) membrane was configured to flutter at its second mode in these experiments. The Reynolds number based on the membrane’s length was 6.35×104 to 1.28×105, resulting in a full view of membrane dynamics in the subcritical, critical, and postcritical regimes. The membrane shape and the terminal voltage were simultaneously measured with a high-speed camera and an oscilloscope, respectively. The influence of the upstream cylinder on the membrane dynamics was discussed in terms of time-mean electricity, instantaneous variations and power spectra of terminal voltage and membrane shape, fluctuating voltage amplitude, and flapping frequency. The experimental results overwhelmingly demonstrated that the terminal voltage faithfully reflected various unsteady events embedded in the membrane’s flapping motion. For all configurations, dependency of the captured electricity on a flow speed beyond the critical status was found to follow the parabolic relationship. In the two configurations in which S/D=0 and 1, the extraneously induced excitation by the Kármán vortex street behind the circular cylinder substantially reduced the critical flow speed, giving rise to effective energy capture at a lower flow speed and a relatively high gain in power output. However, in the configuration in which S/D=2, the intensified excitation by the Kármán vortex street on the membrane considerably reduced the captured energy. Finally, a transient analysis of the membrane’s flapping dynamics in the configuration in which S/D=0 was performed in terms of phase-dependent variations of the membrane segment’s moving speed, membrane curvature, and terminal voltage; the analysis resulted in a full understanding of the energy harvesting process with consecutive inter transfer of elastic, kinetic, and electric energies.

Dynamic stability of a vertically excited non-linear continuous system

Easily deformable tall structures exposed to a strong vertical component of an excitation are endangered by auto-parametric resonance effect. This non-linear dynamic process in a post-critical regime might contribute to various damages caused by a kinematic excitation. Vertical and horizontal response components are independent on the linear level. However their interaction takes place due to non-linear terms in post-critical regime. Two generally different types of the post-critical regimes are presented: (i) post-critical state with possible recovery and (ii) exponentially rising horizontal response leading to a collapse. A special attention is paid to processes of transition from semi-trivial to post-critical state in case of time limited excitation period as it concerns the seismic processes. Solution method combining analytical and numerical approaches is developed and used. Its applicability and shortcomings are commented. A few hints for engineering applications are given.

Feasibility assessment of a leading-edge-flutter wind power generator

This study addresses the preliminary technical feasibility assessment of a mechanical apparatus for conversion of wind energy. The proposed device, designated as “leading-edge-flutter wind power generator”, employs aeroelastic dynamic instability of a blade airfoil, torsionally rotating about its leading edge. Although the exploitation of aeroelastic phenomena has been proposed by the research community for energy harvesting, this apparatus is compact, simple and marginally susceptible to turbulence and wake effects. The objective of this study was to identify a conceptual configuration that could become operational at low and medium wind speeds. Pre- and post-critical regimes are simulated; the prediction of mechanical efficiency in the absence of dissipation is also discussed.

The buckling of elongated rotating drill strings

There is a growing need for the development and implementation of new numerical methods for computer simulation of static and dynamic critical states of drill strings in connection with mastering deeper oil and gas resources. Current methods for analysis of extraordinary behavior of the drill strings, as a rule, concern their post-critical states related to essential change of their shape and contact interaction with borehole wall. In parallel with the post-critical regimes there also are critical modes of operation attended with incipiency of unstable equilibrium states and the DS buckling. Inasmuch as the critical states precede the post-critical ones, sometimes their prognosis can be more important. This paper considers the problem of quasi-static stability of elongated drill string tubes studied with allowance made for destabilizing effects of the longitudinal non-uniform preloading, action of torque, inertia forces of rotation and internal flows of mud. The constitutive equations are deduced, and the methods for their computer solving are proposed. As a result of the completed analysis the critical values of the characteristic parameters are found, and it is shown that the modes of incipient buckling possess complicated shapes associated with the tube spiral curving prevailing in the zone of bottom-hole-assembly.

Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper

The pendulum vibration damper modelled as a two degree of freedom strongly non-linear auto-parametric system is investigated. A kinematic external excitation in the suspension point is applied. The excitation is considered to be horizontal and harmonically variable in time. A semi-trivial solution and its stability are analyzed. Special attention is paid to the resonance domain. In certain domains of pendulum and excitation parameters the semi-trivial solution does not exist in this domain and various post-critical three-dimensional regimes occur. Some of them are non-stationary despite the harmonic excitation. Three different types of the resonance domain are investigated. Their main properties depend significantly on dynamic parameters of the pendulum and of the external excitation amplitude. An analytical and numerical study brings forth several recommendations for designers of these devices. Their aim is to avoid any post-critical response regimes endangering the pendulum functionality.

Postcritical regimes in the nonlinear problem of vortex motion under the free surface of a weighable fluid

An improved Levi-Civita method in which the singularities of the desired function are taken into account by introducing terms containing power singularities is proposed. Results of numerical analysis of the nonlinear problem of a vortex in a bounded flow of an ideal weighable fluid (Fr>1) are given. The following limiting flow regimes are studied: the Stokes waves with one and two crests, emergence of a critical point on the surface, and the detachment of a vortex from a soliton and a uniform flow. It is shown that nonperiodic waves can form in a local zone in the vicinity of the critical point.

Viscous sublayer in the neighborhood of a corner point in supersonic flow past a cooled body

The behavior of the flow functions at the upper edge of the viscous sublayer is subjected to an asymptotic analysis. The results obtained permit a more accurate formulation of the boundary-value problem and an improvement in the quality of numerical calculation of the flow in the viscous sublayer. The heat flux and friction distributions on the surface of the body are obtained for precritical and postcritical interaction regimes; for expansion flows the calculated results are presented in the form of the dependence of the maximum values of the friction and heat flux on the temperature factor and the angle of deflection of the flap; approximate expressions suitable for practical use are proposed.

The effect of surface roughness on the flow regime around circular cylinders

Experimental research has been carried out to study the influence of surface roughness on transitions between the subcritical, critical supercritical and postcritical regimes around isolated circular cylinders in cross-flow. The experiments were performed in an open-jet wind tunnel, which was modified to obtain satisfactorily bidimensional conditions, with Reynolds numbers ranging from ∼2.6 × 10 4 to ∼2.8 × 10 5 . The cylinders were covered with different types of standard commercial emery cloth, and the estimated relative roughness varied from ∼1.0 × 10 −3 to ∼12 × 10 −3 . The signal from a hot-wire probe placed near the cylinder was analysed in order to ascertain the presence of vortex shedding and to obtain the Strouhal number. The mean pressure distribution around most of the cylinders were also measured, and then the drag coefficients were obtained by integration. The boundaries between the various flow regimes as a function of the degree of roughness and of the Reynolds number were tentatively established, assuming the critical regime to be characterised by the absence of regular vortex shedding, and the postcritical one by Reynolds-number independence. The results confirmed the striking influence of roughness on the flow regime; in particular, the critical regime can be impressively reduced, and it may even disappear in the case of highly roughened cylinders. The boundaries between the various regimes, as well as the supercritical and postcritical drag coefficients and Strouhal numbers, seem to be a function of both the size and the type of surface roughness. This is confirmation, therefore, that it is not possible to characterise the flow regime only by means of a Reynolds number based on the size of the roughness. Finally, it was demonstrated that strong supercritical and postcritical vortex shedding can take place from highly, but uniformly, roughened circular cylinders.