Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

Analytical and experimental investigations of the pulsed air–water jet

This paper describes a new way of generating pulsed air–water jet by entraining and mixing air into the cavity of a pulsed water jet nozzle. Based on the theory of hydro-acoustics and fluid dynamics, a theoretical model which describes the frequency characteristic of the pulsed air–water jet is outlined aimed at gaining a better understanding of this nozzle for generating pulses. The calculated result indicates that as the air hold-up increases, the jet oscillation frequency has an abrupt decrease firstly, and then reaches a minimum gradually at α (air hold-up)=0.5, finally it gets increased slightly. Furthermore, a vibration test was conducted to validate the present theoretical result. By this way, the jet oscillation frequency can be obtained by analyzing the vibration acceleration of the equal strength beam affected by the jet impinging. Thereby, it is found that the experimental result shows similar trend with the prediction of the present model. Also, the relationship between vibration acceleration and cavity length for the pulsed water jet follows a similar tendency in accord with the pulsed air–water jet, i.e. there exists a maximum for each curve and the maximum occurs at the ratio of L/d₁ (the ratio of cavity length and upstream nozzle diameter) =2.5 and 2.2, respectively. In addition, experimental results on specimens impinged by the pulsed water jet and pulsed air–water jet show that the erosion depth increases slightly with air addition within a certain range of cavity length. Further, this behavior is very close to the vibration test results. As for erosion volume, the air entrained into the cavity significantly affects the material removal rate.     

Numerical study of oscillating boundary layer flow over a flat plate using k–kL–ω turbulence model

Oscillating boundary layer flow over an infinite flat plate at rest was simulated using the k–k_L–ω turbulence model for a Reynolds number range of 32 ⩽ Re_δ ⩽ 10,000 ranging from fully laminar flow to fully turbulent flow. The k–k_L–ω model was validated by comparing the predictions with LES results and experimental results for intermittently turbulent and fully turbulent flow regimes. The good agreement obtained between the k–k_L–ω model prediction with the experimental and LES results indicate that the k–k_L–ω model is able to accurately simulate transient intermittently turbulent flow and as well as accurately predict the onset of turbulence for such oscillatory flows.     

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj-Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t_v and t_d. Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω, on the ratio M=t_v/t_d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M. Motivated by experiments, we then emphasize the case M?1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1?ωt_d?M^-1 within which the loss varies relatively weakly with ω; and a loss maximum at ωt_d∼M^-1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.     

On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities

This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged-hinged Euler-Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary (Ω≈ω_n) and subharmonic (Ω≈2ω_n) resonances, in which Ω and ω_n are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.     

Eulerian–Lagrangian bridge for the energy and dissipation spectra in isotropic turbulence

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E _E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E _L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E _L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E _L (ω) for a given Eulerian spectrum, E _E (k), as well as the Lagrangian dissipation, ${\varepsilon_{\rm L}(\omega)}$ , for a given Eulerian counterpart, ${\varepsilon_{\rm E} (k)=2\nu k^2 E_{\rm E}(k)}$ . These equations were derived from the Navier–Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E _L (ω) and E _E (k) and between ${\varepsilon_{\rm L} (\omega)}$ and ${\varepsilon_{\rm E} (k)}$ are in very good quantitative agreement with our DNS results and explain how ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies and decays exponentially at large frequencies.     

低频弹性波超材料的若干进展

超材料是一类新兴的具有超常物理性质的人造周期/拟周期材料, 能够改变电磁波、声波以及弹性波等在介质中的传播特性. 因在航天、国防以及民用科学等方面的巨大应用潜力, 超材料自被提出后便受到极大的关注并引发研究热潮. 弹性波超材料是超材料的一种, 能够基于弹性波与超材料结构的相互耦合作用实现对弹性波的操控. 带隙是评估弹性波超材料实现弹性波操控的重要工具, 其性质与超材料的材料参数、晶格常数以及局域振子的固有频率相关. 受制于超材料的承载能力、外观尺寸以及局域振子结构等因素, 利用传统超材料开启低频(约100 Hz)弹性波带隙依然存在较大困难. 文章首先简要介绍超材料开启弹性波带隙的基本原理, 然后从低频弹性波超材料基本结构与低频带隙实现方法、低频带隙优化与调控策略、低频带隙潜在应用等三个方面详细总结低频弹性波超材料的研究工作. 其中, 低频带隙超材料的基本结构主要包括布拉格散射型超材料、传统局域共振型超材料以及准零刚度局域共振超材料. 文章通过总结低频弹性波超材料的研究进展, 分析了目前研究中的不足并对未来低频弹性波的研究方向进行了展望.      

Air concentration and bubble characteristics downstream of a chute aerator

Chute aerators separate the flow of water from the bottom of a chute, and air bubbles generated in the cavity zone must go through the impact zone as they travel downstream. In this study, the air concentration and air bubble characteristics along the chute were investigated systematically by a series of model tests that eliminated the effect of the upper aeration region. It was found that the large amount of air entrained in the cavity zone was only partially entrained into the final flow. Based on the lower air discharge properties, the chute downstream of the aerator was partitioned into four reasonable zones: the cavity zone (0 < x < L), the impact zone (L ≤ x ≤ L_m), the equilibrium zone (L_m ≤ x ≤ L_D), and the far zone (x > L_D). The details of the bubble chord length and bubble frequency distributions in each zone were measured. In the cavity zone, the bubble frequency distribution was related to the air concentration by a parabolic law. In the impact zone, the air concentration decreased sharply while the bubble frequency decreased to a lesser extent. Due to the turbulent fluctuation effect, the probability of smaller bubbles increased while the probability of larger bubbles decreased as they progressed down the chute. In the equilibrium zone, the bubble frequency decreased slightly. At the cross section, the range of probability of bubble chord lengths tended to increase from the bottom to the upper surface. The distributions of the mean chord lengths followed approximately a power low distribution. A formula was provided to predict the maximum air bubble frequency in the impact and equilibrium zones.     

Nonlinear aspects of supersonic flow past a cavity

Study of supersonic flow over wall-mounted cavities for two different length/depth (L/D) ratios is carried out experimentally. Unsteady pressure measurements were made on the front and aft walls of the cavity. Data analysis was performed on the experimental results so obtained. Spectra of the unsteady pressure data exhibit multiple tones. Higher-order spectral technique is implemented on the unsteady pressure data to ascertain whether these multiple tones are due to possible nonlinear interactions between the primary cavity modes (Rossiter modes) or not. Significant nonlinear interactions in the form of both sum and difference frequencies between the cavity modes are observed in both the cavities. The spectra of the cavity with L/D ratio 2 show distinct peaks due to nonlinear interactions while the cavity with L/D ratio 3 does not exhibit observable peaks in the spectra. The spectra of both the cavities show presence of low-frequency peaks of significant amplitudes. These low-frequency modes interact with the primary cavity modes to produce significant bicoherence values. The reasons for their existence could not be predicted. It is identified that the dominant mode in the spectra of the cavities is critical for most of the interactions observed.     

Optimization of Conical Wings in Hypersonic Flow

A method of calculation is presented to determine conical wing shapes that minimize the coefficient of (wave) drag, C _D, for a fixed coefficient of lift, C _L, in steady, hypersonic flow. An optimization problem is considered for the compressive flow underneath wings at a small angle of attack δ and at a high free-stream Mach number M _∞ so that hypersonic small-disturbance (HSD) theory applies. A figure of merit, F=C _D/C _L ^3/2, is computed for each wing using a finite volume discretization of the HSD equations. A set of design variables that determine the shape of the wing is defined and adjusted iteratively to find a shape that minimizes F for a given value of the hypersonic similarity parameter, H= (M _∞δ)⁻², and planform area. Wings with both attached and detached bow shocks are considered. Optimal wings are found for flat delta wings and for a family of caret wings. In the flat-wing case, the optima have detached bow shocks while in the caret-wing case, the optimum has an attached bow shock. An improved drag-to-lift performance is found using the optimization procedure for curved wing shapes. Several optimal designs are found, all with attached bow shocks. Numerical experiments are performed and suggest that these optima are unique. Received 1 May 1998 and accepted 14 October 1998     

Flow-induced vibrations of two circular cylinders in tandem with shear flow at low Reynolds number

The flow-induced vibrations of two elastically mounted circular cylinders subjected to the planar shear flow in tandem arrangement are studied numerically at Re=160. A four-step semi-implicit Characteristic-based split (4-SICBS) finite element method is developed under the framework of the fractional step method to cope with the vortex-induced vibration (VIV) problem. For the computational code verification, two benchmark problems are examined in the laminar region: flow-induced vibration of an elastically mounted cylinder having two degrees of freedom and past two stationary ones in tandem arrangement. Regarding the two-cylinder VIVs in shear flow, the computation is conducted with the cylinder reduced mass M_r=2.5π and the structural damping ratio ξ=0.0. The effects of some key parameters, such as shear rate (k=0.0, 0.05, 0.1), reduced velocity (U_r=3.0–18.0) and spacing ratio (L_x/D=2.5, 3.5, 4.5, 8.0), are demonstrated. It is observed that the shear rate and reduced velocity play an important role in the VIVs of both cylinders at various center-to-center distances. Additionally, in comparison with the single cylinder case, a further study indicated that the gap flow has a significant impact on such a dynamic system, leading it to be more complex. The results show that, the performances of ‘dual-resonant’ are discovered in the shear flow. A valley is formed in transverse oscillation amplitude of DC for each spacing ratio when U_r is about 6.0. For the X–Y trajectories of the circular cylinders, figure-eight, figure-O and oval shape are obtained. Finally, the interactions between cylinders are revealed, together with the wake-induced vibration (WIV) mechanism underlying the oscillation characteristics of both cylinders exposed to shear flow. Besides, the “T+P” wake pattern is discovered herein.     

Lift force acting on single bubbles in linear shear flows

Lift coefficients, C_L, of single bubbles in linear shear flows are measured to investigate effects of the bubble shape, the liquid velocity gradient and the fluid property on C_L. The range of the Morton number, M, tested is from logM = − 6.6 to − 3.2. The shapes of bubbles are spherical and ellipsoidal. A correlation of bubble aspect ratio for single bubbles in infinite stagnant liquids proposed in our previous study can give good evaluations for bubbles in the linear shear flows. The C_L of spherical bubbles at low bubble Reynolds numbers, Re, depend on the dimensionless shear rate Sr and Re and decrease with increasing Re. These characteristics agree with the Legendre-Magnaudet correlation. The use of a single dimensionless group such as Re, the Eötvös number, the Weber number and the Capillary number cannot correlate C_L of non-spherical bubbles. The trend of the critical Re for the reversal of the sign of C_L is the same as that for the onset of oscillation of bubble motion, which supports the mechanism proposed by Adoua et al., at least within the range of −6.6 ≤ logM ≤ −3.2. An experimental database of C_L is provided for validation of available C_L models and CFD.     

Waves Ahead of an Underwater Wing. An Experiment

The results of an experimental study of nonlinear waves, both smooth and breaking, formed during the motion of a symmetric wing in a channel with a plane horizontal bottom are presented. It is found that if the wing velocity U ₀ is close to the limiting speed c ₂ of propagation of free solitary waves, then the following qualitative changes in the flow pattern take place. For U ₀= c ₂ and a strictly specified combination of the other relevant parameters, a forced solitary wave of limiting amplitude is formed above the wing. Behind the wing level depression gives way to elevation. If the blocking ratio is not too large, the waves do not propagate upstream for U ₀ > c ₂. If the blocking ratio is large, the disturbances can propagate far upstream even for U ₀ > c ₂; however, in this case the leading front of the disturbances breaks.     

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

This paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second-order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω ₁, ω ₂, and ω ₃ such that ω ₃?2ω ₂?4ω ₁, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω ₃?ω ₂?2ω ₁. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics and favors the presence of unstable periodic orbits.     

Co-current flow effects on a rising Taylor bubble

The effects of co-current flows on a rising Taylor bubble are systematically investigated by a front tracking method coupled with a finite difference scheme based on a projection approach. Both the upward (the co-current flows the same direction as the buoyancy force) and the downward (the co-current moves in the opposite direction of the buoyancy force) co-currents are examined. It is found that the upward co-current tends to elongate the bubble, while the downward co-current makes the bubble fatter and shorter. For large N_f (the inverse viscosity number), the upward co-current also elongates the skirted tail and makes the tail oscillate, while the downward co-current shortens the tail and even changes a dimpled bottom to a round shape. The upward co-current promotes the separation at the tail, while the downward co-current suppresses the separation. The terminal velocity of the Taylor bubble rising in a moving flow is a linear combination of the mean velocity (U_C) of the co-current and the terminal velocity (U₀) of the bubble rising in the stagnant liquid, and the constant is around 2 which agrees with the literature. The wake length is linearly proportional to the velocity ratio (U_C/U₀). The co-currents affect the distribution of the wall shear stresses near the bubble, but not the maximum.     

Global modes and superdirective acoustic radiation in low-speed axisymmetric jets

Global linear stability theory is used to study the resonances in a slowly diverging axisymmetric jet. The absolute frequency ω₀ is calculated as a function of slow axial position X , and analytic continuation into the complex X -plane allows a saddle point in ω₀ to be identified. A key element in the analysis is the approximation of ω₀(X) by rational functions which identifies a well-defined saddle point and leading-order global frequency. A preferred-mode Strouhal number, S_D=0.44 , is calculated which compares well with existing experimental values. The global frequency has negative imaginary part and the jet is interpreted as being marginally globally stable, so that forcing in the vicinity of the resonance frequency produces a large response above background, rather like that of a slightly damped linear oscillator. The axial shape of the global-mode amplitude is Gaussian and yields a superdirective acoustic field.     

Influence of heat transfer on the aerodynamic performance of a plunging and pitching NACA0012 airfoil at low Reynolds numbers

The unsteady low Reynolds number aerodynamics phenomena around flapping wings are addressed in several investigations. Elsewhere, airfoils at higher Mach numbers and Reynolds numbers have been treated quite comprehensively in the literature. It is duly noted that the influence of heat transfer phenomena on the aerodynamic performance of flapping wings configurations is not well studied. The objective of the present study is to investigate the effect of heat transfer upon the aerodynamic performance of a pitching and plunging NACA0012 airfoil in the low Reynolds number flow regime with particular emphasis upon the airfoil's lift and drag coefficients. The compressible Navier–Stokes equations are solved using a finite volume method. To consider the variation of fluid properties with temperature, the values of dynamic viscosity and thermal diffusivity are evaluated with Sutherland's formula and the Eucken model, respectively. Instantaneous and mean lift and drag coefficients are calculated for several temperature differences between the airfoil surface and freestream within the range 0–100 K. Simulations are performed for a prescribed airfoil motion schedule and flow parameters. It is learnt that the aerodynamic performance in terms of the lift C_L and drag C_D behavior is strongly dependent upon the heat transfer rate from the airfoil to the flow field. In the plunging case, the mean value of C_D tends to increase, whereas the amplitude of C_L tends to decrease with increasing temperature difference. In the pitching case, on the other hand, the mean value and the amplitude of both C_D and C_L decrease. A spectral analysis of C_D and C_L in the pitching case shows that the amplitudes of both C_D and C_L decrease with increasing surface temperature, whereas the harmonic frequencies are not affected.     

Flow evolution of a turbulent submerged two-dimensional rectangular free jet of air. Average Particle Image Velocimetry (PIV) visualizations and measurements

The paper presents average flow visualizations and measurements, obtained with the Particle Image Velocimetry (PIV) technique, of a submerged rectangular free jet of air in the range of Reynolds numbers from Re = 35,300 to Re = 2200, where the Reynolds number is defined according to the hydraulic diameter of a rectangular slot of height H. According to the literature, just after the exit of the jet there is a zone of flow, called zone of flow establishment, containing the region of mixing fluid, at the border with the stagnant fluid, and the potential core, where velocity on the centerline maintains a value almost equal to the exit one. After this zone is present the zone of established flow or fully developed region. The goal of the paper is to show, with average PIV visualizations and measurements, that, before the zone of flow establishment is present a region of flow, never mentioned by the literature and called undisturbed region of flow, with a length, L_U, which decreases with the increase of the Reynolds number. The main characteristics of the undisturbed region is the fact that the velocity profile maintains almost equal to the exit one, and can also be identified by a constant height of the average PIV visualizations, with length, L_CH, or by a constant turbulence on the centerline, with length L_CT. The average PIV velocity and turbulence measurements are compared to those performed with the Hot Film Anemometry (HFA) technique. The average PIV visualizations show that the region of constant height has a length L_CH which increases from L_CH = H at Re = 35,300 to L_CH = 4–5H at Re = 2200. The PIV measurements on the centerline of the jet show that turbulence remains constant at the level of the exit for a length, L_CT, which increases from L_CT = H at Re = 35,300 to L_CT = 4–5H at Re = 2200. The PIV measurements show that velocity remains constant at the exit level for a length, L_U, which increases from L_U = H at Re = 35,300 to L_U = 6H at Re = 2200 and is called undisturbed region of flow. In turbulent flow the length L_U is almost equal to the lengths of the regions of constant height, L_CH, and constant turbulence, L_CT. In laminar flow, Re = 2200, the length of the undisturbed region of flow, L_U, is greater than the lengths of the regions of constant height and turbulence, L_CT = L_CH = 4–5H. The average PIV and HFA velocity measurements confirm that the length of potential core, L_P, increases from L_P = 4–5H at Re = 35,300 to L_P = 7–8H at Re = 2200, and are compared to the previous experimental and theoretical results of the literature in the zone of mixing fluid and in the fully developed region with a good agreement.     

FLOW-INDUCED VIBRATION OF A CIRCULAR CYLINDER AT LIMITING STRUCTURAL PARAMETERS

Transverse oscillation of a dynamically supported circular cylinder in a flow at Re=100 has been numerically simulated using a high-resolution viscous-vortex method, for a range of dynamical parameters. At the limiting case with zero values of mass, damping and elastic force, the cylinder oscillates sinusoidally at amplitudeA /D=0·47 and frequency fD/U_∞=0·156. For zero damping, the effects of mass and elasticity are combined into a new, “effective” dynamic parameter, which is different from the classic “reduced velocity”. Over a range of this parameter, the response exhibits oscillations at amplitudes up to 0·6 and frequencies between 0·15 and 0·2. From this response function, the classic response in terms of reduced velocity can be obtained for fixed values of the cylinder/fluid ratio m^*. It displays “lock-in” at very high values of m^*.     

NEAR-WAKE FLOW STRUCTURE OF A CYLINDER WITH A HELICAL SURFACE PERTURBATION

The near-wake of a circular cylinder having a helical wire pattern about its surface is characterized using a technique of high-image-density velocimetry. Patterns of vorticity in three orthogonal planes show substantial influence of a wire having a diameter an order of magnitude smaller than the cylinder diameter. The distinctive patterns of vorticity in these three planes are associated with lack of formation of large-scale Kármán-like clusters of vorticity (ω_z) in the near-wake region of the cylinder. The instantaneous structure of the separating spanwise vorticity (ω_z) layers on either side of the cylinder involve small-scale concentrations of vorticity analogous to the well-known Kelvin–Helmholtz vortices from a smooth cylinder. Moreover, a dual vorticity layer, i.e., two adjacent layers of like vorticity (ω_z), can form from one side of the cylinder. Along the span of the cylinder, distributions of instantaneous velocity and transverse vorticity (ω_y) show a spatially periodic sequence of wake-like patterns, each of which has features in common with the very near-wake of a two-dimensional bluff body, including a large velocity defect bounded by vorticity layers with embedded small-scale vorticity concentrations. In the cross-flow plane of the wake, patterns of streamwise vorticity (ω_x) show small-scale, counter-rotating pairs of vorticity concentrations (ω_x) emanating from the inclined helical perturbation, rather than isolated concentrations of vorticity of like sign, which would indicate single streamwise vortices. All of the aforementioned patterns of small-scale vorticity concentrations are scaled according to the local wake width/local pitch of the helical wire pattern in the respective plane of observation.