Non-sinusoidal Pitching Research Articles

Power Extraction Performance by a Hybrid Non-Sinusoidal Pitching Motion of an Oscillating Energy Harvester

This study proposes a hybrid pitching motion for oscillating flat plates aimed at augmenting the energy extraction efficiency of an energy harvester. The proposed hybrid pitching motion, within the first half cycle, integrates a non-sinusoidal movement starting at t/T = 0 and progressing to t/T = 0.25, with a sinusoidal movement initiating after t/T > 0.25 and continuing to t/T = 0.5. The second half of the cycle is symmetric to the first half but in the opposite direction. The calculated results show that the proposed hybrid pitching motion outperforms both the sinusoidal and the non-sinusoidal motions. The hybrid pitching motion merges the merits of both the sinusoidal and non-sinusoidal motions to optimize pitch angle variation. This integration is pivotal for enhancing the overall power output performance of an oscillating energy harvester characterized by momentum change that enhances the orientation of the heaving movement, smoother motion transitions, and consistent energy harvesting. The power generation is obtained at wing pitch angles of 55°, 65°, 70°, 75°, and 80° during a hybrid pitching motion. The proposed hybrid pitching motion, set at a pitch angle of 70°, achieves a maximum power output that exceeds the oscillating flat plate using a sinusoidal pitching motion by 16.0% at the same angle.

Dynamic responses of pitching hydrofoil in laminar–turbulent transition regime

Recently, the problem of a transitional boundary layer around a pitching foil has been attracting increasing attention. To determine the underlying physical mechanisms, in this study, a pitching hydrofoil is numerically investigated using the shear stress transport (SST) k–ω turbulence model coupled with the γ–Re˜θt transition model. First, the prediction of the static performance is compared with the experimental measurements and XFOIL data to validate and verify the numerical accuracy of the proposed method. Subsequently, the flow morphologies induced by sinusoidal and non-sinusoidal pitching laws are analyzed and compared in different reduced frequencies. The results indicate that reducing the empirical coefficient, A1, in the SST k–ω turbulence model can generate the flow separation in advance, thus improving the prediction of the hydraulic performance at a high angle of attack. The dynamic behavior of the laminar separation bubble is strongly associated with the pitching method and velocity. The collapse of the laminar separation bubble can destabilize the local boundary layer, generating multi-laminar separation bubbles. When new separation bubbles form and shed, a turbulent boundary layer moves downstream and is adsorbed on the surface simultaneously. The turbulence flow passes through the trailing edge, destroying the local vortex, contributing to a transiently elevated hydrodynamic lift. During a rapid pitching down, a counterclockwise trailing edge vortex becomes increasingly unstable, repeating the processes of inception, development, and shedding. The Theodorsen model fails to produce a reasonable hysteresis loop owing to the assumption of a fully attached flow. The results of a hybrid of the Theodorsen and Øye models show noticeable improvement in the dynamic lift prediction. The empirical coefficient, ks, of the Snel model is optimized, improving the predictions of low-frequency pitching significantly. All dynamic stall models present the same trend: low-frequency motions show better agreement than high-frequency ones. In the former, modeling of a non-sinusoidal pitching is closer to the transient numerical results than that of a sinusoidal motion.

Efficient thrust enhancement by modified pitching motion

Thrust and/or efficiency of a pitching foil (mimicking a tail of swimming fish) can be enhanced by tweaking the pitching waveform. The literature, however, show that non-sinusoidal pitching waveforms can enhance either thrust or efficiency but not both simultaneously. With the knowledge and inspiration from nature, we devised and implemented a novel asymmetrical sinusoidal pitching motion that is a combination of two sinusoidal motions having periods T1 and T2 for the forward and retract strokes, respectively. The motion is represented by period ratio $\mathrm{\mathbb{T}} = {T_1}/T$ , where T = (T1 + T2)/2, with $\mathrm{\mathbb{T}} > 1.00$ giving the forward strokes (from equilibrium to extreme position) slower than the retract strokes (from extreme to equilibrium position) and vice versa. The novel pitching motion enhances both thrust and efficiency for $\mathrm{\mathbb{T}} > 1.00$ . The enhancement results from the resonance between the shear-layer roll up and the increased speed of the foil. Four swimming regimes, namely normal swimming, undesirable, floating and ideal are discussed, based on instantaneous thrust and power. The results from the novel pitching motion display similarities with those from fish locomotion (e.g. fast start, steady swimming and braking). The $\mathrm{\mathbb{T}} > 1.00$ motion in the faster stroke has the same characteristics and results as the fast start of prey to escape from a predator while $\mathrm{\mathbb{T}} < 1.00$ imitates braking locomotion. While $\mathrm{\mathbb{T}} < 1.00$ enhances the wake deflection at high amplitude-based Strouhal numbers (StA = fA/U∞, where f and A are the frequency and peak-to-peak amplitude of the pitching, respectively, and U∞ is the freestream velocity), $\mathrm{\mathbb{T}} > 1.00$ improves the wake symmetry, suppressing the wake deflection. The wake characteristics including wake width, jet velocity and vortex structures are presented and connected with $S{t_d}( = fd/{U_\infty })$ , ${A^{\ast}}( = A/d)$ and $\mathrm{\mathbb{T}}$ , where d is the maximum thickness of the foil.

Effects of Non-Sinusoidal Motion and Effective Angle of Attack on Energy Extraction Performance of a Fully-Activated Flapping Foil

Flapping foil energy harvesting systems are considered as highly competitive devices for conventional turbines. Several research projects have already been carried out to improve performances of such new devices. This paper is devoted to study effects of non-sinusoidal heaving trajectory, non-sinusoidal pitching trajectory, and the effective angle of attack on the energy extraction performances of a flapping foil operating at low Reynolds number (Re=1100). An elliptic function with an adjustable parameter S (flattening parameter) is used to simulate various sinusoidal and non-sinusoidal flapping trajectories. The flow around the flapping foil is simulated by solving Navier–Stokes equations using the commercial software Star CCM+ based on the finite-volume method. Overset mesh technique is used to model the flapping motion. The study is applied to the NACA0015 foil with the following kinetic parameters: a dimensionless heaving amplitude h0 = 1c, a shift angle between heaving and pitching motions = 90°, a reduced frequency f* = 0.14, and an effective angle of attack αmax varying between 15° and 50°, corresponding to a pitching amplitude in the range 0 = 55.51° to 99.51°. The results show that, the non-sinusoidal trajectory affects considerably the energy extraction performances. For the reference case (sinusoidal heaving and pitching motions, Sh = S =1), best performances are obtained for the effective angle of attack, αmax = 40°. At small effective angle of attack αmax 40°), non-sinusoidal pitching motion has a negative effect. Performances improvement is quite limited with the combined motions non-sinusoidal heaving/sinusoidal pitching.

Effects of non-sinusoidal pitching motion on energy input performance of micro-linear turbine cascade

In this paper, a numerical simulation of a micro-linear turbine cascade is presented. The upstream stay vane and downstream guide vane were surrounded by incompressible flow, and the guide vane was modeled with forced pitching motions. The implicit immersed boundary method was used as the moving boundary flow solver. To seek a better understanding of the influence of the motion parameters on hydrodynamic characteristics and input power of the oscillating guide vane, the forced motion of a non-sinusoidal oscillating guide vane was first investigated for different reduced frequencies and profile parameters. Second, the instantaneous hydrodynamic behavior was simulated for different motion combinations and Reynolds numbers. The results showed that the reduced frequencies and profile parameters of the pitch movement had a large influence on the input power, and the peak of the input power at f∗ = 0.22 was nearly 9.41 times that at f∗ = 0.074, but the input power was not sensitive to the motion combination and Reynolds number. However, the Reynolds number had a large influence on the drag, and the maximum drag coefficient for Re = 6.65 × 105 was 4.566; however, it was only 0.5912 for Re = 6.65 × 104.

Effects of non-sinusoidal pitching motion on the propulsion performance of an oscillating foil.

Numerical simulations have been used in this paper to study the propulsion device of a wave glider based on an oscillating hydrofoil, in which the profile of the pitching and heaving motion have been prescribed for the sake of simplicity. A grid model for a two-dimensional NACA0012 hydrofoil was built by using the dynamic and moving mesh technology of the Computational Fluid Dynamics (CFD) software FLUENT and the corresponding mathematical model has also been established. First, for the sinusoidal pitching, the effects of the pitching amplitude and the reduced frequency were investigated. As the reduced frequency increased, both the mean output power coefficient and the optimal pitching amplitude increased. Then non-sinusoidal pitching was studied, with a gradual change from a sinusoid to a square wave as the value of β was increased from 1. It was found that when the pitching amplitude was small, the trapezoidal pitching profile could indeed improve the mean output power coefficient of the flapping foil. However, when the pitching amplitude was larger than the optimal value, the non-sinusoidal pitching motion negatively contributed to the propulsion performance. Finally, the overall results suggested that a trapezoidal-like pitching profile was effective for the oscillating foil of a wave glider when the pitching amplitude was less than the optimal value.

Foil locomotion through non-sinusoidal pitching motion

We investigate the propulsive characteristics of a neutrally buoyant foil that self propels as it pitches about its quarter chord with a prescribed time periodic waveform. For a fixed pitching frequency and amplitude, we find the foil motion to be strongly dependent on the prescribed waveform. The maximum mean self-propelling speed corresponds to a square shaped waveform for which the foil is held at the two extremes for a significant part of the pitching cycle. In contrast, the cost of transport is minimal, and therefore the energetic efficiency of pitching induced self propulsion a maximum, for a triangle shaped profile in which case the foil’s angular speed is largely held constant over the pitching period. Pitching waveform induced variations in the mean self-propelling speed (u¯p) are surprisingly well predicted with a simple power-law relationship Strms∼Re−0.37, where Reynolds number Re is based on u¯p and the Strouhal number Strms is the ratio of the root mean square foil trailing edge speed and u¯p. This power-law relationship is shown to arise naturally from a balance between the cycle-averaged inertial thrust generated from prescribed pitching and the enhanced drag owing to the boundary layer thinning. The imposed pitching motion induces a Reynolds number independent out of phase heave motion features of which are remarkably well predicted with a simple inviscid model for the foil’s transverse motion. For all the prescribed pitching waveforms, a maximum in the energetic efficiency is always attained over the range 0.3≤St≤0.6. The close correspondence between this Strouhal number range for maximum energetic efficiency and the optimal Strouhal number range for swimming and flying animals is indicative of the similarity between self-propelling flapping foil configurations and undulatory biolocomotion. Our results point to the significance of rigid body pitching profile as an important input parameter that can be varied independently to achieve a desired outcome such as maximization of the mean self-propelling speed or energetic efficiency.

Numerical Study on the Effect of Non-Sinusoidal Motion on the Energy Extraction Performance of Parallel Foils

The effect of non-sinusoidal motion which influences the energy extraction performance of foil is considered in this paper. Two oscillation motions, the combined non-sinusoidal plunging and sinusoidal pitching motion, as well as the combined non-sinusoidal pitching and sinusoidal plunging motion, are selected to investigate the oscillation process of two-dimensional parallel foils numerically. The optimal oscillation motion and average power coefficient at different combined motions are gained. The effects of the plunging motion and pitching motion at different oscillation motions are analyzed, and the evolution law of the foil lift force and vortex field are obtained. It is indicated that the non-sinusoidal motion has a significant influence on energy extraction. When the motion is combined (non-sinusoidal plunging and sinusoidal pitching motion), the best extraction performance is gained at Kh = −0.5. The maximal CPm is 0.375 and the maximal η is 0.188. When the motion is combined (non-sinusoidal pitching and sinusoidal plunging motion), the maximal CPm is 0.623 and the maximal η is 0.312 which appear at Kθ = 2. For the same frequency, the more the plunging motion is similar to the sinusoidal motion, the more energy is extracted by foils. While the more the pitching motion approximates to the square wave, the worse the achieved extraction performance is.

Numerical investigations on the force generation and wake structures of a nonsinusoidal pitching foil

We numerically investigate the force generation and wake structures of a nonsinusoidal periodic pitching foil under a thickness-based Reynolds number of ReD=255. We consider various nonsinusoidal periodic trajectories of pitching motion, such as the sawtooth and square trajectories, using an adjustable parameter, K. Similar to previous studies on sinusoidal pitching foils, the present results for the nonsinusoidal pitching foil indicate that the transition from a Bénard–von Kármán wake to a reverse Bé nard–von Kármán wake precedes the actual drag–thrust transition. In addition, the symmetry breaking of a reverse Bé nard–von Kármán wake is observed. As K increases, the drag–thrust transition of the nonsinusoidal pitching foil advances gradually and it becomes easy to destabilize the wake structures produced by the oscillating foil. Since K physically describes how rapidly the foil reverses its pitching direction, we argue that the initial velocity of vortex shedding from the foil increases with K and results in the aforementioned phenomena. The present work provides helpful findings for better understanding the kinematic behaviour of actual animals and for designing bio-inspired robotics.

Numerical study on hydrodynamics for a non-sinusoidal forced oscillating hydrofoil based on an immersed boundary method

Fluid-structure investigations in hydraulic machines using coupled simulations based on dynamical re-mesh are particularly time-consuming. This paper presents an implementation of an immersed boundary (IB) method for dynamical boundary flowing problem in a flow solver based on the fractional time step method and the finite different method, which is validated by a basic numerical example. The hydrofoil, which is surrounded by incompressible flow, is modeled with forced pitching motions. To seek better understanding of the coupled hydrodynamic performance of an oscillating (fully active) hydrofoil, a forced motion of non-sinusoidal oscillating is investigated for different pitching frequencies f and maximum angle amplitude θ0 as well as profile parameter β. Effects of non-sinusoidal pitching motion on the hydrodynamic load and power input performance are investigated under different pitching motional parameters. The results show that the time-averaged input power gradually increases with both the pitching frequency f and profile parameter β while it decreases with the maximum pitching amplitudeθ0. The instantaneous peak value of input power also increases with both the pitching frequency f and profile parameter β rising up. In addition, the flow fields for different motional parameters are discussed. The different vortex evolutions for various β and pitching frequency remind us that the flow around a flapping foil is complicated and strongly related to the unsteady hydrodynamics. The present research results are helpful to control strategy design for the hydrofoil movement in the operating process of turbine unit.

Effects of non-sinusoidal pitching motion on energy extraction performance of a semi-active flapping foil

Numerical simulations are used to study the energy harvester based on a semi-active flapping foil, in which the profile of the pitching motion is prescribed and the heaving motion is activated by the vertical hydrodynamic force. We consider a two-dimensional NACA0015 airfoil with the Reynolds number Re = 1000. First, for the sinusoidal pitching, an optimal combination of the parameters of pitching amplitude θ0 = 75° and reduced frequency f∗ = 0.16 is identified, with the highest energy harvesting efficiency of 32% being recorded. Then we study non-sinusoidal pitching, with a gradual change from a sinusoid to a square wave as β is increased from one. We find that its effect of efficiency enhancement is limited for the parameters approaching their optimal values, and the upper boundary of the efficiency appears not to be increased. In detail, we report that when the pitching amplitude is small, non-sinusoidal pitching motions can indeed improve the performance of the system. However, when both the pitching amplitude and the flapping frequency are close to their optimal values, non-sinusoidal pitching motions contribute negatively to the harvesting efficiency. We suggest that a non-sinusoidal profile, at least a simple trapezoidal-like one is ineffective in the semi-active system reported by the current study.

Flow Structures Around an Oscillating-Wing Power Generator

Power extraction from wind and water streams using flapping wings is known to be an alternative method to harvest renewable energy. The vortical flow structures around and in the wake of a NACA0012 airfoil oscillating with non-sinusoidal pitching and plunging motions are investigated using digital particle image velocimetry and compared with Navier–Stokes computations to give insight into the physics that determine the performance of an oscillating-wing power generator for a plunge amplitude of 1.05 chords, reduced frequency of 0.8, pitch amplitude of 73 deg, pivot points at quarter and mid-chords, phase angles of 90 and 110 deg, and stroke reversal times () of 0.1 (rapid reversal) to 0.5 (sinusoidal reversal) at a Reynolds number of . As the airfoil rotation speed during pitch reversals is increased, vortex shedding occurs earlier with higher strength. As the phase angle by which the pitching motion leads the plunging motion is increased, the shed vortex convection distance is also increased. There is good agreement between the flow structures obtained using 2_D Navier–Stokes simulations and those from particle image velocimetry measurements in terms of the formation, evolution and the timing of shedding of the leading edge and trailing edge vortices. Time-averaged vorticity results reveal that power producing oscillating-airfoils generate a time-averaged wake flow whereas an inner-jet like flow is present for all the cases with negative time averaged power output. Both particle image velocimetry measurements and Navier–Stokes simulations show that the time averaged power output is primarily determined by the timing of the formation of the leading edge vortex, its convection, and its interaction with the airfoil.

Computational Analysis of Propulsion Performance of Modified Pitching Motion Airfoils in Laminar Flow

The thrust generation performance of airfoils with modified pitching motion was investigated by computational fluid dynamics (CFD) modeling two-dimensional laminar flow at Reynolds number of 104. The effect of shift distance of the pitch axis outside the chord line(R), reduced frequency(k), pitching amplitude(θ), pitching profile, and airfoil shape (airfoil thickness and camber) on the thrust generated and efficiency were studied. The results reveal that the increase inRandkleads to an enhancement in thrust generation and a decrease in propulsive efficiency. Besides, there exists an optimal range ofθfor the maximum thrust and the increasingθinduces a rapid decrease in propulsive efficiency. Six adjustable parameters(K)were employed to realize various nonsinusoidal pitching profiles. An increase inKresults in more thrust generated at the cost of decreased propulsive efficiency. The investigation of the airfoil shape effect reveals that there exists an optimal range of airfoil thickness for the best propulsion performance and that the vortex structure is strongly influenced by the airfoil thickness, while varying the camber or camber location of airfoil sections offers no benefit in thrust generation over symmetric airfoil sections.

Nonsinusoidal motion effects on energy extraction performance of a flapping foil

To seek better energy extraction performance of a flapping hydrofoil, various nonsinusoidal motion profiles are employed instead of conventional sinusoidal flapping motions. The effects of nonsinusoidal motions are investigated for four kinds of nonsinusoidal flapping motions, i.e. varying effective angle of attack profile, nonsinusoidal pitching motion combined with sinusoidal plunging, nonsinusoidal plunging motion combined with sinusoidal pitching, and combined nonsinusoidal pitching and nonsinusoidal plunging motion. An adjustable parameters K is used to realize various nonsinusoidal profile varying from a sawtooth wave to a square wave profile. Numerical results show that by imposing a square-like effective angle of attack profile, extraction power and efficiency can be significantly increased compared with sinusoidal flapping motion. By accelerating the formation time and development of leading edge vortex (LEV), the square-like effective angle of attack profile leads to a better synchronization between the vertical force and plunging velocity. Similar effect on the flapping foil energy extraction performance is also found by imposing nonsinusoidal pitching profile. While the output power enhancement is quite limited by using the nonsinusoidal plunging profile. Moreover, the energy extraction performance can be significantly improved with an appropriate combination of nonsinusoidal pitching and nonsinusoidal plunging motion. Of all the nonsinusoidal motions studied, a toothed-like plunging profile together with square-like pitching profile should be selected for the best energy extraction performance.

Nonsinusoidal Path Optimization of a Flapping Airfoil

The path of a flapping airfoil undergoing a combined, nonsinusoidal pitching and plunging motion is optimized for maximum thrust and/or propulsive efficiency. The nonsinusoidal, periodic flapping motion is described using nonuniform rational B splines. A gradient based algorithm is then employed for the optimization of the nonuniform rational B-spline parameters. Unsteady, low speed laminar flows are computed using a Navier-Stokes solver in a parallel computing environment The numerical evaluation of the gradient vector components, which requires unsteady flow solutions, is also performed in parallel. It is shown that the thrust generation may significantly be increased in comparison to the sinusoidal flapping motion. For a maximum thrust generation, the airfoil stays at about a constant angle of attack during the upstroke and the downstroke, and may reach very high effective angle of attack values. The pitching motion mostly occurs at the minimum and maximum plunge positions.

References

Recently, the problem of a transitional boundary layer around a pitching foil has been attracting increasing attention. To determine the underlying physical mechanisms, in this study, a pitching hydrofoil is numerically investigated using the shear stress transport (SST) k–ω turbulence model coupled with the γ–Re˜θt transition model. First, the prediction of the static performance is compared with the experimental measurements and XFOIL data to validate and verify the numerical accuracy of the proposed method. Subsequently, the flow morphologies induced by sinusoidal and non-sinusoidal pitching laws are analyzed and compared in different reduced frequencies. The results indicate that reducing the empirical coefficient, A1, in the SST k–ω turbulence model can generate the flow separation in advance, thus improving the prediction of the hydraulic performance at a high angle of attack. The dynamic behavior of the laminar separation bubble is strongly associated with the pitching method and velocity. The collapse of the laminar separation bubble can destabilize the local boundary layer, generating multi-laminar separation bubbles. When new separation bubbles form and shed, a turbulent boundary layer moves downstream and is adsorbed on the surface simultaneously. The turbulence flow passes through the trailing edge, destroying the local vortex, contributing to a transiently elevated hydrodynamic lift. During a rapid pitching down, a counterclockwise trailing edge vortex becomes increasingly unstable, repeating the processes of inception, development, and shedding. The Theodorsen model fails to produce a reasonable hysteresis loop owing to the assumption of a fully attached flow. The results of a hybrid of the Theodorsen and Øye models show noticeable improvement in the dynamic lift prediction. The empirical coefficient, ks, of the Snel model is optimized, improving the predictions of low-frequency pitching significantly. All dynamic stall models present the same trend: low-frequency motions show better agreement than high-frequency ones. In the former, modeling of a non-sinusoidal pitching is closer to the transient numerical results than that of a sinusoidal motion.