IB–LBM Based High Energy Harvesting Efficiency Prediction of Two Parallel Oscillating Foils

The flapping foil energy harvester is a promising alternative to traditional rotary turbines for power generation in currents. In order to investigate the energy harvesting performance and interaction mechanism of two parallel–arranged flapping foils, a systematic numerical study is conducted using the immersed boundary–lattice Boltzmann method (IB–LBM). Two parameters, the relative gap and motion phase difference, are varied to determine the potential optimal situation. Oscillating frequency is varied to reveal the coupling effects. The results indicate that the optimal two-foil configuration outperforms the optimal single foil case by 10.3% in terms of energy harvesting efficiency, while the average power coefficient is 2.2% lower than its single foil counterpart. These observations are attributable to two interaction processes. Firstly, a pronounced high pressure regime due to the compressing effects always adds energy extraction to motions that tend to relieve it and deprives that of the opposite motions. Secondly, the favorable formation and timing of the leading edge vortices (LEVs) are significantly affected when two foils are in close proximity. The detrimental interaction effects are reinforced with more compact foil arrangements. However, more compact configurations also involve a lower motion range against which the efficiency is measured. Therefore, it is reasonable for two parallel foils to obtain higher energy extracting efficiency with smaller extracted power coefficients. Introduction Expectation and exploitation of renewable and environmental friendly energy are widely recognized human tasks for this new century. Wind, water current and tidal are all well–known sources of renewable and clean energy. Currently, rotary turbines are the dominant technology used to extract flow power. The inherent requirement for relative high stream velocity and large scale for rotary turbines to maintain considerable efficiency, together with the associated noise, impose a undesirable limitation for further wind and water energy exploitation. Flapping foils, a promising alternative concept to extract energy from flowing fluid, have been extensively studied during the last three decades. This concept was inspired by the locomotion of fishes, birds and insects. The first flaping foil energy harvester model was attributed to McKinney and DeLaurier [1]. After that, substantial research progress has been made on this topic. The formation and timing of LEVs are recognized as a critical factor to maximize the power extraction performance [2]. Techniques including non-sinusoidal oscillation motions [3], corrugated foils [6], structural flexibility [7] and multi–wing configurations [3, 8, 9] have been verified to be effective on improving the output power coefficient and efficiency. Among the proposed strategies, multiple wing configurations include tandem and parallel arrangements. Lindsey [10] and Jones et al. [11] studied two tandem wings both numerically and experimentally. However, their simulated power coefficients apparently overestimated the experimental data. Platzer et al. [12] and Ashraf et al. [3] performed systematic numerical investigations on two tandem arranged wings by using a Navier–Stokes equations based solver. Their results showed that the output power coefficient and efficiency are sensitive to the relative spacing and motion phase difference between the considered foils. Kinsey et al. [13, 9] implemented a series study about tandem arranged wings. According to their analysis, the achievable power efficiency is more than 40% for two tandem arranged wings. Parallel arranged multiple wing configurations have also received some attention. Lefrancois [8] numerically investigated two parallel foils with dimensionless frequency fixed at 0.12, pitching amplitude being 67◦, distance between the two foils varying from 1.08c to 3.80c (where c is the chord length), and phase difference between the two foils ranging from 0◦ to 180◦ with an increment of 45◦. The best power efficiency was 30%, which is significantly lower than that of their tandem counterpart, 41%. Abiru and Yoshitake [14] successfully built a twin– flapping–foils model, where the two foils are parallel arranged with a phase difference of 180◦, which stably lit a LED both in an experimental circular water current and a field irrigation stream. To complement and analyze the work in Ref [14], Isogai and Abiru [15, 16] completed numerical simulations for two and three parallel foils using potential flow and Navier Stokes solutions. They reported optimal power efficiency of 33% for anti-phase motions and 21% for in-phase motions. More recently, Wu et al. [17, 18] used parallel arranged small rotating foils to help the main oscillating foil improve power efficiency due to the induced vortex interaction. Inferred from the above listed literature, multiple parallel arranged foils have the potential to improve the power extraction performance, though, compared to tandem foils, the expectation is lower and less attention have been attracted. Considering the optimal dimensionless frequency is confirmed around 0.15, and a single oscillating foil could achieve an efficiency as high as 34% [2], a more detailed parametric study with various frequency for the parallel configurations is valuable, which is the focus of this article. The lattice Boltzmann method combined with the immersed boundary (IB–LBM) technique, which eliminates complex grid deformation for moving boundaries, is used to simulate the laminar flow around two parallel arranged oscillating NACA0012 foils with different parameter setup. The parameters considered includes the dimensionless frequency, the relative spacing distance and oscillating phase difference between the two foils. The article is organized as following: the kinematic conditions are described in section 2; two classic cases with one oscillating foil are presented to validate the IB–LBM solver in section 3, which is followed by the parametric analysis for parallel configurations, which composes the major body of this paper, in section 4; finally some critical conclusions are drawn in section 5. Flapping foil kinematics Here we only consider micro foils, where the Reynolds number is in the order of 1000 and thus turbulence is not considered. Within a given stream setup, the motions for an oscillating foil are given as following: θ(t) = θ0 sin(2π f t) ; h(t) = h0 sin(2π f t +φ) where θ is the pitching angle, θ0 is the pitching amplitude, h is heaving displacement, h0 is the heaving amplitude, f is the dimensionless oscillating frequency, and φ is the phase difference between the two modes of motion. When two parallel foils are considered, two more relative parameters are needed. The first is the half distance between the two balanced positions (where h(t)= 0), which is indicated as hi (HDBP for short in figures) in the following; the other one is the phase difference φ (DPA for short in figures) between motions of the two foils. Therefore, the motion equations for the two foils are written as: θ(t) = θ0 sin(2π f t +φ) ; h(t) = h0 sin(2π f t +φ+φ)+hi θ(t) = θ0 sin(2π f t) ; h(t) = h0 sin(2π f t +φ)−hi For all the following energy extraction cases some parameters are fixed: θ0 = 76.3◦, h0 = 1.0c,φ = 90◦, the distance from the leading edge to the pitching center lc = c/3 where c is the chord length and it is the default length reference without special instructions. These parameters are examined to be an optimal setup for a single oscillating foil in terms of power extraction performance [2]. To assess the power extraction performance, average output power coefficient and efficiency are defined conventionally as following: