Self-propulsion in Pitching Airfoils

Abstract

Self-propelled oscillating foils are simple yet useful models that can be used to explore the physics of oscillatory bio-locomotion. In this investigation, we explore the self-propulsion characteristics of a rigid NACA0015 section airfoil and a composite airfoil subjected to sinusoidal pitching actuation. The composite airfoil consists of a flexible flap appended to the trailing edge of the rigid airfoil. The flap’s length is 0.75 times the chord length (𝐶) of the rigid airfoil, resulting in an effective chord length (𝐿) of 1.75𝐶 for the composite airfoil. We employ a rotary apparatus designed specifically to attain self-propulsion over a wide range of parameters. We employ 2D planar Laser-Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV) to capture flow patterns and the wake velocity fields in the mid-plane of the airfoils. We introduce a wake energy coefficient (𝐶𝑊𝐸) to quantify the energetic efficiency during self-propulsion. 𝐶𝑊𝐸 is essentially the normalised kinetic energy flux in the wake and is a measure of the work done as well as the energy wasted in the wake. The self-propulsion apparatus consists of a long, freely rotating arm pivoted on one end and houses the airfoil on the other. The novelty of the apparatus is in the use of a spiral-springbased crank rocker mechanism employed to provide the necessary long-distance sinusoidal actuation to the airfoil. We first discuss in detail the design and construction of the apparatus, especially the kinematics of a planar four-bar crank-rocker mechanism and identify the conditions under which the rocker motion can be approximated as sinusoidal. For the kinematic constraints considered here, the mechanism can produce sinusoidal actuation up to 20°. With this apparatus, we achieve self-propulsion speeds ranging from 1cm/s to almost 45 cm/s with very high repeatability characterised by a very small variation of 2-5%. We then examine the effect of imposed kinematics – such as the pitching frequency (𝑓), amplitude and pitching point location – on self-propulsion speed (𝑈𝑠) and 𝐶𝑊𝐸. We define a trailing edge Reynolds number 𝑅𝑒𝑇𝐸 = (𝑓𝐴)𝐶/𝜈 where 𝐴 is the trailing edge excursion of the airfoil and 𝜈 is the fluid viscosity. The product 𝑓𝐴 provides a velocity scale considering all the input parameters. We analyse the results in terms of self-propulsion Reynolds number (𝑅𝑒𝑠 = 𝑈𝑠𝐶/𝜈), reduced frequency (𝑘𝑠), Strouhal number (𝑆𝑡), and two non-dimensional speeds, 𝑈𝐵𝐿 ∗ (body length per oscillation) and 𝑈𝐴𝐿 oscillation). ∗ (forward speed in terms of trailing edge excursion per iv For rigid airfoils, three distinct propulsion regimes are identified, each characterised by a specific relationship between 𝑅𝑒𝑠 and 𝑅𝑒𝑇𝐸. When pitching at low-amplitude, close to leading-edge, 𝑅𝑒𝑠 ∼ (1 − 2𝑝)𝑅𝑒𝑇𝐸 3/2 (power scaling). For higher amplitude pitching, 𝑅𝑒𝑠 ∼ 𝑅𝑒𝑇𝐸(1 −2𝑝)1/2(𝐴/𝐶)−1/2 (separable scaling). When pitched close to the midpoint, the airfoil propels beyond a threshold 𝑅𝑒𝑇𝐸,0, and 𝑅𝑒𝑠 increases linearly with 𝑅𝑒𝑇𝐸 (linear scaling). While the power and linear scaling have been reported in earlier studies for pitching airfoils and heaving flat plates, respectively, the separable scaling identified here is seminal as it captures the non-uniform contribution of amplitude and frequency to self-propulsion speed. These relations collectively provide a comprehensive database relating the speed to the imposed kinematics across a wide range of parameters. The wake of a rigid airfoil exhibits different vortical patterns classified into Deflected Vortex Pair (DVP), DVP with a single vortex (DVP-S), reverse von Karman (RvK) and coalescing RvK (RvK-C). The observed pattern depends on 𝐴/𝐶 and 𝑈𝐵𝐿 ∗ , with DVP and DVPS being prominent for 𝐴/𝐶 < 0.2 and 𝑈𝐵𝐿 ∗ <0.4 and RvK-C for 𝐴/𝐶 > 0.3 and 𝑈𝐵𝐿 ∗ > 1. The RvK and RvK-C wake exhibit a bifurcation within a chord-length downstream of the airfoil leading to a wide wake. The rigid airfoil self-propulsion efficiency, as indicated by 𝐶𝑊𝐸, increases as the is minimum when 𝑈𝐴𝐿 pitching point moves closer to the leading edge. Within the range of parameters studied, 𝐶𝑊𝐸 ∗ ≈ 4.5 indicating maximum efficiency corresponding to 𝑆𝑡 ≈ 0.22. In the composite foil, the effect of flexibility is quantified by the phase lag, ϕ, between the rigid airfoil trailing edge and the tip of the flexible flap attached. We observe that for a given material and flap length, ϕ is a function of frequency alone, whereas the maximum ϕ attained for a given amplitude of pitching decreases with increasing amplitude. ϕ is observed to increase with frequency then plateau at a maximum value for 𝑓 > 2Hz. In our experiments, the effect of flexibility is more pronounced for low amplitude pitching where the selfpropulsion speeds (normalized by total chord length, 𝑈𝑠/𝐶𝑒𝑓𝑓) are larger than the rigid counterpart until ϕ reaches maximum. Beyond this, the speeds are comparable. For moderate and higher amplitudes, we note that the introduction of flexibility does not appreciably change the speed of propulsion. However, the wake of a composite airfoil is very different from that of a rigid airfoil. The inclusion of flexibility suppresses wake bifurcation, with the passive bending in the flap directing the momentum and energy along the direction of propulsion, resulting in more v efficient propulsion characterised by lower 𝐶𝑊𝐸 values. For low amplitude pitching, the wake is characterized by a nearly ‘momentum-less’ vortex pattern with counter-rotating vortices aligned very near to the centerline. For higher amplitude pitching, the wake is characterized by multiple smaller scale vortices that continually interact with each other – merging, deflecting, and annihilating. The wake of the composite airfoil is narrower and sustains two-dimensional features. We propose that the modification of the vorticity shedding process as a consequence of inclusion of flexibility leads to efficient propulsion. These findings provide valuable insights into the role of flexibility in enhancing swimming efficiency.