Biomimetics

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Alexander Alexeev Subscribe SciFeed Recommended Articles Related Info Link Google Scholar More by Authors Links on DOAJ Alberti, L. Carnevali, E. Costa, D. Crivellini, A. on Google Scholar Alberti, L. Carnevali, E. Costa, D. Crivellini, A. on PubMed Alberti, L. Carnevali, E. Costa, D. Crivellini, A. /ajax/scifeed/subscribe Article Views Citations - Table of Contents Altmetric share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up ... Endorse textsms ... Comment Need Help? Support

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Get Information clear JSmol Viewer clear first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessArticle A Computational Fluid Dynamics Investigation of a Flapping Hydrofoil as a Thruster by

Luca Alberti ORCID

Emanuele Carnevali,

Daniele Costa and

Andrea Crivellini * ORCID

Department of Industrial Engineering and Mathematical Sciences, Polytechnic Marche University, 60131 Ancona, Italy * Author to whom correspondence should be addressed. Biomimetics 2023, 8(2), 135; https://doi.org/10.3390/biomimetics8020135 Received: 8 February 2023 / Revised: 13 March 2023 / Accepted: 23 March 2023 / Published: 25 March 2023 (This article belongs to the Special Issue Latest Trends in Bio-Inspired Underwater Robotics) Download Download PDF Download PDF with Cover Download XML Download Epub Download Supplementary Material Browse Figures Versions Notes

Abstract: The paper features a computational fluid dynamics study of a flapping NACA0015 hydrofoil moving with a combination of sinusoidal heaving and pitching. Several kinematic configurations are explored, varying sequentially pitch and heave amplitude, Strouhal number and phase angle, in an attempt to determine the influence of each parameter on the propulsive performance. To optimize efficiency the angle of attack should assume the highest value that also avoids the arise of the leading edge vortex generated in the dynamic stall state. At low Strouhal number optimum is reached at high heave amplitudes, which correspond to the configurations minimizing the hysteresis in the ( C y , C x ) plane. The same outcome in terms of hysteresis minimization has been verified to occur when optimal phase shift was considered. Differently, when the Strouhal number and the angle of attack become higher, to exploit efficiently the lift increment owed to dynamic stall it emerged the necessity of adopting low heave amplitude to improve separation resistance, avoiding the occurrence of deep stall. Keywords: carangiform thruster; propulsive performance; computational fluid dynamics; NACA0015; flapping foil; discontinuous Galerkin; Spalart-Allmaras; dynamic stall 1. IntroductionIn the field of autonomous underwater vehicles (AUVs), bio-inspired solutions have been sought in the last three decades as a source of improvement in terms of propulsive efficiency and maneuverability. As a matter of fact, fishes and marine mammals are faster and nimbler than their robotic counterparts: AUVs turnabout radius is normally a multiple of the robot hull length, namely from two to six times, whereas a biological swimmer is capable to reverse its course without breaking, with a radius of curvature of the order of one third of its body length. The cost of transport, which measures the energy spent to cruise at a given speed, is also significantly lower for aquatic animals when compared to the state-of-the-art of modern nautical technology [1]. Therefore, several prototypes of swimming robots have been manufactured by researchers worldwide in the last thirty years, and an extensive review is provided in [2].Despite the efforts made to pursue the considerable potential payoffs of marine animals’ locomotion, the performance of biological systems are still far to reach. Indeed, the possibility to emulate effectively the swimming modes developed by aquatic animals over thousands of years of evolution depends from the understanding of the fluid mechanics principles of swimming locomotion. In order to address this very ambitious objective, computational fluid dynamics (CFD) analysis represents an invaluable tool to investigate the propulsive performances of biological and bio-inspired thrusters.According to swim mechanics, thrust force originates from the momentum transfer due to the interaction between the fish body and the surrounding water [3]. Particularly, body and caudal fin (BCF) swimmers generate thrust by bending their tails and caudal fins following specific undulation patterns. BCF locomotion is further expanded in five swimming modes characterized by the percentage of the body involved in the tail undulations. Thus, carangiform and subcarangiform swimmers generate thrust by bending respectively half and the last third of their tails. The motion law commonly adopted by biologists and roboticists to model the shape of the tail as a function of time is Lighthill’s travelling wave, an harmonic function whose amplitude increases moving towards the caudal fin [4]. On the other hand, in thunniform locomotion, thrust generation is mainly due to the caudal fin motion, where tail motion is mostly confined. Here, the fin traces an undulating path to adjust its angle of attack and prevent flow separation [5]. Indeed, thunniform locomotion is the most efficient swimming mode in BCF locomotion.As stated before, several prototypes of bio-inspired underwater robots propelled both by carangiform and thunniform locomotion have been designed by researchers in the last thirty years. In the literature, the most common solution adopted to manufacture a carangiform swimming robot consists of a sealed forebody hinged to a piecewise flexible tail embodying a multi-joint, open-chain mechanism driven by dedicated servomotors or by a single rotary actuator [6,7,8]. A similar architecture has been employed to drive the caudal fins of thunniform swimming robots [9]. When a multi-joint mechanism is adopted to approximate the tail undulation patterns of the aforementioned swimming modes, it results easy to prove that the links of the system oscillate following an harmonic motion law. As a result, the caudal fin, which coincides with the tail linkage end effector, performs an harmonic roto-translation called flapping, where the individual components of the resulting motion are oscillation functions characterized by different amplitudes but the same frequency, and a constant phase shift. When a flapping caudal fin is employed as the thruster of a bio-inspired underwater robot, the quantification of its performance in terms of thrust generation and propulsive efficiency is a fundamental issue in the design process. As a matter of fact, in order to size the robot actuation system, the propulsive loads generated by the fin must be known as a function of the geometric and kinematic parameters of flapping motion.To this end, this paper presents a detailed investigation of the dynamic performance of a roto-translating foil predicted by means of computational fluid dynamics. Here, the fin is simulated as a stand-alone thruster and the numerical predictions allow a complete characterization of its propulsive behavior. Aside from comparative considerations, the analysis results can be also exploited to compute the dynamics of a swimming robot and size its driving systems [8]. CFD analysis has been extensively employed in the field of biomimetics. In [10], the authors performed a two-dimensional analysis on the base ornithopter configuration of an insect flying robot using commercial CFD codes: the results have yielded deeper insights regarding the influence of varying flapping frequency on critical flow metrics regarding adequate lift and thrust generation. Flying systems have been investigated also in [11], where CFD methods have been employed to model the transitional aerodynamics of the variable camber morphing wing. Numerical simulations have been also exploited in the marine field to model the hydrodynamic performance of manta-like flapping [12].Unlike other papers on the subject, the goal here is to provide a comprehensive performance-oriented manual on rigid flapping thrusters, which could be exploited as a design tool. Moreover, this work advances novel considerations on the physical effects of each kinematic parameter and proposes new perspectives on the cause-and-effect relationships between the wake structure and the propulsive performance, under a broad range of kinematic conditions. The heave amplitude, in particular, was varied in an interval that has seldom been studied in the past.The paper is organized in the following way: Section 2 outlines the kinematics, introducing the fundamental expressions of the flapping motion and the parameters of interest. In Section 3 the numerical setup of the simulations is presented, including the spatial and temporal discretization schemes, the turbulence model implemented and the special treatment reserved to moving boundaries. The results are collected in Section 4, preceded by a brief description of the mesh and the definition of the main propulsive performance indicators. Finally, the most significant conclusions are drawn in Section 5. 2. Motion KinematicsFishes and aquatic mammals arrived to their current locomotion capabilities through a process of biological optimization, driven by natural selection. As a result, a wide and complex variety of swimming behaviors have developed, usually classified into (i) undulatory, (ii) oscillatory, (iii) pulsatile jet-based and (iv) drag-based motions. For a detailed explanation of each class the interested reader can refer to Smits’ well-known review [13]. The species adopting motion behaviors that fall within one or another of these categories further exhibit peculiar body shapes, with geometries and mass distributions that coupled with the adopted swimming configuration have the effect of optimizing their underwater motion.A complete and exact representation of such a rich variety of configurations, if possible, would require the adoption of prohibitively complex models, rendering computational simulations unfeasible. However, it has been showed how for the oscillatory configurations, which include BCF swimmers, significant simplifications may be adopted and still derive a model retaining relevant information about the physics of the problem. For instance, various experimental investigations [5,14,15,16] showed that the approximation of the oscillatory regime through a flapping hydrofoil predicts the maximum efficiency in the same range of kinematic configurations found for cetaceans and carangiform fishes.Within the flapping foil approximation, the body inertia contribution in the propulsion generation is discarded, with the motion assumed to be concentrated at the body end, namely at the propeller tail. The latter is then suitably approximated via hydrofoil profiles, moving according to kinematic laws approximating their natural motion. The analysis has here been restrained to a two-dimensional case, with the propeller approximated using a symmetric NACA0015 profile, whose schematic representation is given in Figure 1 together with some relevant kinematic parameters.The profile is hence prescribed to move according to an harmonic law composed of a rotation around a pivotal point and a vertical translation, following the parametrization presented below: θ ( t ) = θ 0 sin ( 2 π f t ) h ( t ) = h 0 sin ( 2 π f t + ψ ) . According to Equation (1), pitch amplitude and vertical displacement follow a sinusoidal variation with frequency f. The pitching motion occurs around a fixed center of rotation, with the maximum angular displacement given by θ 0 . The simultaneous heaving motion is characterized by an amplitude h 0 and present, with respect to the periodic rotation, a constant phase shift ψ .From the above relations, the vertical and rotational velocities may be read as: ω z t ( t ) = 2 π f θ 0 cos ( 2 π f t ) u y t ( t ) = 2 π f h 0 cos ( 2 π f t + ψ ) , where the superscript ‘t’ identifies the components representing the relative motion between the inertial and the moving reference frames. The velocity induced by the profile vertical motion affects the instantaneous, effective angle of attack, that according to the convention exposed in Figure 1 may be expressed as follows: α ( t ) = θ ( t ) + arctan u y t ( t ) U ∞ , with U ∞ being the horizontal free-stream velocity. The maximum value reached by α in the harmonic period is indicated by α m a x .A global description of the profile kinematics widely adopted in the topic of flapping hydrofoils is provided via the Strouhal number S t = f A U ∞ ≈ f 2 h 0 U ∞ . This non-dimensional parameter groups together the profile oscillation frequency with a length scale that best characterizes the flow. In the context of flapping foils the characteristic length may be associated to the wake width A, here approximated by the total heave amplitude 2 h 0 .The predominance of heaving or pitching in the foil motion can be quantified by means of the dimensionless heave ratio, defined by Akoz et al. [17] as h ∗ = 2 h ( t ∗ ) A T E , where t ∗ is the time instant at which the trailing edge (TE) reaches its highest vertical position and A T E is the total vertical excursion of the TE. The motion is heave-dominated when h ∗ > 0.5 , while h ∗ 0 . This leads to a very small L / D ratio which determines not only a very small C ¯ T , but also an unsatisfactory η p . 4.2. Pitch AmplitudeThe pitch amplitude is also an important quantity, because it does not only set the maximum excursion of the pitch angle, but it also modulates the pitch rate. Figure 3a displays the performance sensitivity to the pitch amplitude θ 0 , maintaining S t = 0.1935 , h 0 = c and ψ = 90 ∘ . The pitch amplitude was varied using the stencil [ 1 ∘ , 5 ∘ , 10 ∘ , 15 ∘ , 20 ∘ , 25 ∘ ] . Note that the S t was set to such value in order to get S t T E = 0.2 for the reference case h 0 = c , θ 0 = 15 ∘ , ψ = 90 ∘ , where S t T E = f A T E / U ∞ may be thought of as the ‘real’ Strouhal number based on the TE total excursion.The thrust coefficient presents a global maximum presumably within the interval 1 ∘

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