1. Introduction

Flapping wings for flying and oscillating fins for swimming stand out as one of the most complex yet widespread propulsion methods found in nature. Natural flyers and swimmers (which have evolved over millions of years) represent illuminating examples of biokinetics, unsteady aerodynamics, high maneuverability, endurance, and large aero/hydrodynamics efficiency.

In the field of flapping flight, biologist, zoologist, and engineers are sharing findings and conducting research together. From the point of view of a biologist or zoologist, studying flapping flight in nature is of great importance for understanding the biology, allometry, flight patterns, flight skills, and the migratory habits of avian life. From an engineering point of view, the main reason for studying flapping flight is the use of animal locomotion as inspiration for improving existing applications or developing new technologies by just mimicking nature evolutionary-optimization process (biomimetics). Such applications may include drag reduction,

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noise reduction, and flow control by using feather-like structures [1] and flippers tubercles [2]; the development of new propulsion/lift generation systems for micro-air-vehicles (MAVs), nanoair-vehicles (NAVs), and autonomous-underwater-vehicles (AUVs) is inspired by flapping wings or oscillating fins [3–8], energy harvesting applications [9], and even robotic extraterrestrial exploring missions [10].

An important aspect of flying using flapping wings or swimming by using oscillating fins or fanning is the ability to generate thrust with relatively high propulsive efficiency. Early attempts at building fish-inspired mechanisms achieved disappointingly low propulsive efficiencies [11]. It was only through a deeper understanding of the vorticity and wake produced by swimming animals, significant progress was achieved [12].

Many researchers [13–15] have found that flying and swimming animals cruise in a narrow range of Strouhal numbers (between 0.2 and 0.4), corresponding to a regime of vortex growth and shedding in which the propulsion efficiency peaks. The Strouhal number St is a dimensionless parameter defined as,

$$St = \frac{fh}{\mathcal{U}}\tag{1}$$

where f is the flapping frequency, h is the peak to peak amplitude of the flapping stroke, and U is the forward velocity. This definition describes a ratio between the oscillating speed (f h) and the forward speed. Another dimensionless parameter that characterizes the aero/hydrodynamic performance and wake signature of flying and swimming animals is the reduced frequency k, which is a measure of the residence time of a vortex (or a particle) convecting over the wing/fin chord compared to the period of motion and is defined as,

$$k = \frac{\pi f c}{U} \tag{2}$$

Hence, it becomes evident that gaining a better understanding of the wing/fin motion parameters driving forces generation, vortices generation and shedding, the manner in which the vortices interact with the moving surfaces and themselves, and how they contribute to lift and propulsion would aid in better understanding the propulsion mechanism of birds, insects, and fishes, independently of their possible practical applications.

In the current numerical study, we aim at performing a comprehensive analysis of the wake signature and aerodynamic performance of finite-span rigid wings undergoing pure heaving motion. The laminar incompressible Navier-Stokes equations are numerically approximated, and all unsteady, viscous, and three-dimensional effects are solved. The simulations are conducted for Strouhal numbers values between 0.15 ≤ St ≤ 0.5, and for two different reduced frequency values, one corresponding to high frequency and the other one to low frequency, this was done to study leading edge vortex shedding dependency.

The remainder of this paper is organized as follows. In Section 2, we give a brief description of the numerical method and gridding methodology. In Section 3, we present a description of the computational domain, case setup, and heaving kinematics. In Section 4, we present a short discussion of the quantitative and qualitative results obtained from a grid dependence study. In Section 5, we present a detailed discussion of the results. Finally, in Section 6, we present the conclusions and future developments.
