# Effect of aspect ratio on the propulsive performance of tandem flapping foils

## Abstract and Figures

In this work, we describe the impact of aspect ratio ( $AR$ ) on the performance of optimally phased, identical flapping flippers in a tandem configuration. Three-dimensional simulations are performed for seven sets of single and tandem finite foils at a moderate Reynolds number, with thrust producing, heave-to-pitch coupled kinematics. Increasing slenderness (or $AR$ ) is found to improve thrust coefficients and thrust augmentation but the benefits level off towards higher values of $AR$ . However, the propulsive efficiency shows no significant change with increasing $AR$ , while the hind foil outperforms the single by a small margin. Further analysis of the spanwise development and propagation of vortical structures allows us to gain some insights into the mechanisms of these wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.

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... Majority of the computational research conducted on flapping foils has been two-dimensional (Wu et al., 2020) and the three-dimensional spanwise as well as end effects have not been taken into consideration. Recent works by Lagopoulos et al. (2021); Arranz et al. (2020);Jurado et al. (2022) have studied the effect of aspect ratio of the foil on propulsive performance. However, the conditions under which three-dimensional (3D) flow effects become important are yet to be studied in detail for tandem foils. ...

In this study, we present two and three-dimensional numerical investigation to understand the combined effects of the non-dimensional heave amplitude varying from 0 to 1 and the pitch amplitude ranging from 0° to 30° on the propulsive performance for a single and tandem foil system at Reynolds number 1100 and reduced frequency 0.2. We initially present a systematic analysis on the thrust generation due to the kinematic parameters for a single foil. The significance of effective angle of attack and the projected area of the foil has been emphasized in explaining the dynamics of lift and drag forces and their relationship with the propulsion. We next investigate the relation between the streamwise gap and kinematic parameters on propulsion for the tandem foil system. We show that the propulsive performance strongly depends on the upstream wake interacting with the downstream foil, and the timing of the interaction due to the gap between the foils. Through a control volume analysis, the time-averaged pressure and streamwise velocity have been investigated to explain the effect of kinematic parameters on the hydrodynamic forces. Typically in the literature, the formation of jet in the wake has been attributed to thrust generation. However, in this study, we emphasize and show the significance of the time-averaged pressure in the wake apart from the streamwise velocity (jet) for predicting the thrust forces. The study is concluded with a three-dimensional demonstration of the tandem foils to understand the possible three-dimensional effects due to the large amplitude flapping and wake-foil interaction.

Immersed boundary methods are extensively used for simulations of dynamic
solid objects interacting with fluids due to their computational efficiency and
modelling flexibility compared to body-fitted grid methods. However, thin
geometries, such as shells and membranes, cause a violation of the boundary
conditions across the surface for many immersed boundary projection algorithms. Using a one-dimensional analytical derivation and multi-dimensional numerical simulations, this manuscript shows that adjustment of the Poisson matrix itself is require to avoid large velocity, pressure, and force prediction errors when the pressure jump across the interface is substantial and that these errors increase with Reynolds number. A new minimal thickness modi
cation is developed for the Boundary Data Immersion Method (BDIM-),
which avoids these issues while still enabling the use of efficient projection
algorithms for high-speed immersed surface simulations.

We report direct numerical simulations of a pair of wings in horizontal tandem configuration to analyze the effect of their aspect ratio on the flow and the aerodynamic performance of the system. The wings are immersed in a uniform free stream at the Reynolds number Re = 1000, and they undergo heaving and pitching oscillation with the Strouhal number St = 0.7. The aspect ratios of forewing and hindwing vary between 2 and 4. The aerodynamic performance of the system is dictated by the interaction between the trailing edge vortex (TEV) shed by the forewing and the induced leading-edge vortex formed on the hindwing. The aerodynamic performance of the forewing is similar to that of an isolated wing irrespective of the aspect ratio of the hindwing, with a small modulating effect produced by the forewing–hindwing interactions. On the other hand, the aerodynamic performance of the hindwing is clearly affected by the interaction with the forewing's TEV. Tandem configurations with a larger aspect ratio on the forewing than on the hindwing result in a quasi-two-dimensional flow structure on the latter. This yields an 8% increase in the time-averaged thrust coefficient of the hindwing, with no change in its propulsive efficiency.

The importance of the leading-edge sweep angle of propulsive surfaces used by unsteady swimming and flying animals has been an issue of debate for many years, spurring studies in biology, engineering, and robotics with mixed conclusions. In this work, we provide results from three-dimensional simulations on single-planform finite foils undergoing tail-like (pitch-heave) and flipper-like (twist-roll) kinematics for a range of sweep angles covering a substantial portion of animals while carefully controlling all other parameters. Our primary finding is the negligible 0.043 maximum correlation between the sweep angle and the propulsive force and power for both tail-like and flipper-like motions. This indicates that fish tails and mammal flukes with similar range and size can have a large range of potential sweep angles without significant negative propulsive impact. Although there is a slight benefit to avoiding large sweep angles, this is easily compensated by adjusting the fin’s motion parameters such as flapping frequency, amplitude and maximum angle of attack to gain higher thrust and efficiency.

Pakistan Understanding the connection between physiology and kinematics of natural swimmers is of great importance to design efficient bio-inspired underwater vehicles. This study looks at high-fidelity three-dimensional numerical simulations for flows over an undulating American eel with prescribed anguilliform kinematics. Particularly, our work focuses on why natural anguilliform swimmers employ wavelengths shorter than their bodylengths while performing wavy kinematics. For this purpose, we vary the undulatory wavelength for a range of values generally observed in different aquatic animals at Strouhal numbers 0.30 and 0.40. We observe that our anguilliform swimmer is able to demonstrate more suitable hydrodynamic performance for wavelengths of 0.65 and 0.80. For longer wavelengths , the swimmer experiences large frictional drag, which deteriorates its performance. The wake topology was dominated by hairpin-like structures, which are closely linked with the underlying physics of anguilliform swimming found in nature.

The propulsive performance of a flexible foil with prescribed pitching and heaving motions about any pivot point location and passive chordwise flexural deflection is analysed within the framework of the linear potential flow theory and the Euler–Bernoulli beam equation using a quartic approximation for the deflection. The amplitude of the flexural component of the deflection and its phase, the thrust force, input power and propulsive efficiency are computed analytically in terms of the stiffness and mass ratio of the plate, frequency, pivot point location and remaining kinematic parameters. It is found that the maximum flexural deflection amplitude, thrust and input power are related to the first fluid–structure natural frequency of the system, corresponding to the deflection approximation considered. The same relation is observed for the propulsive efficiency when an offset drag is included in the analytical expressions. These results, which are valid for small amplitude and sufficiently large stiffness of the foil, are compared favourably with previous related results when the foil pivots about the leading edge. The configurations generating maximum thrust and efficiency enhancement by flexibility are analysed in relation to those of an otherwise identical rigid foil.

The role of aspect ratio on the dynamic stall process of an unswept finite wing is investigated using high-fidelity large-eddy simulations. Three aspect ratios (AR=4, 8, and 16) are explored for wings (NACA 0012 cross section) at chord Reynolds number Rec=2×105 and freestream Mach number M∞=0.1. The wings pitch sinusoidally from initial incidence of 4° to a maximum angle of attack of 22° with reduced frequency k=πfc/U∞=π/16 over one pitching cycle. The three-dimensional unsteady flowfields show similarity among the three wings through laminar separation bubble formation/bursting. The flow topology during dynamic stall exhibits distinctly different evolutions at the higher aspect ratio relative to the lower, baseline aspect ratio. Rather than evolving into a Λ vortex (AR=4), the higher-aspect-ratio wings show dramatic three-dimensional deformation of the vortex tube that resembles cellular structures. The vortical structure eventually interacts with the trailing-edge vortex, which contrasts with the lower aspect ratio. Examination of the unsteady loads shows an increase in lift slope, average loads, peak loads, and earlier stall with aspect ratio.

The present study investigates the mechanisms of wake-induced flow dynamics in tandem National Advisory Committee for Aeronautics 0015 flapping foils at low Reynolds number of Re = 1100. A moving mesh arbitrary Lagrangian–Eulerian framework is utilized to realize the prescribed flapping motion of the foils while solving the flow via incompressible Navier–Stokes equations. The effect of the gap between the two foils on the thrust generation is studied for gaps of 1–10 times the chord of the downstream foil. The mean thrust as well as the propulsive efficiency vary periodically with the gap indicating alternate regions of higher and lower thrust generation, emphasizing the profound effect of upstream foil's wake interaction with the downstream foil. Five crucial wake–foil interactions leading to either favorable or unfavorable conditions for thrust generation are identified and different modes depending on the interactions are proposed for the tandem flapping foils. It is observed that the effect of the wake of the upstream foil on the downstream foil decreases with increasing gap. The study also focuses on the effect of the chord sizes of the upstream and the downstream foils on the propulsive forces, where the chord of the upstream foil is selected as 0.25–1 times the downstream foil's chord length. The effect of the chord size on the thrust is noticed to diminish as the chord size of the upstream foil decreases. Furthermore, the effect of the phase difference between the kinematics of the upstream and the downstream foils on flow dynamics is also explored along with its relationship with the chord sizes. For a fixed chord size, the effect of the phase difference on the propulsive performance is observed to be similar to that by varying the gap between the foils due to similar type of vortex interactions. The mechanisms of vortex interactions are linked to provide a comprehensive and generic understanding of the flow dynamics of tandem foils.

Animals and bio-inspired robots can swim/fly faster near solid surfaces, with little to no loss in efficiency. How these benefits change with propulsor aspect ratio is unknown. Here we show that lowering the aspect ratio weakens unsteady ground effect, thrust enhancements become less noticeable, stable equilibrium altitudes shift lower and become weaker, and wake asymmetries become less pronounced. Water-channel experiments and potential flow simulations reveal that these effects are consistent with known unsteady aerodynamic scalings. We also discovered a second equilibrium altitude even closer to the wall (<0.35 chord lengths). This second equilibrium is unstable, particularly for high-aspect-ratio foils. Active control may therefore be required for high-aspect-ratio swimmers hoping to get the full benefit of near-ground swimming. The fact that aspect ratio alters near-ground propulsion suggests that it may be a key design parameter for animals and robots that swim/fly near a seafloor or surface of a lake.

Flapping flight and swimming are increasingly studied due to both their intrinsic scientific richness and their applicability to novel robotic systems. Strip theory is often applied to flapping wings, but such modeling is only rigorously applicable in the limit of infinite aspect ratio (AR) where the geometry and kinematics are effectively uniform. This work compares the flow features and forces of strip theory and three-dimensional flapping foils, maintaining similitude in the rolling and twisting kinematics while varying the foil AR. We find the key influence of finite AR and spanwise varying kinematics is the generation of a time-periodic spanwise flow which stabilizes the vortex structures and enhances the dynamics at the foil root. An aspect-ratio correction for flapping foils is developed analogous to Prandtl finite wing theory, enabling future use of strip theory in analysis and design of finite aspect ratio flapping foils.

Based on a high-order implicit discontinuous Galerkin method, numerical simulations of a two-dimensional oscillating foil are performed to explore the origin of basic aspects of the flow such as the generation of interesting flow structures in the wake and the associated aerodynamic forces. Dimensional arguments suggest that the flow is characterized by non dimensional aerodynamic coefficients depending on the kinematics of the oscillation, such its frequency and amplitude, and on the dynamics of the flow, such as the Reynolds number. Most of the studies have concentrated their attention on the role played by the kinematic of the oscillation with less or no attention to the effect of the Reynolds number. Here, we show that this effect cannot be neglected in the study of the phenomena at the basis of the generation of lift and thrust. We found that the Reynolds number plays a fundamental role for the development of thrust by defining critical values Rec for the switch from drag to thrust conditions. It is also shown that for Re>Rec, the Reynolds number defines additional subcritical values which are at the basis of flow instabilities leading to smooth and sharp transitions of the structure of the wake and of the related aerodynamic forces. For the analysis of the behaviour of the flow, the space of phases composed by the instantaneous lift and thrust (cL,cT) is introduced. It is shown how the orbits in the (cL,cT)-space allow us for a clear understanding of the physical evolution of the flow system and of the cyclical phenomena composing it.