## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

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.

To read the full-text of this research,

you can request a copy directly from the authors.

... 57 Apart from a single flapping foil, the effects of gap, amplitude ratios, and phase difference between the motion of foils in tandem configuration have also been studied. [58][59][60][61][62] The ease of manufacturing and simplicity of rigid foils has led to their extensive investigation, both experimentally and computationally. However, aquatic animals using oscillatory or undulatory motion for propulsion have flexibility in the motion of their fins or body. ...

... The discretization of the domain for the present study satisfies the criterion of converged mesh from the previous studies. 62,94 A representative mesh is shown in Fig. 3 where the discretization in the far-field as well as the boundary layer and the trailing edge can be observed. A refined region is constructed close to the surface of the foil that helps to capture the viscous boundary layer. ...

... This mesh resolution has already been utilized for validation of flapping rigid foils in the previous study. 62 The unstructured mesh consists of 69 165 nodes and 68 830 four-node quadrilateral elements. ...

Locomotion of aquatic animals involves flapping of their body to generate lift and thrust. Through evolution, they have mastered their ability to move through complex environments in an energy-efficient manner. A crucial component of this movement is the ability to actively bend their bodies to generate maximum thrust. This motion is widely termed as morphing. A simplification of this motion is implemented for a foil in this study to realize a thrust-generating bio-inspired device. The propulsive performance of the heaving foil undergoing a prescribed trailing-edge morphing is numerically studied by a stabilized finite element moving mesh formulation. The effects of the morph position and amplitude on the flow dynamics and propulsion of the foil are investigated in the present work. The position of trailing-edge morphing varies from the leading edge to half of the foil's chord, whereas the morph amplitude varies from [Formula: see text] to [Formula: see text] at the trailing edge. The instantaneous thrust is analyzed with vorticity plots and surface pressure diagrams. Within the parametric space, it is found that the foil is highly efficient in generating propulsive forces at high morph amplitudes and low morph positions. The interplay between the thrust-generating leading-edge vortex (LEV) and the drag-inducing trailing-edge vortex (TEV), which governs the thrust cycle of a morphing–heaving foil, is elucidated. It is observed that the LEV-induced thrust is higher at low morph positions, while the TEV-induced drag is dominant at high morph amplitudes. An ideal balance of these opposing effects of LEV and TEV occurs at the lowest morph position and intermediate morph amplitudes, emphasizing the optimal flexibility for the maximum propulsive performance of the foil.

... The various mechanisms which lead to the generation of thrust have been studied in [4,3,9,11,16], where they were characterized as constructive or destructive. A recent study by the authors in [7,8] discussed in detail the flow dynamics across the tandem flapping configuration of rigid foils. The study focused on the effects of large range of gap between the foils and their chord sizes on the propulsive performance. ...

... The present study is an attempt to expand the parametric space studied by the authors in [8]. Here, the effect of the upstream foil's heave amplitude on the propulsive performance of the downstream foil is investigated. ...

... The schematic of the numerical steps within a time step is depicted in Fig. 1. The presented numerical framework has been utilized for conducting mesh and time convergence as well as validation for the flapping of a single foil in [8]. ...

The current study investigates the flow dynamics around rigid flapping NACA0015 foils in tandem configuration at low Reynolds number. The effect of the heave amplitude of the upstream foil on the propulsive performance of the downstream foil is studied for heave amplitudes of 0.2-1 times the downstream foil chord length. The tandem foils are modelled as rigid foils with prescribed sinusoidal motion and the flow is formulated numerically by variational stabilized Petrov-Galerkin finite element technique with the help of moving mesh arbitrary Lagrangian-Eulerian framework. The heave amplitude of the upstream foil is found to have significant impact on the propulsion of the downstream foil. The trend of the propulsive performance can be explained by the type of vortex interaction, which may be favourable or unfavourable for thrust generation.KeywordsFlapping foilsBio-mimeticsFluid-structure interactionComputational fluid dynamics

... Biologically inspired unmanned underwater vehicles have experimented with flapping motion as a source for thrust, however, the current applications are limited to rigid foils where the flapping motion is approximated by combined heaving and pitching motions [7][8][9]. Extensive studies have been performed to understand the flapping dynamics of a single foil [10] and the effect of various parameters like chord ratio, phase difference, and gap ratio for the tandem foil configuration [11][12][13][14][15][16][17][18][19][20][21][22]. ...

... Temporal discretization for the governing equations is carried out using the Generalized-α method [35], and finite element variational stabilized method is employed for the spatial discretization. The variational formulation for the flow equations can be found in [22,36]. ...

... The verification of the formulation along with mesh convergence and validation has been carried out in the previous works [21,22] and will not be dealt with in the current work. The mesh convergence study performed is applicable for the mesh utilized for the current study. ...

Biological locomotion, observed in the flexible wings of birds and insects, bodies and fins of aquatic mammals and fishes, consists of their ability to morph the wings/fins. The morphing capability holds significance in the ability of fishes to swim upstream without spending too much energy and that of birds to glide for extended periods of time. Simplifying the wing or fins to a foil, morphing refers to the ability of the foil to change its camber smoothly, without sharp bends on the foil surface. This allows precise control over flow separation and vortex shedding. Compared to conventional trailing-edge extensions or flaps, used in rudders and elevators in submarines and ships, morphing foils provide better control of thrust and lift characteristics. This study aims at understanding the importance of the morphing of foil combined with a sinusoidal heaving motion on thrust generation. A two-dimensional variational stabilized Petrov-Galerkin moving mesh framework is utilized for modelling the incompressible low Reynolds number flow across the flapping foil. The morphing motion is characterized by the extent of morphing, measured as an angle of deviation from the initial camber, and the point
of initiation of morphing on the foil as a percentage of its chord length. The effect of the foil morphing and the heaving motion on the propulsive performance are investigated. The extent of morphing is varied from -30◦ and 30◦, and the point of initiation
ranges from 15% to 50% of the chord. The Reynolds and Strouhal numbers for the study are 1100 and 0.2, respectively. The results from the current work can pave the way for enhanced engineering designs in bio-mimetics and give insights into design conditions for optimal thrust performance.

... For the structure of tandem flapping wings, the effects of the phase and separation distance between the fore and hind wing on aerodynamic performance were investigated. [24][25][26][27] In these studies, Muscutt et al. 26 and Joshi and Mysa 27 both emphasized the role of performance augmentation in tandem flapping wings that the hind foil in the tandem configuration can even produce almost twice as much thrust as the fore/single flapping foil. Aside from that Joshi and Mysa 27 and Pan et al. 28 both agreed that the wake formation of the tandem flapping wings is a characterization of the propulsive performance, and the wake of tandem flapping wings could be classified to three modes, corresponding to high, neutral, and low propulsion, respectively. ...

... For the structure of tandem flapping wings, the effects of the phase and separation distance between the fore and hind wing on aerodynamic performance were investigated. [24][25][26][27] In these studies, Muscutt et al. 26 and Joshi and Mysa 27 both emphasized the role of performance augmentation in tandem flapping wings that the hind foil in the tandem configuration can even produce almost twice as much thrust as the fore/single flapping foil. Aside from that Joshi and Mysa 27 and Pan et al. 28 both agreed that the wake formation of the tandem flapping wings is a characterization of the propulsive performance, and the wake of tandem flapping wings could be classified to three modes, corresponding to high, neutral, and low propulsion, respectively. ...

... [24][25][26][27] In these studies, Muscutt et al. 26 and Joshi and Mysa 27 both emphasized the role of performance augmentation in tandem flapping wings that the hind foil in the tandem configuration can even produce almost twice as much thrust as the fore/single flapping foil. Aside from that Joshi and Mysa 27 and Pan et al. 28 both agreed that the wake formation of the tandem flapping wings is a characterization of the propulsive performance, and the wake of tandem flapping wings could be classified to three modes, corresponding to high, neutral, and low propulsion, respectively. ...

In the present work, we propose an optimization framework based on the active learning method, which aims to quickly determine the conditions of tandem flapping wings for optimal performance in terms of thrust or efficiency. Especially, multi-fidelity Gaussian process regression is used to establish the surrogate model correlating the kinematic parameters of tandem flapping wings and their aerodynamic performances. Moreover, the Bayesian optimization algorithm is employed to select new candidate points and update the surrogate model. With this framework, the parameter space can be explored and exploited adaptively. Two optimization tasks of tandem wings are carried out using this surrogate-based framework by optimizing thrust and propulsion efficiency. The response surfaces predicted from the updated surrogate model present the influence of the flapping frequency, phase, and separation distance on thrust and efficiency. It is found that the time-average thrust of the hind flapping wing increases with the frequency. However, the increase in frequency may lead to a decrease in propulsive efficiency in some circumstances. Published under an exclusive license by AIP Publishing. https://doi.

... A numerical study by Pan and Dong 25 at Re = 1000 identified that an increase in the streamwise spacing in a school of flapping foils reduced the influence of the lateral neighbors on the performance of the flapping foils. Recently, Joshi and Mysa 26,27 studied the effect of gap and the chord sizes for the tandem foils on the propulsion at Re = 1100. The mechanism of wake interaction with the downstream foil was identified and studied in detail. ...

... The nonlinear Navier-Stokes equations are solved by the Newton-Raphson iterative technique. The coupling between the flow equations and the moving mesh framework is carried out in a partitioned iterative manner, the details of which can be found in the works 27,50 . The above formulation has been verified and validated, consisting of mesh convergence and time convergence studies in the earlier work 27 for the single and tandem flapping foils and will not be discussed in the present work for brevity. ...

... The coupling between the flow equations and the moving mesh framework is carried out in a partitioned iterative manner, the details of which can be found in the works 27,50 . The above formulation has been verified and validated, consisting of mesh convergence and time convergence studies in the earlier work 27 for the single and tandem flapping foils and will not be discussed in the present work for brevity. ...

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.

... Biologically inspired unmanned underwater vehicles have experimented with flapping motion as a source for thrust, however, the current applications are limited to rigid foils where the flapping motion is approximated by combined heaving and pitching motions [7][8][9]. Extensive studies have been performed to understand the flapping dynamics of a single foil [10] and the effect of various parameters like chord ratio, phase difference, and gap ratio for the tandem foil configuration [11][12][13][14][15][16][17][18][19][20][21][22]. ...

... Temporal discretization for the governing equations is carried out using the Generalized-α method [35], and finite element variational stabilized method is employed for the spatial discretization. The variational formulation for the flow equations can be found in [22,36]. ...

... The verification of the formulation along with mesh convergence and validation has been carried out in the previous works [21,22] and will not be dealt with in the current work. The mesh convergence study performed is applicable for the mesh utilized for the current study. ...

Biological locomotion, observed in the flexible wings of birds and insects, bodies and fins of aquatic mammals and fishes, consists of their ability to morph the wings/fins. The morphing capability holds significance in the abilities of fishes to swim upstream without spending too much energy and that of birds to glide for extended periods of time. Simplifying the wing or fins to a foil, morphing refers to the ability of the foil to change its camber smoothly, without sharp bends on the foil surface. This allows precise control over flow separation and vortex shedding. Compared to conventional trailing-edge extensions or flaps, used in rudders and elevators in submarines and ships, morphing foils provide better control of thrust and lift characteristics. This study aims at understanding the importance of the morphing of foil combined with a sinusoidal heaving motion on thrust generation. A two-dimensional variational stabilized Petrov-Galerkin moving mesh framework is utilized for modelling the incompressible low Reynolds number flow across the flapping foil. The morphing motion is characterized by the extent of morphing, measured as an angle of deviation from the initial camber, and the point of initiation of morphing on the foil as a percentage of its chord length. The effect of the foil morphing and the heaving motion on the propulsive performance are investigated. The extent of morphing is varied from -30 deg and 30 deg, and the point of initiation ranges from 15% to 50% of the chord. The Reynolds and Strouhal numbers for the study are 1100 and 0.2, respectively. The results from the current work can pave the way for enhanced engineering designs in bio-mimetics and give insights on design conditions for optimal thrust performance.

... For those studies on tandem wings, the researchers mainly explored the effects of spacing and the phase difference between the front and rear wings on aerodynamic performance and flow structures. [6][7][8][9][10][11][12][13][14][15][16][17][18] For example, the experiments of Usherwood and Lehmann 7 showed that the aerodynamic power requirements can be reduced by 22% compared to a single wing by appropriate front and rear wing phase adjustments, which is achieved by recovering energy from the wake. Experiments by Boschitsch et al. 9 demonstrated that the thrust and propulsive efficiency of the downstream foil could be as high as 1.5 times or as low as 0.5 times those of an isolated foil depending on the streamwise spacing and phase differential between the two in-line tandem foils. ...

... 14 In addition, Joshi and Mysa 15 investigated the effect on thrust generation when the wing spacing is 1-10 times the chord length of the rear foil, and they identified five crucial wake-foil interactions, which can lead to either favorable or unfavorable conditions for thrust generation. 15 In addition to the studies of the influence of spacing and phase difference on wing-wing interference, some other work has also been carried out on the performance of tandem foils with an asymmetric pitching motion or with stroke time asymmetry. 19,20 Despite much knowledge that has been gained from the above research on tandem flapping wings, we are far from fully understanding the complex aerodynamic interference of wings in collective motion, especially when considering the three-dimensional (3D) situation. ...

Collective movements are common in nature, such as the swimming of fish schools and the flight of birds in formation. The aero/hydrodynamic performance of such movements is a research hotspot at present. As a continuation of the previous research [X. G. Meng et al., "Aerodynamic performance and flow mechanism of multi-flapping wings with different spatial arrangements," Phys. Fluids 34, 021907 (2022)], this study examined the aerodynamic interference effect of three tandem flapping wings at different morphological and kinematic parameters. Computational fluid dynamics were used with the aspect ratio ( AR) of the wing ranging from 2.75 to 4.75, stroke amplitude (Φ) from 60{degree sign} to 120{degree sign}, advance ratio ( J) from 0.25 to 0.6, and Reynolds number ( Re) from 200 to 2000. The aerodynamic interference for the tandem flapping wings includes three effects, namely, the narrow channel effect, the downwash effect, and the wake capture effect. The AR, Φ, and J can significantly influence the evolution of the vortex structures of the three-flapping-wing system, especially the velocity of wake vortices developing downstream. As a result, the downwash effect in the downstroke and the wake capture effect in the upstroke change obviously with these parameters. Due to the decreasing viscous effect with the increase of Re, the wake capture effect, which can improve the thrust of the wings, is more obvious at higher Re. This study further deepens our insights into the flow physics of the multi-flapping wings and provides a theoretical basis for improving the aerodynamic performance of multi-flapping-wing vehicles in the future.

... Other examples of tandem foil propulsion in nature include fish schooling [26][27][28][29] and bird flocking, 30,31 where favorable interactions within the flow allow neighboring animals to enhance the propulsive performance and efficiency by adjusting their flapping behavior (synchronization) and distance from the neighbor. Studies on tandem pitching foils 32,33 and tandem pitching and heaving foils [34][35][36][37][38] have shown that the propulsive performance of the downstream foil is primarily determined by the phase difference and spacing of the foils. For either case, the parameters adjust the time-of-arrival of the vortices shed from the upstream foil arriving at the downstream foil. ...

... To synthesize the observations in previous section, it has been shown that the wake of the fore foil can significantly affect the performance of the hind foil, similar to the observations of previous studies. [32][33][34][35][36][37]39 It has been demonstrated that in the high-thrust cases, a vortex pair is formed at the middle of the stroke and is shed at the end of the stroke far from the centerline. This is the underlying mechanism of thrust enhancement for both Ar 0.5 and Ar 1 configurations. ...

Identical tandem flippers of plesiosaurs, which are unique among all animals, have been a source of debate regarding the role of hind flippers in their locomotion. Here, inspired by the kinematics of plesiosaur flippers, the effect of amplitude ratio on the propulsive performance of in-line tandem pitching foils is investigated through a series of particle image velocimetry experiments. Three leader-to-follower amplitude ratios are considered for the foils pitching over a range of 0-2π phase difference. For the first time, it is shown that the amplitude ratio can significantly affect the performance of the hind foil at spacing larger than one chord length. It is found that the thrust generation of the hind foil at the optimum phase difference augments by 130% when it is pitching at the twice angular amplitude of the upstream foil. Although the total performance of the rear-biased and equal amplitude models reaches to similar values, thrust production of the hind foil in the equal amplitude model increases only by 23%. In contrast, the performance of the forward-biased model decreases drastically for all phase differences due to the destructive wake-foil interaction of the hind foil. Studying the instantaneous wake-foils interactions, it is found that high thrust generation is associated with the formation of a vortex pair on the suction side of the hind foil, which causes stronger trailing edge vortices to shed with a greater total wake spacing. Finally, through scaling analysis, high-thrust configurations of tandem models are ranked based on the total efficiency of the system.

... It has been found that the spacing between the two foils, the phase differential, and the Strouhal number could affect the interception time and location of the vortex shed from the fore-foil onto the hind-foil and, thus, strongly affect the hydrodynamic performance of the hind-foil as well as the whole system. [6][7][8][9][10][11][12][13][14][15][16][17][18][19] For instance, Boschitsch et al. 11 found that the thrust and propulsive efficiency of the downstream foil could be as high as 1.5 times or as low as 0.5 times those of an isolated foil depending on two different wake interaction modes, namely, the coherent mode and the branched mode. The work of Kurt and Moored 17 also showed the similar two interaction modes, which are linked to peak thrust performance and minimum power consumption, respectively. ...

... Re was set to be 200 in the following simulations, and the flow around the wing was considered to be laminar. The selected Re (¼200) falls into the range of the Re that small insects operate, i.e., fruit fly (%130) 43 and cranefly (%250), 44 and it also falls in the range of Re (100-30 000) of previous works, [6][7][8][9][10][11][12][13][14][15][16][17][18][19][26][27][28][29][30][31][32][33][34] which studied the tandem wing configuration. The numerical method used for solving Eqs. ...

In nature, the phenomenon of cluster movements of fish, birds, and insects is universal, which constantly inspires people to explore its advantages. In this study, the aerodynamic performance of three three-dimensional flapping wings under different spatial arrangements was numerically investigated at a low Reynolds number and the interactions among the individuals and the associated underlying fluid mechanisms were explored. In addition, the effects of the number of individuals on the aerodynamic performance of the group as well as the individual were also considered based on the spatial arrangement when the three-wing group produces the maximum thrust. The results show that the spatial arrangement between flapping wings has an important impact on the aerodynamic performance of the whole group and individuals. At a specific spacing (in-line tandem arrangement), the overall thrust of the group can be increased by about 40%, while the overall lift has little change. It is also found that the overall lift of the group decreases with the increase in the number of individuals in the group, and the thrust remains unchanged. The detailed analysis of the wake flow reveals that the downwash of the vortex wake plays a dominant role in the aerodynamic interference.

... [18][19][20] Due to the interaction of the fore and hind wing, the vortices near the tandem wings are coupled with each other, with new vortices constantly generating and dissipating in the process of flapping movement. [21][22][23] In the past, the identity of the wake type of the flapping wings relied on naked eye observation, according to the number, intensity, trajectory, and interaction of vortices. 5,24,25 However, this method has low efficiency and limited scope of application, which is difficult to be effectively applied to complex wake analysis. ...

Bionic flapping wing vehicles have great potential for civil and defense applications due to their flexibility and concealment at low Reynolds numbers. Since traditional flow field pattern recognition methods are difficult to identify effective information from the measured local flow field and deduce the state information of the moving body, this study uses an artificial intelligence method to establish the internal correlation between flow field pattern and state information. Specifically, a fully connected neural network is adopted to recognize the tandem flapping wings' flow field pattern by using different data acquisition methods and detector array distribution methods. Compared with the neural network based on time series data, the neural network based on spatial distribution data can realize the real-time judgment of flow field environment, which is closer to the real-time requirements in practical applications. In the paper, the intelligent perception of multi-flapping wings' flow field environment with sparse detectors is carried out and lays the theoretical foundation for autonomous navigation and obstacle avoidance of flapping wing aircrafts.

... Li et al. [17,18] performed fluid chromatography experiments on vortices generated by parallel robot fish swimming and analyzed that whether following the fish saves energy depends on whether the direction of its tail fin swing coincides with the direction of the induced flow in the vortex street generated by the leading fish. Also, in terms of anterior-posterior fin interactions, the area of enhanced low-pressure suction generated by the vortex at the trailing edge of the anterior fin at the leading edge of the posterior fin improves the propulsive performance of the posterior fin [19][20][21]. In addition, pectoral fin stiffness [22,23], pectoral fin trailing edge amplitude [24], phase difference between anterior and posterior fin movements [25], amplitude [26] and the spacing between the two fish [27][28][29] also affect propulsive performance. ...

Rays have shown superior ocean swimming performance, but the underwater robots that use rays as bionic objects still need to be improved in terms of forward speed. In this paper, we observe and investigate the structure and motion characteristics of rays, and design a bionic ray robot driven by two pairs of serial pectoral fins. In Addition, theoretical analysis of the deformation of the pectoral fin motion is conducted, and fluid simulation and experimentation studies are used to examine the forward propulsion capabilities of the anterior and posterior fins under various phase differences. The enhanced propulsion performance of the posterior fin is explained from the perspective of fluid vortices, which are related to the jet direction and position of the front fin vortices. The results show that the phase difference has a significant impact on the forward speed of the robot. The maximum forward speed of the robot fish is up to 1.2 m/s at 30° phase difference, i.e., 2.86 body length per second (BL/s), which is ahead of the existing robots of the same type.

... Specifically, the break-up of the LEV shed from the front foil means that the back foil does not experience a coherent wake across its span, which limits the benefits derived from wake recapture. Therefore, although a ∼30 % increase in thrust can be noteworthy (Figure 10a for AR = 2), it is still far away from the optimal cases reported here or found in the literature (Akhtar, Mittal, Lauder, & Drucker, 2007;Boschitsch, Dewey, & Smits, 2014;Joshi & Mysa, 2021;Lagopoulos, Weymouth, & Ganapathisubramani, 2020;Muscutt et al., 2017aMuscutt et al., , 2017bXu et al., 2017). It should also be noted that a similar performance deterioration of inline flapping due to 3-D associated effects has been witnessed within insect-like concepts, where lower Re C and S C have been used (Arranz et al., 2020). ...

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.

... Two critical parameters, i.e., the phase angle in flapping motion (u) and the dimensionless inter-foil spacing (G/c) were found to modulate the aerodynamic performance of a tandem airfoil/wing system. 15,[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] For a tandem airfoil/wing system with vertical flapping and symmetric pitching, an enhancement of propulsion thrust is achieved under the in-phase mode (u ¼ 0 ), while the interaction tends to be detrimental to the thrust generation as u deviates from 0, i.e., the out-of-phase or antiphase mode. [24][25][26] For example, Lua et al. found that the overall thrust generation of two tandem airfoils is higher than that of an isolated airfoil (SA) with À90 < u < 90 . ...

The flapping wings or fins in an in-line arrangement are a common scene in flocks and schools, as well as flying creatures with multiple pairs of wings, e.g., dragonflies. Conventional studies on these topics are underpinned by tandem plunging airfoils in either a vertical or a declined stroke plane. The former model mostly considers a symmetrical pitching motion, and the latter model fails to separate the effect of the asymmetric pitching from that of the declined incoming flow. However, our study focuses on the tandem airfoils with vertical plunging and asymmetric pitching in a horizontal freestream and, therefore, explains the effects of asymmetric pitching on tandem plunging airfoils. Using numerical methods, the aerodynamic performance and vortical structures of the tandem airfoils are examined, and the effects of the non-zero geometric angle of attack (α0), phase angles in the plunging and pitching motion (φ and θ), and inter-foil spacing (G/c) are discussed. Our results show that the tandem arrangement is beneficial to enhance the propulsion thrust while retaining the lifting capacity of the airfoil at a non-zero α0. The effects of φ and G/c are coupled since they both determine the interaction between the hind airfoil and the leading-edge vortex in the wake and the out-of-phase mode is suggested for the tandem airfoils at G/c = 1 to enhance both lift and thrust. For a tandem airfoil with in-phase mode, the optimal G/c is around 1.5 to 2. Moreover, the asymmetric pitching of the in-phase plunging airfoils should be synchronized to retain the enhanced performance.

Recent studies on the snap-through motion of elastic sheets have attracted intense interest in energy harvesting applications. However, the effect of boundary conditions on energy extraction performance still remains an open question. In this study, we explored the snapping dynamics and energy-harvesting characteristics of the buckled sheet at various conditions using fluid-structure interaction (FSI) simulations at a Reynolds number Re = 100. It was found that the front boundary condition (BC) dramatically affects the sheet's snapping dynamics, e.g., the pinned or relatively soft front BC triggers the sheet's instability easily and thus boasts the collection of potential energy. In the snap-through oscillation state, a stiffer rear BC results in a larger improvement in the sheet's energy collection compared with a minor effect of front BC. Meanwhile, the enhancement can also be achieved by adjusting the rear rotational spring stiffness up to 1.125×10-4 , after which it remains nearly constant, as observed in the case EI * = 0.004. This introduction of an elastic BC with * = 1.125×10-4 not only efficiently enhances energy extraction but significantly reduces stress concentration and, as a result, greatly prolongs the sheet's fatigue durability, especially for the stiffer sheet with EI * = 0.004. The effect of three other governing parameters, including length ratio ΔL * , sheet's bending stiffness EI * and mass ratio m * , on the sheet's energy harvesting performance were also explored. The result shows that increasing ΔL * and EI * could improve the total energy harvested, primarily by enhancing the elastic potential energy, particularly in the aft half of the sheet. In contrast, increasing m * mainly enhances the kinetic energy collected by the sheet's central portion, thus improving the total energy-extracting performance. This study provides an in-depth insight into the dynamics of a buckled sheet under various BCs, which may offer some guidance on the optimization of relevant energy harvesters.

The unsteady aerodynamics play an important role in fluid-structure interaction due to its engineering applications. These types of research can be utilized to improve the fundamental analysis of bluff body-airfoil wake interaction and to achieve better design and more efficiency in engineering applications. This paper aims to conduct experimental tests over parameter space to investigate flow around a NACA0012 pitching airfoil in the wake of a circular cylinder. The one-dimensional hot-wire anemometer probe was used in an open-type wind tunnel to examine the effect of the cylinder-airfoil gap, the amplitude of pitching oscillation, and reduced frequency on the time-averaged flow characteristics in Reynolds number
. The results show that the increase of the reduced frequency at the
has a negligible effect on the flow characteristics, while its increase at the
is consequential. Pitching oscillation creates a region with a higher fluctuation in the wake center of the upstream cylinder, whose width grows with the increase in the amplitude of pitching oscillation. In addition, the distributions of the turbulent flow characteristics for
cases in
are approximately similar but different from the
case. The analytical results presented in this paper provide basic information about cylinder-pitching airfoil wake interaction.

It has always been a challenge to implement the natural flyer and swimmer kinematics into human-made aero/hydro vehicles for the enhancement of their performance. The propulsive performance of underwater vehicles can be enhanced by following the fishtailed kinematics. In the present study, a two-dimensional simulation has been performed on a tandem flapping foil by altering the simple flapping trajectory motion to a fishtailed trajectory by varying the Strouhal number ( St) in the range of 0.1–0.5. The effect of the inter-foil spacing and phasing between the foils on wake interaction is also investigated. The results show that fishtailed trajectory motion and inter-foil spacing of 2 c m –3 c m (where c m is the mean chord length) between the foils would enhance the propulsive efficiency of the downstream foil by up to 41%. The unfavorable spacing between the foils results in adverse wake interaction, which reduces the propulsive efficiency compared to solo flapping foil.

Fish schooling is a common phenomenon that is assumed to have energy advantages, and various fish schooling configurations have been observed. However, the underlying mechanism of fish schooling remains an open question. To understand the relationship between energetically advantageous formations and swimming modes, we numerically investigated two self-propelled fish at different initial spacings. Stable beneficial formations were classified into three types, based on the benefiting individuals. The analysis focused on the intrinsic mechanisms of individual benefit for the three types of energetically beneficial formations in terms of the swimming performance, velocity clouds, and vortex evolution. For the first type, both fish benefited from parallel formations as the carangiform swimming mode promoted fluid flow and reduced lateral power because of the channel effect. For the second type, the front fish gained an efficiency advantage from pressure and vortex mechanisms, while the rear fish was almost indistinguishable from the swimming-alone condition in the sinusoidal and anguilliform swimming modes. For the last type, similar to the tandem queue, the rear fish modulated the vortex and gained an energy advantage in the carangiform and anguilliform swimming modes. This study provides new insights into the appropriate configuration of fish for different swimming modes in nature.

Dragonfly-like flapping wings and foils take an in-line arrangement and their performance is strongly affected by the wing–wing interaction. In this study of tandem pitching-plunging foils, the effects of flapping deviation on the hovering performance are studied numerically. A two-dimensional model is established, and the flapping deviation is represented by the non-dimensional deviation distance ΔL/c, the effects of which are investigated, together with those of the phase angle φ in plunging motion and the non-dimensional inter-foil spacing G/c. In the hovering status, our results support that the wing–wing interaction at ΔL/c=0 is detrimental to the lifting capacity of tandem pitching-plunging foils. More importantly, we prove that a modification of ΔL/c in which the fore foil is below the hind foil during hovering can compensate for the loss of lift and may even provide a lift greater than that generated by a single foil. The mechanism of this compensation is dependent on the interaction between the wake vortices of the fore foil and the leading-edge vortex of the hind foil. Our results also indicate that the effects of ΔL/c and φ are coupled. The lift enhancement by a negative ΔL/c is maintained within the ranges φ = 45°–90° and G/c<2, beyond which an abrupt drop may occur. These findings suggest that the control of flapping deviation may enhance the performance of dragonfly-like micro air vehicles.

The fluid dynamics of flapping foils are reviewed in this article. A very wide range of researches are conducted for the two-dimensional flapping foil which has a relatively simple geometry. However, for a three-dimensional foil, the aspect ratio and shape take effects and completely distinct fluid dynamics are revealed compared with the two-dimensional one. This review gives a summary on the experimental techniques and numerical methods used in the researches on the fluid dynamics of flapping foils. The effects of some key parameters including Reynolds number, reduced frequency, flapping amplitude and three-dimensional effect on the fluid dynamics of flapping foils are reviewed. The researches focusing on the wake structures, aero/hydrodynamic characteristics and energy harvesting efficiency are discussed. Finally, some conclusions are drawn and potential future research directions are recommended.

The present paper aims to investigate numerically small amplitude oscillation of NACA 0012 airfoil at Re = 1000. The airfoil is sinusoidally pitching around the quarter chord point with 1° pitch amplitude about a mean angle of attack. The computations are performed for mean angles of attack ranging from 0° to 60° and for pitching frequencies of 1 Hz and 4 Hz. The effect of the mean angle of attack and pitching frequency on the instantaneous forces as well as the vortex structure is investigated in comparison with the non-oscillatory conditions. It was shown that airfoil oscillations at the investigated conditions change the amplitude of oscillation of the aerodynamic loads. The instantaneous drag coefficient is always positive for pitching airfoil at 1 Hz. In the meantime, there are time intervals where instantaneous drag coefficient becomes negative for pitching motion at 4 Hz for mean angles of attack from 3° to 36°.

Fish use coordinated motions of multiple fins and their body to swim and maneuver underwater with more agility than contemporary unmanned underwater vehicles (UUVs). The location, utilization and kinematics of fins vary for different locomotory tasks and fish species. The relative position and timing (phase) of fins affects how the downstream fins interact with the wake shed by the upstream fins and body, and change the magnitude and temporal profile of the net force vector. A multifin biorobotic experimental platform and a two-dimensional computational fluid dynamic simulation were used to understand how the propulsive forces produced by multiple fins were affected by the phase and geometric relationships between them. This investigation has revealed that forces produced by interacting fins are very different from the vector sum of forces from combinations of noninteracting fins, and that manipulating the phase and location of multiple interacting fins greatly affect the magnitude and shape of the produced propulsive forces. The changes in net forces are due, in large part, to time-varying wakes from dorsal and anal fins altering the flow experienced by the downstream body and caudal fin. These findings represent a potentially powerful means of manipulating the swimming forces produced by multifinned robotic systems.

In this study, using a hydraulic system couples 2 hydrofoils and fulfills the fully flow‐induced oscillating motion. Two hydrofoils have a tandem spatial configuration because in such arrangement allows the turbine to reach a higher efficiency. An iteration scheme is compiled to the solver to solve the coupling equations which are built based on the hydrofoil motions and hydrodynamic. Numerical investigations are carried out to analyze the response of 2 tandem oscillating hydrofoils. The objective of this study is to review the wake interactions between 2 tandem hydrofoils rather than to pursue the maximum power efficiency. It is found that the heaving amplitude and energy extraction performance of the upstream hydrofoil are greater than that of downstream hydrofoil, because the downstream hydrofoil operates in the wake of upstream hydrofoil. For the fully flow‐induced motion model presented in this study, both favorable and unfavorable strong wake vortices interactions are not observed, because the change in system response will weaken the strong wake interaction. Moreover, the difference between 2 hydrofoil responses does not necessarily decline with increasing inter‐hydrofoil spacing. The response of 2 flow‐induced oscillating hydrofoils with a tandem arrangement is reviewed. Strong wake interactions are not observed for the fully flow‐induced system. Increasing inter‐hydrofoil spacing does not necessarily narrow the difference of 2 hydrofoil response.

We study numerically the propulsive wakes produced by a flapping foil. Both pure pitching and pure heaving motions are considered, respectively, at a fixed Reynolds number of Re = 1700. As the major innovation of this paper, we find an interesting coincidence that the efficiency maximum agrees well with the 2D-3D transition boundary, by plotting the contours of propulsive efficiency in the frequency-amplitude parametric space and comparing to the transition boundaries. Although there is a lack of direct 3D simulations, it is reasonable to conjecture that the propulsive efficiency increases with Strouhal number until the wake transits from a 2D state to a 3D state. By comparing between the pure pitching motion and the pure heaving motion, we find that the 2D-3D transition occurs earlier for the pure heaving foil than that of the pure pitching foil. Consequently, the efficiency for the pure heaving foil peaks more closely to the wake deflection boundary than that of the pure pitching foil. Furthermore, since we have drawn the maps on the same parametric space with the same Reynolds number, it is possible to make a direct comparison in the propulsive efficiency between a pure pitching foil and a pure heaving foil. We note that the maximum efficiency for a pure pitching foil is 15.6%, and that of a pure heaving foil is 17%, indicating that the pure heaving foil has a slightly better propulsive performance than that of the pure pitching foil for the currently studied Reynolds number.

The modified immersed boundary method is introduced and applied to study the propulsive mechanism of a tandem flapping wings system. The effects of tandem wings distance and phase lag between the two flapping wings are investigated. Thrust force of the upstream wing is nearly constant and close to the magnitude of single flapping wing system. Thrust force of second wing is influenced by the distance and phase lag. With specific parameters, the second wing can obtain a maximum thrust which is larger than the one of first wing. The flow structures of the wake flow are classified into three different formations, and they are correlated to the trends of thrust force. The effects of distance and phase lag are coupled other than isolated. It is possible to lower down the power consumption of this tandem flapping wings system and enhance the total thrust force of the system at the same time.

We performed numerical experiments on a one-dimensional elastic solid oscillating in a two-dimensional viscous incompressible fluid with the intent of discerning the interplay of vorticity and elastodynamics in flapping wing propulsion. Perhaps for the first time, we have established the role of foil deflection topology and its influence on vorticity generation, through spatially and temporally evolving foil slope and curvature. Though the frequency of oscillation of the foil has a definite role, it is the phase relation between foil slope and pressure that determines thrust or drag. Similarly, the phase difference between flapping velocity, and pressure and inertial forces, determine the power input to the foil, and in turn drives propulsive efficiency. At low frequencies of oscillation, the sympathetic slope and curvature of deformation of the foil allow generation of leading-edge vortices that do not separate; they cause substantial rise in pressure between the leading edge and mid-chord. The circulatory component of pressure is determined primarily by the leading-edge vortex and therefore thrust too is predominantly circulatory in origin at low frequencies. In the intermediate and high-frequency range, thrust and drag on the foil spatially alternate and non-circulatory forces dominate over circulatory and viscous forces. For the mass ratios we simulated, thrust due to flapping varies quadratically as a function of Strouhal number or trailing-edge flapping velocity; further, the trailing edge flapping velocities peak at the same set of frequencies where the thrust is also a maximum. Propulsive efficiency, on the other hand, is roughly a mirror image of the thrust variation with respect to Strouhal number. Given that most instances of flapping propulsion in nature are primarily through distributed muscular actuation that enables precise control of deformation shape, leading to high thrust and efficiency, the results presented here are pointers towards understanding some of the mechanisms that drive thrust and propulsive efficiency.

Fish schools and bird flocks are fascinating examples of collective behaviours in which many individuals generate and interact with complex flows. Motivated by animal groups on the move, here we explore how the locomotion of many bodies emerges from their flow-mediated interactions. Through experiments and simulations of arrays of flapping wings that propel within a collective wake, we discover distinct modes characterized by the group swimming speed and the spatial phase shift between trajectories of neighbouring wings. For identical flapping motions, slow and fast modes coexist and correspond to constructive and destructive wing-wake interactions. Simulations show that swimming in a group can enhance speed and save power, and we capture the key phenomena in a mathematical model based on memory or the storage and recollection of information in the flow field. These results also show that fluid dynamic interactions alone are sufficient to generate coherent collective locomotion, and thus might suggest new ways to characterize the role of flows in animal groups.

The phase difference and spacing distance between two plunging wings in tandem affect the interaction of the vortices between the forewing and hindwing. An experimental study is performed in a low Reynolds number water tunnel to investigate how this interaction changes the mean thrust coefficient and propulsive efficiency. A three-dimensional force sensor and two-dimensional digital particle image velocimetry are used to measure the wing thrust, lift force, and leading edge vortex around the wings. The mean thrust coefficient of the forewing nearly changes sinusoidally with the phase difference in the range of 0–360 deg. The increase in the mean thrust coefficient of the forewing is caused by the leading edge vortex and stagnation region of the hindwing, enhancing the jet velocity behind the forewing and its effective angle of attack. The curve of the mean thrust coefficient of the hindwing has a V-shaped bottom. The decrease in the mean thrust coefficient of the hindwing is caused by the vortex shed from the forewing restraining the leading edge vortex formation of the hindwing, reducing its effective angle of attack. The propulsive efficiency of the hindwing shows a large variation range compared to that of the forewing.

Numerical simulations have been used to analyze the equivalence of pitching and plunging motions found in a flapping NACA0012 airfoil. Two-dimensional incompressible Navier–Stokes equations are solved at Reynolds number of 1,000 over a range of Strouhal numbers. A novel criterion based on the Strouhal number is proposed which provides equivalence of pitching and plunging motions using the length scale traversed by the trailing edge in each case. Aerodynamic coefficients are found to match well for both the kinematics in temporal as well as spectral domains. Detailed analysis provides contribution of different mechanisms, such as vortex shedding, added mass, interaction of leading and trailing edge vortices, in the overall aerodynamic forces produced by a pitching or plunging airfoil. Wake deflection is observed for a plunging airfoil at high Strouhal numbers resulting in a bias in the lift coefficient. Further investigation reveals the dominance of second harmonic of the fundamental frequency in the lift spectrum emphasizing the role of quadratic nonlinearity in the observed phenomenon.

The phenomenon of low amplitude self-sustained pitch oscillations in the transitional Reynolds number regime is studied experimentally, and numerically through unsteady, two-dimensional simulations. Experimental results on the effects of inertia and boundary layer tripping are shown. Using the SST k - ω model, numerical results on the Reynolds number effects are presented. This paper reveals how viscous effects play a critical role on the aeroelastic oscillations. Laminar separation at the trailing edge initiates the oscillations and, as the separation moves upstream over one side of the airfoil, it contributes to limit the amplitude of oscillation.

Force and particle image velocimetry measurements were conducted on a NACA 0012 airfoil undergoing small-amplitude sinusoidal plunge oscillations at a poststall angle of attack and Reynolds number of 10,000. With increasing frequency of oscillation, lift increases and drag decreases due to the leading-edge vortices shed and convected over the suction surface of the airfoil. Within this regime, the lift coefficient increases approximately linearly with the normalized plunge velocity. Local maxima occur in the lift coefficient due to the resonance with the most unstable wake frequency, its subharmonic and first harmonic, producing the most efficient conditions for high-lift generation. At higher frequencies, a second mode of flowfield occurs. The leading-edge vortex remains nearer the leading edge of the airfoil and loses its coherency through impingement with the upward-moving airfoil. To capture this impingement process, high-fidelity computational simulations were performed that showed the highly transitional nature of the flow and a strong interaction between the upper and lower-surface vortices. A sudden loss of lift may also occur at high frequencies for larger amplitudes in this mode.

It is the objective of this paper to review recent developments in the understanding and prediction of flapping-wing aerodynamics. To this end, several flapping-wing configurations are considered. First, the problem of single flapping wings is treated with special emphasis on the dependence of thrust, lift, and propulsive efficiency on flapping mode, amplitude, frequency, and wing shape. Second, the problem of hovering flight is studied for single flapping wings. Third, the aerodynamic phenomena and benefits produced by the flapping-wing interactions on tandem wings or biplane configurations are discussed. Such interactions occur on dragonflies or on a recently developed micro air vehicle. The currently available two- and three-dimensional inviscid and viscous flapping-wing flow solutions are presented. It is shown that the results are strongly dependent on flapping frequency, amplitude, and Reynolds number. These findings are substantiated by comparison with the available experimental data.

A wing that is heaving and pitching simultaneously may extract energy from an oncoming How, thus acting as a turbine. The theoretical performance of such a concept is investigated here through unsteady two-dimensional laminar-flow simulations using the commercial finite volume computational fluid dynamics code FLUENT. Computations are performed in the heaving reference frame of the airfoil, thus leaving only the pitching motion of the airfoil to be dealt with through a rigid-body mesh rotation and a circular nonconformal sliding interface. Unsteady aerodynamics basics of the oscillating airfoil are first exposed, with a description of the operating regimes. Effects of unsteadiness are stressed and the inadequacy of a quasi-steady approach to take them into account is exposed. We present a mapping of power-extraction efficiency for a single oscillating airfoil in the frequency and pitching-amplitude domain: 0 < fc/U-infinity < 0.25 and 0 < theta(0) < 90 deg for a NACA 0015 airfoil at a Reynolds number of Re = 1100, a heaving amplitude of one chord (H-0 = c), and a pitching axis at the third chord (x(p) = c/3). Remarkably, efficiency as high as 34% is observed, as well as a large parametric region at theta(0) > 55 deg in which efficiencies are higher than 20%. Results from a parametric study are then provided and discussed. It is found that motion-related parameters such as heaving amplitude and frequency have the strongest effects on airfoil performances, whereas geometry and viscous parameters turn out to play a secondary role.

Motivated by our interest in unsteady aerodynamics of insect flight, we devise a computational tool to solve the Navier Stokes equation around a two-dimensional moving wing, which mimics biological locomotion. The focus of the present work is frequency selection in forward flapping flight. We investigate the time scales associated with the shedding of the trailing- and leading-edge vortices, as well as the corresponding time-dependent forces. We present a generic mechanism of the frequency selection as a result of unsteady aerodynamics.

Flapping hydrofoils in tandem configuration find applications in wave gliders, dragonfly, dorsal-tail fin interaction in fishes, among others. The flapping motion consists of a combination of heaving and pitching motion. This type of motion involves complex interaction of the vortices shed from the upstream hydrofoil with the downstream hydrofoil, thus influencing the performance of the downstream hydrofoil. A two-dimensional stabilized finite element moving mesh framework is utilized for the current study. The important parameters which influence the flow interactions are the chord size ratio and the gap between the hydrofoils. The size ratio is defined as the ratio of the chord of the upstream hydrofoil to that of the downstream hydrofoil. The size ratio is varied from 0.25 to 1. The gap is varied from one chord length to 3 chord lengths of the downstream foil. The study focuses on the effect of the size ratio, gap and flapping kinematics based on sinusoidal heaving and pitching motion on the detailed flow dynamics of the tandem hydrofoils. The effect on the thrust coefficient and hydrodynamic efficiency is explored and compared with that of an isolated hydrofoil. The results obtained from the study can pave way for a better understanding with regard to engineering designs based on biomimetics.

Fish schooling with stable configurations is intriguing. How individuals benefit from hydrodynamic interactions is still an open question. Here, fish are modeled as undulatory self-propelled foils, which is more realistic. The collective locomotion of two foils in a tandem configuration with different amplitude ratios Ar and frequency ratios Fr is considered. Depending on Ar and Fr, the two foils without lateral or yaw motion may spontaneously form stable configurations, separate, or collide with each other. The phase diagram of the locomotion modes in the (Fr, Ar) plane is obtained, which is significantly different from that in Newbolt et al. [“Flow interactions between uncoordinated flapping swimmers give rise to group cohesion,” Proc. Natl. Acad. Sci. U. S. A. 116, 2419 (2019)]. For stable configurations, the gap spacing may be almost constant [stable position (SP) mode] or change dynamically and periodically [stable cycle (SC) mode]. In our diagram, the fast SP mode is found. Besides, the border between the separation and SP/SC modes is more realistic. In the fast SP cases, analyses of hydrodynamic force show the phenomenon of inverted drafting, in which the leader achieves hydrodynamic advantages. For the SC mode, the cruising speed increases piecewise linearly with FrAr. When Ar < 1, the linear slope is identical to that of the isolated leader, and the follower-control mechanism is revealed. Our result sheds some light on fish schooling and predating.

Over the past few decades, energy generation from piezoelectric patches mounted on a flexible flat plate exhibiting flapping motion has gained attention. Piezoelectric patches are generally multilayered consisting of piezoelectric, substrate, and electrode layers placed on top of each other. Although the flapping dynamics of single-layered structures have been extensively studied, understanding the flapping dynamics of multilayered structures is minimal. We propose a quasi-monolithic formulation with exact interface tracking to simulate the fluid–multilayered structure interactions. The proposed formulation is validated by considering a simple two-layered plate-like structure with identical material properties against a single-layered plate. We then use this formulation to perform parametric simulations by providing different material properties to each layer of the plate to understand the effect of differences in the material properties on the flapping dynamics. The simulations are performed by selecting different values of Young’s modulus and density for each of the layers such that the average structure-to-fluid mass ratio m*avg=0.1 and the average non-dimensional bending stiffness KBavg=0.0005 remain constant for a Reynolds number Re = 1000. First, the effects of difference in elasticity between the two layers on the flapping amplitude, frequency, forces, and vortex shedding patterns are investigated. Following this, the effect of differences in elastic properties on the onset of flapping is investigated for a case with Re = 1000, m*avg=0.1, and KBavg=0.0008, for which a single-layered plate does not undergo self-sustained flapping. Two distinct response regimes are observed depending on the difference in elastic properties between the two layers: (I) fixed-point stable and (II) periodic limit cycle oscillations. Finally, we look into the effects of structural density differences on the flapping dynamics of a two-layered plate.

We show how phasing between tandem bioinspired fins flapping at high-stroke amplitudes modulates rear fin thrust production and wake characteristics. Load cell thrust measurements show that the rear fin generates 25% more thrust than the front fin when it lags the latter by a quarter cycle, and performs 8% worse when it leads the front fin by the same amount. The flow interactions between the fins responsible for these observations are analyzed using two-dimensional particle image velocimetry measurements and three-dimensional computational fluid dynamics simulations. Distributions of velocity elucidate variations in the effective flow induced on the rear fin for different phase offsets. Vortex structure interactions and particle rakes reveal the contributions of the leading- and trailing-edge vortices shed by the front fin in modulating the suction at the rear fin leading edge. Furthermore, the wake structure far downstream of the fins changes in its coherence, axial and radial extents for the different phase offsets. These findings are relevant for the design and performance optimization of various unmanned underwater vehicles that utilize such tandem systems.

Symmetric flapping foils are known to produce deflected jets at high frequency–amplitude combinations even at a zero mean angle of attack. This reduces the frequency range of useful propulsive configurations without side force. In this study, we numerically analyse the interaction of these deflected jets for tandem flapping foils undergoing coupled heave-to-pitch motion in a two-dimensional domain. The impact of the flapping Strouhal number, foil spacing and phasing on wake interaction is investigated. Our primary finding is that the back foil is capable of cancelling the wake deflection and mean side force of the front foil, even when located up to five chord lengths downstream. This is achieved by attracting the incoming dipoles and disturbing their cohesion within the limits of the back foil’s range of flapping motion. We also show that the impact on cycle-averaged thrust varies from high augmentation to drag generation depending on the wake patterns downstream of the back foil. These findings provide new insights towards the design of biomimetic tandem propulsors, as they expand their working envelope and ability to rapidly increase or decrease the forward speed by manipulating the size of the shed vortices.

Unique propulsion systems have evolved in fish that interact with the effects on the surrounding fluid of upstream fish. The downstream fish utilize these complex interactions to swim efficiently. The immersed boundary method is used to explore the phase-mediated locomotion of two self-propelled flexible plates in a tandem arrangement. The interactions caused by the phase difference are elucidated, and the hydrodynamic benefits obtained from the phase-mediated interactions are scrutinized. The variations with the phase difference (Δϕ) and initial gap distance (Gx,0) in the average cruising speed (ŪC), the average input power (P¯), the swimming efficiency (η), and the equilibrium gap distance (G¯x) are determined. Three flapping modes are identified: a tandem flapping mode, a closely mediated flapping mode, and an interfered flapping mode. The propulsion mechanisms in these modes are analyzed in detail in terms of Δϕ and Gx,0. ŪC and η are increased by more than 45% in the closely mediated flapping mode. The vortical structures are visualized to characterize the three flapping modes qualitatively.

Hydrodynamic behavior of two-dimensional tandem-arranged flapping flexible foils in uniform flow is investigated numerically by an immersed boundary-lattice Boltzmann method. The leading edge of the leading foil is forced to undergo both heave and pitch motions, while the leading edge of the trailing foil is forced to undergo heave motion only. Of particular interests are the effects of stream-wise gap distance Gx (Gx/c = 0.25–1.75, where c denotes the length of the foil) and the phase difference Φ between the heave motions of the foils (Φ/π = −1.00 to 1.00) on the hydrodynamic characteristics of the foils, such as the propulsive force, the propulsive efficiency, the passive deformation, and the flow field around the foils. For the leading foil, because of the existence of the trailing foil and the resulting gap flow between the foils, the propulsive performance is noticeably influenced by Φ at small Gx/c values and such an influence is weakened with increasing Gx/c. For the trailing foil, the propulsive performance is primarily affected by Φ, and the physics behind such a strong effect is that Φ dictates the manner by which the vortices shed from the leading foil interact with the trailing foil. In contrast, the interaction of the vortices shed from the leading foil with the trailing foil is not significantly affected by Gx/c because the trailing foil experiences similar vortices shed from the leading foil, regardless of Gx/c. With different Gx/c and Φ/π values, three distinct deformation states of the foils, namely, the symmetric periodic state, the asymmetric periodic state, and the irregular state, are identified and are mapped out in the (Φ/π, Gx/c) space. Good correlation between the deformation state of the foils and the propulsive performance of the trailing foil has been observed.

The excellent performance of many creatures using flapping wings has attracted a lot of research on the performance of a single flapping wing. However, many species generally choose highly organized movements rather than alone in the animal world; there is a very popular and interesting biological clustering phenomenon known as schooling. Understanding the flow mechanisms and thrust performance of flapping multiwings in a schooling could be applied to novel bionic flapping wing aircraft formation design. We perform numerical simulations employing the immersed boundary-lattice Boltzmann method for flow over a single flapping wing and the flapping multiwings in a diamond schooling at different St numbers. Meanwhile, the effects of the difference in individual flapping frequency on the overall propulsive performance of the schooling were investigated. We present the spectra of aerodynamic forces for a single flapping wing and each wing in a diamond schooling at different individual flapping frequencies. Numerical results indicate that the flapping frequency has great effects on the thrust performance of a single wing and the multiwings in a schooling. The average thrust coefficient of a single flapping wing grows with the increase in the St. However, there is an optimal St number to obtain the maximum propulsive efficiency. For a schooling that maintains the same flapping frequency, the overall schooling or each wing in a schooling shows the same trend as a single wing. For a schooling with different individual flapping frequencies, the aerodynamic characteristics of the last downstream wing are more affected by the frequency difference.

The effect of phase difference on the collective locomotion of two tandem flapping foils is numerically studied in this paper. The numerical results indicate that the collective locomotion is greatly affected by the phase difference. Two distinct collective modes are observed, i.e., the fast mode and the slow mode. The fast mode is only observed in part of the range of the phase difference (ϕ), i.e., ϕ=0.0−0.1π and 1.5π–1.9π with appropriate initial distance, and the slow mode appears in the range of ϕ=0.3π−1.4π. Meanwhile, the follower of the two foils has hydrodynamic benefit in both fast and slow modes, and it can obtain the highest efficiency in the fast mode at ϕ=1.6π. However, the leader can only achieve hydrodynamic benefit in the fast mode, and the highest efficiency occurs at ϕ=0.1π. In addition, the stable distance between two foils in the slow mode can be quantized with the phase difference. Furthermore, the fluid-structure interactions between two foils are also analyzed. Two distinct vortex interactions are observed in the fast mode, i.e., merging interaction and broken interaction, which, respectively, result in the highest propulsive efficiency for the follower and the leader. In the merging interaction, the leading edge vortex of the leader is captured by the follower, which results in the weak trailing edge vortex of the leader but a strong trailing edge vortex of the follower. In the broken interaction, the leading edge vortex of the follower sheds into the wake together with the trailing edge vortex of the leader, and induces the trailing edge vortex of the follower to be broken into two parts. Which kind of vortex interaction occurs depends on the phase difference. The results obtained here may provide some light on understanding the coordinated behavior of biological collectives.

The hydrodynamics of a three-dimensional self-propelled flexible plate in a quiescent flow were simulated using the immersed boundary method. The clamped leading edge of the flexible plate was forced into a prescribed harmonic oscillation in the vertical direction but was free to move in the horizontal direction. Several types of trapezoidal plates were simulated by changing the shape ratio (S = Wt/Wl), where Wt is the trailing edge width and Wl is the leading edge width. The aspect ratio was fixed at AS = (Wl + Wt)/2L = 0.4, where L is the length of the plate. To explore the hydrodynamics of a rectangular plate (S = 1.0), the average cruising speed (UC), the input power (P), and the swimming efficiency (η) were determined as a function of the flapping frequency (f). The kinematics of the plate, the maximum angle of attack (φmax), and the mean effective length (Leff) were examined to characterize the hydrodynamics, including the peak-to-peak amplitude (At/A) and the Strouhal number (St=fAt/Uc). Next, the effect of S on the hydrodynamics was explored for 0.1 ≤ S ≤ 3.0. The swimming efficiency was found to be the highest at S = 0.5. The maximum thrust (Ft,max) of S = 0.5 decreased by 15% compared to that of S = 1.0, and the maximum lateral force (Fl,max) decreased by more than 50%. The velocity field behind the plate and the vortical structures around the plate were visualized. The influence of the tip vortex on the swimming efficiency was examined.

We present experiments that examine the modes of interaction, the collective performance and the role of three-dimensionality in two pitching propulsors in an in-line arrangement. Both two-dimensional foils and three-dimensional rectangular wings of $AR = 2$ are examined. \kwm{In contrast to previous work, two interaction modes distinguished as the coherent and branched wake modes are not observed to be directly linked to the propulsive efficiency, although they are linked to peak thrust performance and minimum power consumption as previously described \cite[]{boschitsch2014propulsive}.} \kwm{In fact, in closely-spaced propulsors peak propulsive efficiency of the follower occurs near its minimum power and this condition \kwm{ reveals a} branched wake mode. Alternatively, for propulsors spaced far apart peak propulsive efficiency of the follower occurs near its peak thrust and this condition \kwm{reveals a} coherent wake mode.} By examining the collective performance, it is discovered that there is an optimal spacing between the propulsors to maximize the collective efficiency. For two-dimensional foils the optimal spacing of $X^* = 0.75$ and the synchrony of $\phi = 2\pi /3$ leads to a collective efficiency and thrust enhancement of 50\% and 32\%, respectively, as compared to two isolated foils. In comparison, for $AR = 2$ wings the optimal spacing of $X^* = 0.25$ and the synchrony of $\phi = 7\pi /6$ leads to a collective efficiency and thrust enhancement of 30\% and 22\%, respectively. In addition, at the optimal conditions the collective lateral force coefficients in both the two- and three-dimensional cases are negligible, while operating off these conditions can lead to non-negligible lateral forces. Finally, the peak efficiency of the collective and the follower are shown to have opposite trends with increasing spacing in two- and three-dimensional flows. This is correlated to the breakdown of the impinging vortex on the follower wing in three-dimensions. These results can aid in the design of networked bio-inspired control elements that through integrated sensing can synchronize to three-dimensional flow interactions.

The present study investigates the complex vortex interactions in two-dimensional flow-field behind a symmetric NACA0012 airfoil undergoing a prescribed periodic pitching-plunging motion in low Reynolds number regime. The flow-field transitions from periodic to chaotic through a quasi-periodic route as the plunge amplitude is gradually increased. This study unravels the role of the complex interactions that take place among the main vortex structures in making the unsteady flow-field transition from periodicity to chaos. The leading-edge separation plays a key role in providing the very first trigger for aperiodicity. Subsequent mechanisms like shredding, merging, splitting, and collision of vortices in the near-field that propagate and sustain the disturbance have also been followed and presented. These fundamental mechanisms are seen to give rise to spontaneous and irregular formation of new vortex couples at arbitrary locations, which are the primary agencies for sustaining chaos in the flow-field. The interactions have been studied for each dynamical state to understand the course of transition in the flow-field. The qualitative changes observed in the flow-field are manifestation of changes in the underlying dynamical system. The overall dynamics are established in the present study by means of robust quantitative measures derived from classical and non-classical tools from the dynamical system theory. As the present analysis involves a high fidelity multi-unknown system, non-classical dynamical tools such as recurrence-based time series methods are seen to be very efficient. Moreover, their application is novel in the context of pitch-plunge flapping flight. Published by AIP Publishing. https://doi.

We present a novel partitioned iterative formulation for modeling of fluid-structure interaction in two-phase flows. The variational formulation consists of a stable and robust integration of three blocks of differential equations, viz., incompressible viscous fluid, a rigid or flexible structure and two-phase indicator field. The fluid-fluid interface between the two phases, which may have high density and viscosity ratios, is evolved by solving the conservative phase-field Allen-Cahn equation in the arbitrary Lagrangian-Eulerian coordinates. While the Navier-Stokes equations are solved by a stabilized Petrov-Galerkin method, the conservative Allen-Chan phase-field equation is discretized by the positivity preserving variational scheme. Fully decoupled implicit solvers for the two-phase fluid and the structure are integrated by the nonlinear iterative force correction in a staggered partitioned manner. We assess the accuracy and stability of the new phase-field/ALE variational formulation for two- and three-dimensional problems involving the dynamical interaction of rigid bodies with free-surface. We consider the decay test problems of increasing complexity, namely free translational heave decay of a circular cylinder and free rotation of a rectangular barge. Through numerical experiments, we show that the proposed formulation is stable and robust for high density ratios across fluid-fluid interface and for low structure-to-fluid mass ratio with strong added-mass effects. Using three-dimensional unstructured meshes, we demonstrate the second-order temporal accuracy of the coupled phase-field/ALE method. Finally, we demonstrate the three-dimensional phase-field FSI formulation for a practical problem of internal two-phase flow in a flexible circular pipe subjected to vortex-induced vibrations due to external fluid flow.

We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio and using meshes of arbitrary topology. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed mesh with a mass conservative and energy-stable discretization. Mass is conserved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the solution of the phase-field equation. The spatial part of the Lagrange multiplier is written as a mid-point approximation to make the scheme energy-stable. This enables us to form a conservative, energy-stable and positivity preserving scheme. The proposed variational technique reduces spurious and unphysical oscillations in the solution while maintaining second-order spatial accuracy. To model a generic two-phase free-surface flow, we couple the Allen-Cahn phase-field equation with the Navier-Stokes equations. Comparison of results between standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. Standard two-phase flow benchmarks such as Laplace-Young law and sloshing tank problem are carried out to assess the convergence and accuracy of the coupled Navier-Stokes and Allen-Cahn solver. Two- and three-dimensional dam break problem are then solved to assess the scheme for the problem with topological changes of the air-water interface on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical problem of wave-structure interaction in offshore engineering using general three-dimensional unstructured meshes.

The propulsive performance of two flapping foils with the tandem configuration has been analysed. The trajectories of the foils are prescribed with typical propulsive motions. The hind foil performs oscillatory motion in the wake of the fore foil. The local flow around the hind foil and the effective attack angle have been changed by the vortex street. The velocity potential theory and the boundary element method are introduced to study the interactions of the vortices and the foils. The propulsive performance of the tandem flapping foils is affected significantly when various longitudinal distances and phase differences are adopted. The typical vortex interaction modes are investigated in terms of global phase shift. The thrust coefficient and the propulsive efficiency of tandem NACA0012 foils at typical global phases are analysed. The optimal global phase shifts for the highest thrust and highest efficiency have been found.

The propulsive performance of a pair of tandem flapping foils is sensitively dependent on the spacing and phasing between them. Large increases in thrust and efficiency of the hind foil are possible, but the mechanisms governing these enhancements remain largely unresolved. Two-dimensional numerical simulations of tandem and single foils oscillating in heave and pitch at a Reynolds number of 7000 are performed over a broad and dense parameter space, allowing the effects of inter-foil spacing ( $S$ ) and phasing ( $\unicode[STIX]{x1D711}$ ) to be investigated over a range of non-dimensional frequencies (or Strouhal number, $St$ ). Results indicate that the hind foil can produce from no thrust to twice the thrust of a single foil depending on its spacing and phasing with respect to the fore foil, which is consistent with previous studies that were carried out over a limited parameter space. Examination of instantaneous flow fields indicates that high thrust occurs when the hind foil weaves between the vortices that have been shed by the fore foil, and low thrust occurs when the hind foil intercepts these vortices. Contours of high thrust and minimal thrust appear as inclined bands in the $S-\unicode[STIX]{x1D711}$ parameter space and this behaviour is apparent over the entire range of Strouhal numbers considered $(0.2\leqslant St\leqslant 0.5)$ . A novel quasi-steady model that utilises kinematics of a virtual hind foil together with data obtained from simulations of a single flapping foil shows that performance augmentation is primarily determined through modification of the instantaneous angle of attack of the hind foil by the vortex street established by the fore foil. This simple model provides estimates of thrust and efficiency for the hind foil, which is consistent with data obtained through full simulations. The limitations of the virtual hind foil method and its physical significance is also discussed.

This paper reports a fundamental investigation on the aerodynamics of two-dimensional flapping wings in tandem configuration in forward flight. Of particular interest are the effects of phase angle (φ) and center-to-center distance (L) between the front wing and the rear wing on the aerodynamic force generation at a Reynolds number of 5000. Both experimental and numerical methods were employed. A force sensor was used to measure the time-history aerodynamic forces experienced by the two wings and digital particle image velocimetry was utilized to obtain the corresponding flow structures. Both the front wing and the rear wing executed the same simple harmonic motions with φ ranging from -180° to 180° and four values of L, i.e., 1.5c, 2c, 3c, and 4c (c is the wing chord length). Results show that at fixed L = 2c, tandem wings perform better than the sum of two single wings that flap independently in terms of thrust for phase angle approximately from -90° to 90°. The maximum thrust on the rear wing occurs during in-phase flapping (φ = 0°). Correlation of transient thrust and flow structure indicates that there are generally two types of wing-wake interactions, depending on whether the rear wing crosses the shear layer shed from the front wing. Finally, increasing wing spacing has similar effect as reducing the phase angle, and an approximate mathematical model is derived to describe the relationship between these two parameters.

The two dimensional incompressible viscous flow past a flapping rigid foil immersed in a uniform
stream is studied using a lattice-Boltzmann model. When the foil’s center of mass is fixed in space,
numerical results reproduce the transition from the von Kármán (vKm) to the inverted von Kármán
wake [T. Schnipper, A. Andersen, and T. Bohr, “Vortex wakes of a flapping foil,” J. Fluid Mech. 633,
411 (2009) and A. Das, R. K. Shukla, and R. N. Govardhan, “Existence of a sharp transition in the peak
propulsive efficiency of a low pitching foil,” J. Fluid Mech. 800, 307 (2016)]. Beyond the inverted
vKm transition, the foil was released. The numerical results show that the hydrodynamic forces on the
flapper are oscillatory functions of time with amplitudes and mean values that scale with the square
of the Strouhal number, defined with either the flapping amplitude or the flapper length that decays
an order of magnitude when the foil is freed to swim. Upstream swimming consisted of a uniform
horizontal motion and a vertical heaving. The swimming speed showed a linear dependence on the
Strouhal number, defined with the amplitude of oscillation of the foil tip. As a consequence, thrust
generated by the free flapper is related to the square of the swimming speed for moderate Reynolds
numbers. Published by AIP Publishing.

Publisher Summary Discussion of the hydromechanics of swimming depends on the main methods of propulsion, in terms of the modes of movements employed by the body and appendages, as well as other important flow parameters. The hydromechanics of the swimming of fishes and cetaceans has so far been developed on the basis of small amplitude theory. The undulatory motions that most aquatic animals make to propel themselves through water may be divided into two components: one perpendicular and the other tangential to the animal's instantaneous longitudinal axis of centroid. Under natural conditions, the amplitude of the undulatory transverse waves of the body motion is usually moderate or large, even during the optimum performance at high hydromechanical efficiencies. While the simplified slender-body theory and the two-dimensional theory can provide useful information and preliminary physical insight that would be valuable for deeper understanding, it is nevertheless difficult to give an estimate of their degree of accuracy. Further developments in theoretical and experimental approaches are certainly important and desirable.

A numerical investigation based on 2D URANS simulations is performed in order to seek an optimal spatial configuration for two oscillating foils within a hydrokinetic turbine. The objective of the study is to maximize the power extraction efficiency of the turbine. Tandem spatial configurations are considered because in such arrangement both hydrofoils are sharing the same flow window, which allows the turbine to reach higher efficiencies. The relative positioning of the downstream foil oscillating in the wake shed by the upstream hydrofoil is seen to be critical. Indeed, favorable interactions between the downstream foil and the wake vortices may lead to unexpectedly high power-extraction efficiencies (up to 64%), while unfavorable interactions may cause the downstream foil to contribute negatively to the total power extracted. A global phase shift parameter is introduced to characterize the tandem configuration. This parameter combines the inter-foil spacing and motion phase-shift into a single term. It is found useful to predict additional favorable configurations based on known results for cases with similar upstream-foil wake behavior. A comparison with experimental data is provided. Numerical predictions are seen to overpredict the power extraction performance in some cases. This is likely due to the broken 2D coherence of vortices in the 3D reality which affects the vortex-induced velocities and the subsequent foil-wake interactions. [DOI: 10.1115/1.4005423]

In the present study the power extraction possibility by a number of flapping hydrofoils in tandem formation is investigated. A code is developed to predict power extraction capacity for the various number of flapping hydrofoils based on the kinematic and hydrodynamic models. The selected hydrodynamic model follows two dimensional quasi-steady hydrodynamic instability formulation. It is shown that the power extraction is also possible from water stream with the low Reynolds number. As a result of power extraction at low speed flows, the predicted maximum power efficiency is also in lower flapping frequencies. Furthermore, it is found that there are limited number of required flapping hydrofoils in tandem formation, in which the power influence rate drops notably after the second flapping hydrofoil. The flapping hydrofoils at downstream also experience higher hydrodynamic forces, while the flapping hydrofoil kinematics is the key parameter to harness extracted power. As a result of this investigation, the introduced model and code can be used as one of initial tools to predict power capacity for obtaining vast concept regarding tidal sites with the flapping foil hydrokinetic turbines.

A new family of time integration algorithms is presented for solving structural dynamics problems. The new method, denoted as the generalized-alpha method, possesses numerical dissipation that can be controlled by the user. In particular, it is shown that the generalized-alpha method achieves high-frequency dissipation while minimizing unwanted low-frequency dissipation. Comparisons are given of the generalized-alpha method with other numerically dissipative time integration methods, these results highlight the improved performance of the new algorithm. The new algorithm can be easily implemented into programs that already include the Newmark and Hilber-Hughes-Taylor-alpha time integration methods.

The effects of asymmetric sinusoidal motion on pitching airfoil
aerodynamics were studied by numerical simulations for 2-D flow around a
NACA0012 airfoil at Re=1.35×105. Various unsteady
parameters (amplitude of oscillation, d; reduced frequency, k) were
applied to investigate the effect of asymmetry parameter S on the
instantaneous force coefficients and flow patterns. The results reveal
that S has a noticeable effect on the aerodynamic performance, as it
affects the instantaneous force coefficient, maximum lift and drag
coefficient, hysteresis loops and the flow structures.

A number of flying insects make use of tandem-wing configurations, suggesting that such a setup may have potential advantages over a single wing at low Reynolds numbers. Dragonflies, which are fast and highly maneuverable, demonstrate well the potential performance of such a design. In this paper, a tandem-wing flapping configuration is simulated at a Reynolds number of 10,000 using an incompressible Navier-Stokes solver and an overlapping grid method. The flapping motion consists of a simple sinusoidal pitch and plunge motion with a spacing of one chord length between both wings. The arrangement was tested at a Strouhal number of 0.3 for three different phase angles: 0, 90, and 180 deg. The aerodynamics of the hindwing was compared in detail to a single wing, with the same geometry and undergoing the same flapping kinematics, to determine the effect of vortex shedding from the forewing on the hindwing, as well as how the phase angle affects the interaction. The average lift, thrust, and power coefficients and the average efficiency of the fore- and hindwings were compared with a single wing to determine how the tandem-wing interaction affects performance. The results show that adjusting the phase angle allows the tandem wing to change the flight mode. At 0 deg phase lag, the tandem wing produces high thrust at high propulsive efficiency, but low lift efficiency. Switching to 90/180 deg phase lag decreases the thrust production and propulsive efficiency but greatly increases the lift efficiency. At 90/180 deg, the power coefficient is much lower than at 0 deg, due to the hindwing extracting energy from the wake of the forewing.

The flow over an airfoil undergoing pure plunging motion was simulated to study the effect of airfoil thickness, camber and Reynolds number on the thrust generation and propulsive efficiency. The plunging motion of the airfoil was implemented by the introduction of a source term in the Navier-Stokes equations. At a Reynolds number of 20,000, for symmetric airfoil shapes, it was found that increasing the thickness of the airfoil can benefit both thrust generation and propulsive efficiency. When the airfoil thickness was increased from 6% to 15%, the time-averaged thrust coefficient and propulsive efficiency were increased by almost 200%. For a cambered airfoil, around 6-10% loss in both thrust coefficient and propulsive efficiency was found compared to a symmetric airfoil. By varying Reynolds number from 20 to 200,000 for flow over a plunging NACA0015 airfoil, it has been found that both the thrust coefficient and propulsive efficiency increase with increasing Reynolds number.

We simulate the incompressible, viscous flow over a two-dimensional NACA0012 airfoil oscillating in heave at mean incidences 12°<α¯<20° and Reynolds numbers 800 ⩽ Re ⩽ 104. The two-dimensional Navier–Stokes equations are solved using a Spectral/hp Element Method for the spatial discretization and a high-order splitting scheme for the evolution in time. A moving-frame of reference technique accounts for the airfoil motion. We consider the effects on the aerodynamical flow and the force coefficients caused by the variation of the mean incidence, the Reynolds number and the sinusoidal heave motion of the airfoil. The numerical simulations are in good agreement with previously published experimental and computational work, in particular the increase in the force coefficients due to the increase in the Reynolds number and/or the mean incidence are confirmed by the present study. Furthermore, we present here new details of the spatio-temporal non-linear flow pattern evolution where for the first time the Spectral/hp Element Method associated with the moving frame of reference is used for this kind of flow.

Quantitative evaluation of time dependent flow structures around and in the near-wake of an oscillating airfoil is investigated using the Digital Particle Image Velocimetry (DPIV) technique to perform a detailed categorization of vortex formations in the reduced frequency range of 0.16≤k≤6.26 corresponding to Strouhal number range of 0.05≤St≤1.0. The SD7003 airfoil model known to be optimized for low Reynolds number flows undergoes a combined motion where the pitch leads the plunge motion by ψ=π/2 in a steady current. Five flow structure categories are identified depending on the role of separated vortex structures from the leading and trailing edges. The occurrence of flow structure categories on different two-dimensional parameter spaces is obtained. It is also found that the categorizations are independent of the Reynolds number for the investigated range.

The thrust producing performance and efficiency of an SD8020 foil hydrofoil that undergoes rotational and translational oscillating motions was studied and optimized through force and torque measurement and dye flow visualization, in the water tunnel at low Reynolds number of 13,000-16,000. The foil was set into pitching and heaving motion under different oscillation patterns to mimick the flapping and swimming motion of the marine creatures. The force and moment data were collected and used as optimization basis for best flapping motion combination. The propulsive efficiency and thrust coefficient of the pitching foil were determined as a function of the Strouhal number, pitch amplitude and angular frequency. Based on the force and efficiency data collected for the pure pitching motion, increasing pitch amplitude and angular frequency was associated with a decrease in propulsive efficiency and an increase in thrust forces produced. A high propulsive efficiency of 70%, accompanied by a thrust coefficient of order one was found at a pitch amplitude of 30 and angular frequency of 0.873 rad/s, Strouhal number of 0.24, and freestream of 0.1368 m/s (Reynolds number of 16416). This presented the best conditions for thrust production observed at low Strouhal and Reynolds numbers.

Water tunnel tests of a NACA 0012 airfoil that was oscillated sinusoidally in plunge are described. The flowfield downstream of the airfoil was explored by dye flow visualization and single component laser Doppler velocimetry (LDV) measurements for a range of freestream speeds, frequencies, and amplitudes of oscillation. The dye visualizations show that the vortex patterns generated by the plunging airfoil change from drag-producing wake flows to thrust-producing jet flows as soon as the ratio of maximum plunge velocity to freestream speed, i.e., the nondimensional plunge velocity, exceeds approximately 0.4. The LDV measurements show that the nondimensional plunge velocity is the appropriate parameter to collapse the maximum streamwise velocity data covering a nondimensional plunge velocity range from 0.18 to 9.3. The maximum streamwise velocity at a given streamwise distance downstream starts to exceed the freestream speed as soon as the nondimensional plunge velocity exceeds 0.25, Furthermore, this maximum jet velocity has been shown to be a linear function of the nondimensional plunge velocity.

A set of pure sinusoidal pitch and plunge oscillations for a symmetric airfoil is studied in this paper A gridless Lagrangian vorticity particle-based solver is used to simulate the unsteady flowfield Various kinematics chosen here are inspired by the experimental and computational studies presented in earlier works The Lagrangian tool is used to accurately reproduce the flowfield obtained by earlier experimental results It follows the experimental trend better compared with some of the grid-based solvers' results from earlier studies The strength of this tool is its grid-free nature, as the resolution of the grid is a crucial parameter in resolving the unsteady flowfield accurately This study also investigates the effects of the starting condition in dictating the wake deflection mode at short and long terms and confirms the findings of the earlier works The effect of mean angle of attack on the wake deflection is also highlighted One of the main questions considered here is whether there exists a kinematic equivalence between sinusoidal pitch and plunge Effective angle of attack, quasi-steady criterion, and Theodorsen's criterion of unsteady aerodynamics have been considered here for kinematic equivalence with plunge The latter two give a reasonably good match both in terms of flowfield and load as was also reported in the literature This behavior is seen in two different plunge stroke amplitude cases studied here The success of the quasi-steady and Theodorsen's approach indicates the general wake behavior being an inviscid phenomenon, at least for the chosen range of parameters

SYNOPSIS. Locomotion is the result of transfer of momentum from the fish musculature to the surrounding water. The present paper discusses some basic principles of this momentum transfer and shows the effects of various adaptations of body shape and fin shape, size and positioning.
Muscles take up a large part of the fish body volume in many cases. The effects of distribution of muscle mass on external shape, and drag (with its reciprocal influence on the muscular system) are analysed. Fins provide an effective means of momentum transfer, by allowing large amounts of water to be moved by small body masses. Fin shape, variable flexibility and positioning all interact to influence thrust producing performance. A framework for understanding the various combinations of fins, their shapes and motion is presented. Reasons for shifting the center of propulsion to the rear part of the fish, in anguilliform, and much more so in carangiform swimmers are discussed. This shift is shown to result from considerations of propulsive efficiency. Double-tailed fin configurations, defined as dorsal and ventral fins placed at the same longitudinal positions so as to produce a “continuous” are analysed. Examples of both fast starters (such as esocids) and cruising species (scombrids, etc.) are used to point out the advantages of such fin placement.

The starting flows past a two-dimensional oscillating and translating airfoil are investigated by visualization experiments and numerical calculations. The airfoil, elliptic in cross-section, is set in motion impulsively and subjected simultaneously to a steady translation and a harmonic oscillation in pitch. The incidence of the airfoil is variable between 0° and 45° and the Reynolds number based on the chord length is between 1500 and 10000. The main object of the present study is to reveal some marked characteristics of the unsteady vortices produced from the oscillating airfoil set at large incidences in excess of the static stall angle. Another purpose is to examine, in some detail, the respective and combined effects of the major experimental parameters on the vortex wake development. It is shown that, in general, the dominant parameter of the flow is the reduced frequency not only when the airfoil oscillates at incidences close to the static stall angle but also at larger incidences. It is also demonstrated that, as the pitching frequency is increased, the patterns of the vortex wake are dependent on the product of the reduced frequency and the amplitude rather than on the frequency itself. It is noted that the combined effect of a high reduced frequency and a large amplitude can give rise to cyclic superposition of leading-edge vortices from which a gradually expanding standing vortex is developed on the upper surface.