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Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude and frequency because the transition also depends sensitively on other details of the kinematics. In this work we replace with the cycle-averaged swept trajectory of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal . Physically, the product expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds . The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.

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... Lighthill [5] studied the biofluiddynamic behaviour of aquatic animal propulsion modes, and concluded that the formation of the jet-like wake, also known as the reverse Von Karman vortex street [6], is an important phenomenon in the generation of 25 thrust. This idea was subsequently confirmed by Koochesfahani et al. [7], Jones et al. [8], Buchholz et al. [9], Godoy-Diana et al. [10] and Lagopoulos et al. [11]. ...

... When a reversed Bénard von Kármán vortex street develops (RvK), the foil experiences thrust due to the 115 higher velocity of the flow in the wake than the ambient/oncoming flow. This drag-to-thrust wake transition stage has been extensively studied for flapping foils in uniform flows, and the foil transition towards thrust occurs with a delay relative to the drag-to-thrust wake transition stage [11,36,37,38]. Fig.7 illustrates three fundamental wake patterns, where U wake is the velocity of the wake and U ∞ is the characteristic flow velocity. ...

... RvK street, with a single factor. Recently, a path-length-based Strouhal number Eq.8c was proposed by Lagopoulos et al. [11], which should be higher than 1 in order for the foil to produce thrust. In contrast, Read et al. [34] reported optimum thrust generation for 0.2 < St A < 0.4. ...

... The jet-like wake of Lighthill is the well known reverse Von Karman vortex street, proposed by Von Karman and Burgers [7]. Following Lighthill [6], this presence of this mechanism was numerically and experimentally confirmed by several researchers like, Koochesfahani et al. [8], Jones et al. [9], Buchholz et al. [10], Godoy-Diana et al. [11], Lagopoulos et al. [12] and Ma et al. [13]. ...

... These findings were aligned with the observations and the results of Isshiki [14] and Grue et al. [17]. Even more recently Lagopoulos et al. [12] applied a boundary data immersion method (BDIM) to study the drag-to-thrust wake transition for hydrofoils submerged in uniform flows. St T , a new version of Strouhal's number using the foil kinematics was proposed for predicting the wake conditions. ...

... While, upon the development of a reversed Bénard von Kármán vortex street (RvK) the foil experiences thrust since the velocity of the flow in the wake becomes larger then the ambient / oncoming flow velocity. For flapping foils in uniform flows, these drag-to-thrust wake transition stages have been extensively studied and the foil's transition towards thrust has been shown to occur with a delay compared with the drag-to-thrust wake transition [12,40,41,42]. The three fundamental wake patterns are illustrated in Fig.5. ...

Submerged flapping foils can convert wave energy directly into thrust, which could be potentially utilised for green marine propulsion. This study analyses the wave-induced flapping hydrofoil propulsion using an in-house developed, new computational fluid dynamics (CFD) framework. The numerical model was initially validated against a few benchmarked problems and then used for the numerical investigation of wave-induced flapping hydrofoil propulsion. The transition between drag and thrust can be observed from the vortex flow pattern. The pitch stiffness and other physical parameter were non-dimensionalised for the first time. The optimal wave conditions and the optimal pitch stiffness are given for the future green marine system design.

... The jet-like wake of Lighthill is the well known reverse Von Karman vortex street, proposed by Von Karman and Burgers [7]. Following Lighthill [6], this presence of this mechanism was numerically and experimentally confirmed by several researchers like, Koochesfahani et al. [8], Jones et al. [9], Buchholz et al. [10], Godoy-Diana et al. [11], Lagopoulos et al. [12] and Ma et al. [13]. ...

... These findings were aligned with the observations and the results of Isshiki [14] and Grue et al. [17]. Even more recently Lagopoulos et al. [12] applied a boundary data immersion method (BDIM) to study the drag-to-thrust wake transition for hydrofoils submerged in uniform flows. St T , a new version of Strouhal's number using the foil kinematics was proposed for predicting the wake conditions. ...

... While, upon the development of a reversed Bénard von Kármán vortex street (RvK) the foil experiences thrust since the velocity of the flow in the wake becomes larger then the ambient / oncoming flow velocity. For flapping foils in uniform flows, these drag-to-thrust wake transition stages have been extensively studied and the foil's transition towards thrust has been shown to occur with a delay compared with the drag-to-thrust wake transition [12,40,41,42]. The three fundamental wake patterns are illustrated in Fig.5. ...

Submerged flapping foils can convert wave energy directly into thrust, which could be potentially utilised for green marine propulsion. This study analyses the wave-induced flapping hydrofoil propulsion using an in-house developed, new computational fluid dynamics (CFD) framework. The numerical model was initially validated against a few benchmarked problems and then used for the numerical investigation of wave-induced flapping hydrofoil propulsion. The transition between drag and thrust can be observed from the vortex flow pattern. The pitch stiffness and other physical parameter were non-dimensionalised for the first time. The optimal wave conditions and the optimal pitch stiffness are given for the future green marine system design.

... We further limited the detailed analyses of the wake characteristics to the cases Re = 1000 and 4000, where the wake features and transition effects were more profound without the additional complexities associated with higher Reynolds number flows (Zurman-Nasution, Ganapathisubramani & Weymouth 2020). This choice of Re also followed closely with assessments conducted by Godoy-Diana et al. (2008), Schnipper et al. (2009), Deng, Sun & Shao (2015, Das et al. (2016), Lagopoulos et al. (2019) and Cimarelli et al. (2021). ...

... This further justified the parameter space considered in the current study. Note that the most propulsively efficient phase offset of 90 • according to Anderson et al. (1998), which was also employed by Zheng et al. (2019) and Lagopoulos et al. (2019), is equivalent to φ = 270 • in the current study following the reference coordinate system employed by Van Buren et al. (2019). ...

... Figure 6(d) depicts further variations in the peak angle of attack (α o ) with respect to increasing phase offset and St c . The effective angle of attack for a foil with combined heaving and pitching motion is the resultant of both pitch-and heave-induced angle of attack (Lagopoulos et al. 2019). This is expressed mathematically as ...

Wake evolution of an oscillating foil with combined heaving and pitching motion is evaluated numerically for a range of phase offsets ( $\phi$ ), chord-based Strouhal numbers ( $St_c$ ) and Reynolds numbers ( $Re$ ). The increase in $\phi$ from $90^\circ$ to $180^\circ$ at a given $St_c$ and $Re$ coincides with a transition of pitch- to heave-dominated kinematics that further reveals novel transitions in wake topology characterized by bifurcated vortex streets. At $Re= 1000$ , each of the dual streets constitutes a dipole-like paired configuration of counter-rotating coherent structures that resemble qualitatively the formation of $2P$ mode. A new mathematical relation between the relative circulation of coherent dipole-like paired structures and kinematic parameters is proposed, including heave-based ( $St_h$ ), pitch-based ( $St_{\theta }$ ) and combined motion ( $St_A$ ) Strouhal numbers, as well as $\phi$ . This model can predict accurately the wake transition towards $2P$ mode characterized by a bifurcation, at low $Re= 1000$ . At $Re= 4000$ , however, the relationship was inaccurate in predicting the wake transition. A shear splitting process is observed at $Re= 4000$ , which leads to the formation of reverse Bénard–von Kármán mode in conjunction with $2P$ mode. Increasing $\phi$ further depicts a consistent prolongation of the splitting process, which coincides with a unique transition in terms of absence and reappearance of bifurcated dipole-like pairs at $\phi = 120^\circ$ and $180^\circ$ , respectively. Changes in the spatial arrangement of $2P$ pairs observed consistently for oscillating foils with the combined motion constitute a novel wake transition that becomes more dominant at higher Reynolds numbers.

... A recent field of investigation on the flapping foil/plate systems concerns the thrust generation and its relation to wake pattern evolution, the major interrogation being about how to parameterize the drag-to-thrust transitions (DTT) and the wake transitions (WT) (Andersen et al. 2017;Lagopoulos, Weymouth & Ganapathisubramani 2019;Chao, Alam & Ji 2021). Although the inviscid theory predicts a DTT as the wake changes from the von Kármán wake to reversed von Kármán wake (von Kármán & Burgers 1935), the observations in experiments and simulations have revealed a lag between the WT and the DTT, due to the need to overcome the viscous and differential pressure resistance Andersen et al. 2017;Lagopoulos et al. 2019). ...

... A recent field of investigation on the flapping foil/plate systems concerns the thrust generation and its relation to wake pattern evolution, the major interrogation being about how to parameterize the drag-to-thrust transitions (DTT) and the wake transitions (WT) (Andersen et al. 2017;Lagopoulos, Weymouth & Ganapathisubramani 2019;Chao, Alam & Ji 2021). Although the inviscid theory predicts a DTT as the wake changes from the von Kármán wake to reversed von Kármán wake (von Kármán & Burgers 1935), the observations in experiments and simulations have revealed a lag between the WT and the DTT, due to the need to overcome the viscous and differential pressure resistance Andersen et al. 2017;Lagopoulos et al. 2019). For example, the study presented by Andersen et al. (2017) has shown that, for a flapping rigid foil in a two-dimensional flow, the WT occurs at St A = 0.18, which precedes the DTT at St A = 0.28. ...

... The amplitude-based Strouhal number St A = 2Af /U, where U is the speed of uniform inflow, A and f are the flapping amplitude and frequency, respectively. Lagopoulos et al. (2019) further scaled the WT between the classical von Kármán wake and the reversed von Kármán wake for the flapping rigid foils with various kinematics, by introducing the chord-averaged trajectory length, T . Results from the wake maps indicated that the onset of WT leads to St T = T f /U → 1, which offers a physical insight into the wake development. ...

The propulsion of a pitching flexible plate in a uniform flow is investigated numerically. The effects of bending stiffness ( $K$ ), pitching amplitude ( $A_L$ ) and frequency ( $St$ ) on the wake patterns, thrust generations and propulsive performances of the fluid–plate system are analysed. Four typical wake patterns, i.e. von Kármán, reversed von Kármán, deflected and chaotic wakes, emerge from various kinematics, and the $St-A_L$ wake maps are given for various $K$ . The drag-to-thrust transitions (DTT) and the wake transitions (WT) between the von Kármán and reversed von Kármán wakes are examined. Results indicate that the WT and DTT boundaries can be scaled by the chord-averaged distance of travel, $\mathcal {L}$ , which leads to $\mathcal {L}\times St \approx 1$ and $\mathcal {L}\times St \approx 1.2$ , respectively. Further, the resonance mechanism for the performance enhancement is revealed and confirmed in a wide range of parameters. The dimensionless average speed of plate, $\mathcal {U^*}\left (=\mathcal {L}\times St\right )$ , is adopted merely to characterize the propulsive performances. For the first time, the $\mathcal {U^*}$ -based scaling laws for the thrust and power are revealed in pitching rigid and flexible plates for various $A_L$ and $St$ . This study may deepen our understanding of biological swimming and flying, and provide a guide for bionic design.

... Lagopoulos et al. [20] recently described transitions of a flapping foil in a pure pitching, heaving, and coupled oscillatory motion at a phase offset of 90 • . Analyses at Re = 1173 and 11730 depicted the universality of the Strouhal number, based on trajectory (Sr τ ), to effectively model and represent the transitions in propulsive performance and wake modes for various motion configurations of a flapping foil. ...

... The range of Reynolds number in the current paper spanned between 1000 to 16 000, which aligns with observations that propulsively efficient swimming motion largely occurs within Re = 100−10 000 [23]. A detailed analysis, however, is focused on Re = 1000, which is consistent with various studies conducted on transitions in the wake, as highlighted and discussed earlier [7,8,17,20]. This parameter space, therefore, provides a venue to compare current observations with those previously reported in literature for validation and consistency purposes. ...

... However, there has not been any attempt to characterize the effects of φ on the transition of propulsive performance and shedding of vortical structures. The chord-based Reynolds number in the current paper range from 1000 to 16 000, whereas detailed analyses of the wake and performance are focused on Re = 1000, following closely with studies of Godoy-Diana et al. [7], Schnipper et al. [8], Deng et al. [17], and Lagapaolos et al. [20]. As discussed earlier, these studies assessed the transitions in the wake of purely pitching foils, while also describing the quasi-two-dimensional and three-dimensional transitions [17]. ...

Transition in the propulsive performance and vortex synchronization of an oscillating foil in a combined heaving and pitching motion is numerically investigated at a range of reduced frequencies (0.16 ≤f∗≤0.64), phase offsets (0∘≤ϕ≤315∘), and Reynolds number (1000≤Re≤16000). Focusing on the common case of Re=1000, the drag to thrust transition is identified on a ϕ−f∗ phase map. Here, the range of 90∘≤ϕ≤225∘ depicted a drag-dominated regime for increasing reduced frequency. However, thrust-dominated regimes were observed for ϕ<90∘ and ϕ>225∘, where increasing the reduced frequency led to an increased thrust production. The isoline-depicting drag-thrust boundary was further observed to coincide with transitions in the characteristic near-wake modes with increasing reduced frequency, which ranged from 2P+2S to 2P and reverse von Kármán modes. However, evaluation of the wake with changing phase offsets at individual reduced frequencies only depicted effects on the spatial configuration of the vortex structures, while the number of vortices shed in one oscillation period was unchanged. The existence of similar wake modes with significantly different propulsive performance clearly suggests that transitions of the wake topology may not always be a reliable tool for understanding propulsive mechanisms of fish swimming or development of underwater propulsion systems. We further assessed a possible route to drag production via investigation into the mean velocity fields at increasing phase offset and at intermediate reduced frequencies ranging from 0.24 to 0.40. This revealed bifurcation of a velocity jet behind the foil on account of the wake topology and dynamics of shed vortex structures. The changes posed by increasing ϕ on wake structure interactions further hints at potential mechanisms that limit the achievement of optimum efficiency in underwater locomotion.

... Due to the level of complexities involved with this, it is also a common practice to use harmonically oscillating foils; as a simplified model for bioinspired propulsive mechanisms, to quantify their hydrodynamic performance and to investigate the underlying fluid mechanics [11][12][13][14][15][16][17]. This traditional strategy helped establish various similarities with the physics associated with the actual natural swimming. ...

... This traditional strategy helped establish various similarities with the physics associated with the actual natural swimming. Godoy Diana et al. [12,13], Deng et al. [14], and Lagopoulos et al. [17] developed transition maps for flows past simply oscillating foils in the phase planes constituted by oscillation amplitudes and Strouhal number; a nondimensional measure of excitation frequency. A common observation was the transformation of the Benard-von Karman vortex street (BvKVS) into reverse Benard-von Karman vortex street (RBvKVS) preceding the production of thrust for the body. ...

... We begin our discussion with phase maps in λ * − f * plane as shown in Fig forming; the phenomenon generally used as an indicator for thrust production by a swimming body, before the swimmers start producing thrust. This was also observed and reported previously by Streitlien and Triantafyllou [36], Godoy-Diana et al. [12], Bohl and Koochesfahani [37], and Lagopoulos et al. [17] for airfoils undergoing simple harmonic oscillations; known as pitching or plunging. This happens because of the contribution from the momentum-surfeit wake in the downstream direction of the body is not enough to overcome the profile or frictional drag. ...

Natural swimmers usually perform undulations to propel themselves and perform a range of maneuvers. These include various biological species ranging from micro-sized organisms to large- sized fishes that undulate at typical kinematic patterns. In this paper, we consider anguilliform and carangiform swimming modes to perform numerical simulations using an immersed-boundary method based computational solver at various Reynolds number (Re) regimes. We carry out thorough studies using wavelength and Strouhal frequency as the governing parameters for the hydrodynamic performance of undulating swimmers. Our analysis shows that the anguilliform kinematics achieves better hydrodynamic efficiency for viscous flow regime, whereas for flows with higher Re, the wavelength of a swimmer’s wavy motion dictates which kinematics will outperform the other. We find that the constructive interference between vortices produced at anterior parts of the bodies and co-rotating vortices present at the posterior parts plays an important role in reversing the direction of Benard-von Karman vortex street. Since most of the thrust producing conditions appear to cause wake deflection; a critical factor responsible for degrading the hydrodynamic efficiency of a swimmer, we discuss the underlying mechanics that would trigger this phenomenon. Moreover, we also find that resistive thrust force due to the frictional drag for anguilliform swimmers plays a substantial role in their propulsion at a low Reynolds number. As we approach more inertial flow conditions, its role is minimum and our carangiform swimmers primarily takes advantage of the thrust force due to the added mass effect under all the flow and kinematic conditions. Our findings are expected to provide a guideline on their selection for bio-inspired underwater vehicles.

... Due to the level of complexities involved with this, it is also a common practice to use harmonically oscillating foils; as a simplified model for bioinspired propulsive mechanisms, to quantify their hydrodynamic performance and to investigate the underlying fluid mechanics [11][12][13][14][15][16][17]. This traditional strategy helped establish various similarities with the physics associated with the actual natural swimming. ...

... This traditional strategy helped establish various similarities with the physics associated with the actual natural swimming. Godoy Diana et al. [12,13], Deng et al. [14], and Lagopoulos et al. [17] developed transition maps for flows past simply oscillating foils in the phase planes constituted by oscillation amplitudes and Strouhal number; a nondimensional measure of excitation frequency. A common observation was the transformation of the Benard-von Karman vortex street (BvKVS) into reverse Benard-von Karman vortex street (RBvKVS) preceding the production of thrust for the body. ...

... We begin our discussion with phase maps in λ * − f * plane as shown in Fig forming; the phenomenon generally used as an indicator for thrust production by a swimming body, before the swimmers start producing thrust. This was also observed and reported previously by Streitlien and Triantafyllou [36], Godoy-Diana et al. [12], Bohl and Koochesfahani [37], and Lagopoulos et al. [17] for airfoils undergoing simple harmonic oscillations; known as pitching or plunging. This happens because of the contribution from the momentum-surfeit wake in the downstream direction of the body is not enough to overcome the profile or frictional drag. ...

Natural swimmers usually perform undulations to propel themselves and perform a range of maneuvers. These include various biological species ranging from micro-sized organisms to large-scaled fishes. In this paper, we consider anguilliform and carangiform swimming modes to perform numerical simulations using an immersed-boundary methods based computational solver. We carry out thorough studies using wavelength and Strouhal frequency as the governing parameters for the hydrodynamic performance of undulating swimmers. Our analysis shows that the anguilliform kinematics is a better choice for viscous flow regime, whereas for flows with higher Reynolds number, the wavelength of a swimmer's wavy motion dictates which kinematics will outperform the other. We find that the constructive interference between vortices produced at anterior parts of the bodies and co-rotating vortices present at the posterior parts plays an important role in reversing the direction of Benard von Karman vortex street. Since most of the thrust producing conditions appear to cause wake deflection; a critical factor responsible for degrading the hydrodynamic efficiency of a swimmer, we discuss the underlying mechanics that would trigger this phenomenon. We demonstrate that the choice of kinematic and flow conditions may be restricted for the natural swimmers due to their morphological structures, but our findings provide a guideline on their selection for bio-inspired underwater vehicles.

... distance and plunging phase angles, the effects of some other variables have also been analyzed. These include the chord length [17], plunge amplitude [18,19], pitch angle [20,21], and wing corrugation [21,22]. ...

... The NACA 0012 airfoil was employed in this study because the NACA-series airfoils have been commonly used in previous works [20,23,24]. The wing motion consists of plunge and pitch, and the functions of motion are defined as ...

Tandem flapping-wing micro aerial vehicles, which are inspired by dragonflies, are unique in that they have two pairs of wings, and this could increase their maneuverability during flight. In this study, a series of tandem flapping-wing arrangements were designed in polar coordinates, and the polar distance and polar angles of the hindwing were used as design variables. Such a consideration of variations in both the horizontal and vertical positions of the hindwing in the design of tandem-wing arrangements has the potential to elucidate new interaction mechanisms, leading to improvements in the aerodynamic characteristics of tandem flapping-wings. Two-dimensional numerical analyses of these arrangements were conducted. The results of these analyses showed that, compared with a single flapping wing, tandem wings can increase the lift coefficient by 78.1% with an appropriate arrangement, but decrease the lift coefficient by 51.6% with an inappropriate arrangement. The vorticity distributions for typical wing arrangements were examined to establish the effect of the design variables on the aerodynamic characteristics of the tandem wings in forward flight. Two types of wing–wake interaction mechanisms were revealed based on this analysis. Both of these interactions can significantly reduce the negative force during the upstroke, resulting in an increase in the time-averaged lift.

... The experimental observations conclude with a wake structure with a distribution of interrelated vortex loops through consequent transportation of circulation [15]. In all the above investigations, both drag and thrust cases yield reverse von Kármán vortex shedding, which was also recently supported by the 2D simulation results on the drag-thrust transition for pitching NACA 0012 foils [16,17]. ...

... At a given R, the drag coefficient becomes zero beyond a threshold limit of St c = St C T =0 with increasing θ max > 5 • . For an increase in St > St c , this figure shows the crossover from drag to thrust, which causes C T > 0. Although earlier experimental [19,38] and 2D numerical simulations [16,17] reported drag-thrust transition for aerofoils, here we report such an explicit transition event for the 3D rigid pitching plate. Nevertheless, both geometries are aerodynamically similar. ...

The spatial transition of the wake behind a thin pitching plate in the thrust regime is studied. The drag-to-thrust transition is seen to occur at a threshold pitching frequency which becomes smaller for higher pitching angle and aspect ratio. The reverse von Kármán wake shows deflected asymmetric vortex pairing in two dimensions, while a bifurcated wake with a swirling ring is the signature in three dimensions, both resulting in nearly the same scaling for the power coefficient. The wake transition in the thrust regime occurs through three spatial regions comprising the reverse von Kármán vortex street, transitional zone, and twin jet region. Near the trailing edge, reverse von Kármán shedding is marked by linear decay of the spanwise wake width and growth of secondary instability. A large volume of fluid resulting from inward horizontal acceleration renders a weak form drag in the subsequent transition zone which is followed by vertical bifurcation into twin jet formation in the far wake. The spanwise pressure gradient is seen to guide the wake compression, with a streamwise adverse pressure gradient aiding it in the near wake.

... For a single flapping foil the onset of thrust generation is marked by a reverse von Kármán street (see figure 1b) downstream of its trailing edge (T.E.) (Von Karman 1935), although a lag between the two conditions exist (Godoy-Diana et al. 2008;Bohl & Koochesfahani 2009;Lagopoulos et al. 2019). This wake pattern is determined by the oscillating T.E. ...

... A grid convergence analysis was conducted to identify the appropriate resolution. As documented in Lagopoulos et al. (2019), a grid spacing of δx = δy = C 192 results in force predictions with less than 3% error compared to a grid with twice the resolution in each direction and is therefore used for all simulations in this manuscript. Table 1. ...

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.

... The onset of thrust generation is marked by a reverse von Kármán street downstream of a flapper (Von Karman 1935) although a lag between the two conditions exist (Godoy-Diana et al. 2008;Bohl & Koochesfahani 2009;Lagopoulos et al. 2019). This wake pattern is determined by the oscillating trailing edge (T.E.) amplitude A and frequency f of the motion which, form together an amplitude based Strouhal number St A = (2f A)/U ∞ as described by Triantafyllou et al. (1991). ...

... Grid density is indicated by the number of grid points per chord. A uniform grid of δx = δy = C/192 is used for the numerical experiments, which showed that this resolution resulted in force predictions with less than 3% error compared to the converged values (Lagopoulos et al. 2019). pure heave (Cleaver et al. 2012;Koz lowski & Kudela 2014). ...

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 analyse numerically the interaction of these deflected jets for tandem flapping foils, undergoing coupled heave to pitch motion. 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 completely cancelling the deflected wake and mean side force of the front foil, even when located up to 5 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 by expanding their working envelope and ability to rapidly increase or decrease the forward speed by manipulating the size of the shed vortices.

... More specifically, pitch refers to the sinusoidal rotation about the pivot point P = 0.25 (normalised by C) while heave is a sinusoidal, vertical translation with respect to the centreline. Thus, the combined motion of the TE can be described as Lagopoulos, G. D. Weymouth and B. Ganapathisubramani 2A Lagopoulos, Weymouth, and Ganapathisubramani (2019). ...

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.

... The flow dynamics past single flapping foils have been extensively studied (Wu et al. 2020) but rarely in the presence of an upstream bluff body wake. For a single pitching foil with high span-to-chord ratio, four different regimes are present depending on the peak-to-peak amplitude of oscillations and the Strouhal number (Godoy-Diana et al. 2009;Andersen et al. 2017;Lagopoulos, Weymouth & Ganapathisubramani 2019). In the first regime, characterised by positive drag, the Bénard-von Kármán (BvK) street is observed. ...

A numerical study on the response of a two-dimensional bluff body wake subjected to harmonic forcing imposed by two rear pitching flaps is performed. The wake is generated by a rectangle at a height-based Reynolds number $Re=100$ , characterised by laminar vortex shedding. Two forcing strategies are examined corresponding to in-phase ‘snaking’ and out-of-phase ‘clapping.’ The effects of the bluff body aspect ratio ( $AR=1,2,4$ ), flapping frequency, flapping amplitude, flap length and Reynolds number are investigated. For the snaking motion, a strong fundamental resonance of the root mean square (r.m.s.) drag is observed when the wake is forced near the vortex shedding frequency. For the clapping motion, a weak subharmonic resonance is observed when the forcing is applied near twice the vortex shedding frequency resulting in an increase of the lift r.m.s. whereas the drag r.m.s. remains unaffected. Both resonances intensify the vortex shedding and a concomitant mean drag increase is observed for the snaking motion. Forcing away from the resonant regimes, both motions result in considerable drag reduction through wake symmetrisation and propulsion mechanisms. The formation of two vortex dipoles per oscillation period due to the flapping motion, which weaken the natural vortex shedding, has been identified as the main symmetrisation mechanism. A single scaling parameter is proposed to collapse the mean drag reduction of the forced flow for both motions over a wide range of flapping frequencies, amplitudes and flap lengths. Finally, the assessment of the performance of the forcing strategies has revealed that clapping is more effective than snaking.

... In other words, tapered attachments can be used to maximize the tip displacement for a wide range of actuation frequencies, which in turn enhances the hydrodynamic thrust. The dependence of the thrust on tip displacement is consistent with the literature data for uniform propulsors (Lighthill 1970;Dewey et al. 2013;Floryan et al. 2017;Ayancik et al. 2019;Lagopoulos, Weymouth & Ganapathisubramani 2019). Furthermore, we find that our simulation results for F are in good agreement with the scaling derived by Gazzola, Argentina & Mahadevan (2014) and confirmed experimentally by Gibouin et al. (2018) (right inset in figure 8a). ...

Using fluid–structure interaction computational modelling, the hydrodynamic performance of bio-inspired elastic propulsors with tapered thickness that oscillate in an incompressible Newtonian fluid at Reynolds number $Re = 2000$ is investigated. The thickness tapering leads to an acoustic black hole effect at the trailing edge of the propulsor that slows down and attenuates flexural waves, thereby minimizing the flexural wave reflection and enhancing travelling wave propulsion. The simulations reveal that, by tuning the propulsor thickness profile modulating the acoustic black hole effect, the tapered propulsors can be designed to vastly outperform the uniformly thick propulsors in terms of the hydrodynamic efficiency and thrust, especially for the post-resonance frequencies. The enhanced hydrodynamic performance is directly linked to the ability of the tapered propulsors to generate travelling waves with a large amplitude displacement at the trailing edge. The results have implications for the development of highly efficient bio-mimetic robotic swimmers and, more generally, the better understanding of the undulatory aquatic locomotion.

... As was shown in figure 8, φ = π/2 and φ = 3π/2 at κh = 0.375 manifest different dynamics (periodic and chaotic, respectively), though both have the same A D value of 6.62. A τ D versus 1/St D plot has also been made here in figure 23(c), as proposed by Lagopoulos et al. (2019). No clear demarcation between the different dynamical states can be found here either. ...

The present study focuses on identifying dynamical transition boundaries and presents an order-to-chaos map for the unsteady flow field of a flapping foil in the low Reynolds number regime. The effect of an extensive parametric space, covering a large number of kinematic conditions, has been investigated. It is shown that the conventional non-dimensional parameters cannot effectively capture the changes in the flow field due to the variations in the relevant kinematic parameters and are unable to demarcate the dynamical transition boundaries. Two new non-dimensional measures-maximum effective angle of attack and a leading-edge amplitude-based Strouhal number-are proposed here, which can capture the physical effect of the parametric variations on the wake dynamics. The study proposes generalised transition boundaries and an order-to-chaos map through a transitional regime in terms of these two newly proposed parameters. Published data from the existing literature have also been tested to verify the proposed transition model. It is seen that despite the wide variety of the parametric combinations, the dynamical states from both the new and the published data corroborate well the proposed boundaries, giving credibility to the order-to-chaos map.

... Figure oscillations. Since α • is proportional to instantaneous α, this implies that the reasoning for increasing κ with α is similar to other reported deviations in previous literature 14,36,37 . ...

The variation in thrust generation with respect to Reynolds number was numerically evaluated for an oscillating foil with combined pitching and heaving motion at a range of reduced frequencies, amplitudes, and phase offsets. Laminar scaling ([Formula: see text]) was found accurate for a reasonable range of average angle of attack ([Formula: see text]). However, quantitative evaluation of laminar scaling using statistical measures indicates that its capability in predicting thrust variation weakens at higher reduced frequencies and amplitudes. This coincides with an increase in [Formula: see text] above 20°. Evaluation of the pressure and viscous forces revealed a dominance of the former toward total thrust generated at high frequencies for all cases, which also coincided with lower coefficient of determination ( R ² ) for laminar scaling. The chordwise variation of pressure and skin friction coefficient provided further evidence indicating that pressure, in contrast to the skin friction, did not achieve an asymptotic trend with increasing Reynolds number, especially at higher frequencies and for all phase offsets. Qualitative evaluation of the developing leading edge vortex structure at increasing reduced frequencies and Reynolds numbers also supported the quantitative assessment of chordwise pressure variations. Empirical incorporation of Reynolds number into the complete scaling model was hence completed, which further validates the laminar scaling ([Formula: see text]) of propulsive thrust generation in oscillating foils with a coupled motion.

... -Diana et al. (2008),Schnipper et al. (2009), Deng et al. (2015 andLagopoulos et al. (2019). These studies described transitions in the wake of purely pitching foils and provided a insight into the quasi-twodimensional and three-dimensional characteristics of wake structures(Deng et al. 2015). ...

The association of lift generation and evolution of wake topology behind an oscillating foil with combined heaving and pitching motion is investigated numerically at a range of reduced frequency (0.16 < f* < 0.48), phase offset (0 deg < \phi < 315 deg) and Reynolds number (1000 < Re < 4000). The pitch-dominated kinematics that coincide with the range of \phi < 120 deg and \phi > 225 deg suggests that leading edge vortices are suppressed while trailing edge vortices dominate the wake with increasing reduced frequency. This corresponds to a transition in wake topology from a 2P to a reverse von K´ arm´an wake mode. Contrarily, heave dominated kinematics (120 deg <\phi < 225 deg) did not exhibit any wake topology transition with increasing f*. The temporal lift variation associated with heave-dominated regime further revealed a symmetric feature in terms of the time taken to attain peak lift generation within an oscillation cycle. This temporal symmetry was, however, lost as kinematics transitioned from heave- to pitch-dominated regime. Analyzing the wake evolution and lift features at quarter phase of an oscillation cycle revealed the existence of a correspondence between the two processes during the heave- and pitch-dominated kinematics.

... Thus, for the sake of simplicity, we use the AR definition of rectangular flippers (explained in section 1) and we set our baseline test case at AR = 2 proceeding towards AR = 8 in increments of AR = 1. (Lagopoulos et al., 2019). ...

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 aspect ratio - AR) is found to improve thrust coefficients and thrust augmentation but the benefits level off towards higher values of AR. On the other hand, 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 on the mechanisms of these wake interactions and provide valuable information for the design of novel biomimetic propulsion systems.

... Above a critical frequency the vortices move away from the centerline to the opposite sides and create a reverse von Karman street. The jet-like mean velocity profile creates mean thrust in this configuration [5,[10][11][12][13][14][15]. ...

The flow around, and in the wake of, pitching airfoils has received renewed interest due to its potential for thrust production at low Reynolds numbers. Past work has centered on the flow fields generated by symmetric pitching of the airfoil. Studies investigating the effects of asymmetric motion are more limited. This work focuses on the wake patterns developed due to asymmetric pitching. Particle Image Velocimetry (PIV) is used to quantify the flow field around a NACA0012 airfoil undergoing small amplitude, high frequency asymmetric pitching. The airfoil is pitched about the quarter chord point with an amplitude of ±4° at reduced frequencies of k = 2.6–5.8 at a Rec = 12000. Pitching symmetries of 50/50, 40/60 and 30/70 are studied, where the symmetry is defined by the fraction of the cycle spent in the pitch down versus pitch up motion. The data show that for the 50/50 (symmetric) motions two alternating sign vortices, with equivalent strength, are formed as expected. The asymmetric cases show that a single vortex is formed during the “fast” portion of the pitching motion. Multiple vortices are formed during the “slow” portion of the pitching motion. The number of secondary vortices and the downstream evolution of the vortices depends on the symmetry value. In some cases they remain isolated but orbit other vortical structures, while in other cases they pair with other vortical structures, and finally when the reduced frequency and asymmetry values are high enough the vortex array shows interaction between cycles.

... Similar to the inspiration of the horizontal-axis turbine from the propeller, the feasibility of energy-harvesting from the water flow of an oscillating hydrofoil was studied analytically and experimentally [10]. Two states of oscillating hydrofoil were identified: the thrust state and the energy-harvesting state [11]. Parametric studies on the hydrodynamic performance and wake characteristics were conducted [12,13]. ...

The aim of this paper is to experimentally and numerically study a coupled-pitching hydrofoil under the fully activated mode. In the experimental study, the operating zones for the energy-releasing state and the energy-capturing state were identified. The phase relationship between the hydrodynamic torque and the angular velocity, which is determined by the combination of the primary and secondary pitching amplitudes, plays a key role in the switch of the energy releasing/capturing states. The primary pitching motion dominates the energy conversion of the hydrofoil under the fully-activated mode. The peak values of the averaged power coefficient and energy converting efficiency are achieved at 1.17 and 0.40, respectively. The numerical study found that the effective angle of attack determines the generation of the leading-edge and trailing-edge vortexes, and in turn affects the pressure distribution of the hydrofoil surface and torque generation. Compared to the traditional heaving-pitching mode, the coupled-pitching mode is horizontally affected by the negative torque because of the restriction of the primary pitching system, resulting in a lower energy capturing efficiency.

... In recent decades, with the development of bionics, the aerodynamic mechanisms behind flapping creatures are drawing considerable attention [1][2][3]. Extensive studies are conducted to explore the basic aspects of the unsteady wake formation and associated aerodynamic forces around the flapping airfoil [4][5][6][7][8], especially the optimal solutions of the kinematic parameters on achieving specific aerodynamic performance. ...

We construct a multifidelity framework for the kinematic parameter optimization of flapping airfoil. We employ multifidelity Gaussian process regression and Bayesian optimization to effectively synthesize the aerodynamic performance of the flapping airfoil with the kinematic parameters under multiresolution numerical simulations. The objective of this work is to demonstrate that the multifidelity framework can efficiently discover the optimal kinematic parameters of the flapping airfoil with specific aerodynamic performance using a limited number of expensive high-fidelity simulations combined with a larger number of inexpensive low-fidelity simulations. We efficiently identify the optimal kinematic parameters of an asymmetrically flapping airfoil with various target aerodynamic forces in the design space of heaving amplitude, flapping frequency, angle of attack amplitude, and stroke angle. Notably, it is found that the angle of attack can significantly affect the magnitude of aerodynamic forces by facilitating the generation of the leading-edge vortex. In the meanwhile, its combination effect with the stroke angle can determine the attitude and trajectory of the flapping airfoil, thus further affect the direction of the aerodynamic forces. With the influence of the streamwise in-line motion, the asymmetrical vortex structures emerge in the wake fields because the streamwise velocities of shedding vortices are different in the upstroke and downstroke. Furthermore, we conduct the kinematic parameter optimization for a three-dimensional asymmetrically flapping wing. Compared to the two-dimensional simulations, we further investigate the flow induced by the vortex ring and its unsteady effects on the vortex structure and aerodynamic performance.

... Furthermore, even "drag-producing" wakes have been observed to produce thrust [14]. In this context, the recent work by Lagopoulos et al. [17] offers an alternative method to distinguish drag-and thrust-producing behavior based on kinematic inputs instead of the vortex arrangement. Despite intuition, vortex spacing, either in the horizontal or vertical direction, is not a reliable indicator of thrust production. ...

The structure of swimmers' wakes is often assumed to be an indicator of swimming performance. Here, we discuss three cases where this assumption fails. In general, great care should be taken in deriving any conclusions about swimming performance from the wake flow pattern.

The Reynolds number, which describes the relative importance of viscous and inertial contributions is commonly used to analyze forces on fish and other aquatic animals. However, this number is based on steady, time-independent conditions, while all swimming motions have a periodic component. Here we apply periodic flow conditions to define a new non-dimensional group, which we name the “Periodic Swimming Number, P”, which rectifies this lacuna. This new non-dimensional number embodies the periodic motion and eliminates the arbitrariness of choosing a length scale in the Reynolds number for Body –Caudal-Fin (BCF) swimming. We show that the new number has the advantage of compressing known data on fish swimming to two orders of magnitude, vs. over six required when using the existing Reynolds number and can point to a new comparison of swimming effectiveness for swimming modes.

Fishes have learned how to achieve outstanding swimming performance through the evolution of hundreds of millions of years, which can provide bio-inspiration for robotic fish design. The premise of designing an excellent robotic fish include fully understanding of fish locomotion mechanism and grasp of the advanced control strategy in robot domain. In this paper, the research development on fish swimming is presented, aiming to offer a reference for the later research. First, the research methods including experimental methods and simulation methods are detailed. Then the current research directions including fish locomotion mechanism, structure and function research and bionic robotic fish are outlined. Fish locomotion mechanism is discussed from three views: macroscopic view to find a unified principle, microscopic view to include muscle activity and intermediate view to study the behaviors of single fish and fish school. Structure and function research is mainly concentrated from three aspects: fin research, lateral line system and body stiffness. Bionic robotic fish research focuses on actuation, materials and motion control. The paper concludes with the future trend that curvature control, machine learning and multiple robotic fish system will play a more important role in this field. Overall, the intensive and comprehensive research on fish swimming will decrease the gap between robotic fish and real fish and contribute to the broad application prospect of robotic fish.

The structure of swimmers' wakes is often assumed to be an indicator of swimming performance, that is, how momentum is produced and energy is consumed. Here, we discuss three cases where this assumption fails. In general, great care should be taken in deriving any conclusions about swimming performance from the wake flow pattern.

The effects of flexibility on the wake structures of a foil under a heaving motion in a viscous uniform flow are numerically studied using an immersed boundary method. An inspection of the phase diagram of the wake structures in a map of the chord-length-based dimensionless heaving amplitude (AL) and Strouhal number (StL) shows that the wake transition boundaries of the rigid foil are well predicted by constant amplitude-based Strouhal number (StA) lines, similar to previous studies. However, the wake transition boundaries of the flexible foil are predictable by constant StA lines only for high StL cases. A large deformation angle of a flexible foil by the amplitude difference and phase difference between the leading and trailing edge cross-stream displacements reduces the effective leading edge velocity, with an accompanying decrease in the leading edge circulation. However, the trailing edge circulation for a flexible foil is increased due to increased trailing edge amplitude. The sum of the leading and trailing edge circulations plays an important role in determining the wake pattern behind a rigid and flexible foil, and wake transitions are observed beyond critical circulations. The decrease in the thrust coefficient for large values of StL and AL is closely associated with the generation of a complex wake pattern behind a foil, and the complex wake is a direct consequence of sufficiently large leading edge circulation. A critical effective phase velocity in a vortex dipole model is proposed to predict the maximum thrust coefficient without a complex wake pattern.

Force and particle image velocimetry measurements were conducted on a NACA 0012 aerofoil undergoing small-amplitude high-frequency plunging oscillation at low Reynolds numbers and angles of attack in the range 0–. For angles of attack less than or equal to the stall angle, at high Strouhal numbers, significant bifurcations are observed in the time-averaged lift coefficient resulting in two lift-coefficient branches. The upper branch is associated with an upwards deflected jet, and the lower branch is associated with a downwards deflected jet. These branches are stable and highly repeatable, and are achieved by increasing or decreasing the frequency in the experiments. Increasing frequency refers to starting from stationary and increasing the frequency very slowly (while waiting for the flow to reach an asymptotic state after each change in frequency); decreasing frequency refers to impulsively starting at the maximum frequency and decreasing the frequency very slowly. For the latter case, angle of attack, starting position and initial acceleration rate are also parameters in determining which branch is selected. The bifurcation behaviour is closely related to the properties of the trailing-edge vortices. The bifurcation was therefore not observed for very small plunge amplitudes or frequencies due to insufficient trailing-edge vortex strength, nor at larger angles of attack due to greater asymmetry in the strength of the trailing-edge vortices, which creates a preference for a downward deflected jet. Vortex strength and asymmetry parameters are derived from the circulation measurements. It is shown that the most appropriate strength parameter in determining the onset of deflected jets is the circulation normalized by the plunge velocity.

Foils oscillating transversely to an oncoming uniform flow produce, under certain conditions, thrust. It is shown through experimental data from flapping foils and data from fish observation that thrust develops through the formation of a reverse von Kármán street whose preferred Strouhal number is between 0.25 and 0.35, and that optimal foil efficiency is achieved within this Strouhal range.

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.

It has been known for some time that two-dimensional numerical simulations of flow over nominally two-dimensional bluff bodies at Reynolds numbers for which the flow is intrinsically three dimensional, lead to inaccurate prediction of the lift and drag forces. In particular, for flow past a normal flat plate (InternationalSymposiumonNonsteadyFluidDynamics, edited by J. A. Miller and D. P. Telionis, 1990, pp. 455–464) and circular cylinders [J. Wind Eng. Indus. Aerodyn. 35, 275 (1990)], it has been noted that the drag coefficient computed from two-dimensional simulations is significantly higher than what is obtained from experiments. Furthermore, it has been found that three-dimensional simulations of flows lead to accurate prediction of drag [J. Wind Eng. Indus. Aerodyn. 35, 275 (1990)]. The underlying cause for this discrepancy is that the surface pressure distribution obtained from two-dimensional simulations does not match up with that obtained from experiments and three-dimensional simulations and a number of reasons have been put forward to explain this discrepancy. However, the details of the physical mechanisms that ultimately lead to the inaccurate prediction of surface pressure and consequently the lift and drag, are still not clear. In the present study, results of two-dimensional and three-dimensional simulations of flow past elliptic and circular cylinders have been systematically compared in an effort to pinpoint the exact cause for the inaccurate prediction of the lift and drag by two-dimensional simulations. The overprediction of mean drag force in two-dimensional simulations is directly traced to higher Reynolds stresses in the wake. It is also found that the discrepancy in the drag between two-dimensional and three-dimensional simulations is more pronounced for bluffer cylinders. Finally, the current study also provides a detailed view of how the fluctuation, which are associated with the Ka´rma´n vortex shedding in the wake, affect the mean pressure distribution and the aerodynamic forces on the body.

A finite element flow solver based on unstructured grids is employed for studying the unsteady flow past oscillating wings. In order to understand the basis of lift and thrust generation mechanisms, we have performed computational studies on the flapping wing of the fruit fly, Drosophila. The computational model is based on the experimental setup of Dickinson et al [1]. Computations are performed for various phase angles between the rotation and translation motions and the time history of the unsteady forces are compared with the experiments. Good agreement is obtained for the thrust and drag forces. Also, a grid refinement study is performed to validate the computational results. The unsteady flow is discussed in detail.

▪ Abstract “What force does an insect wing generate?” Finding answers to this enduring question is an essential step toward our understanding of interactions of moving objects with fluids that enable most living species such as insects, birds, and fish to travel efficiently and us to follow similar suit with sails, oars, and airfoils. We give a brief history of research in insect flight and discuss recent findings in unsteady aerodynamics of flapping flight at intermediate range Reynolds numbers (10–104). In particular, we examine the unsteady mechanisms in uniform and accelerated motions, forward and hovering flight, as well as passive flight of free-falling objects. The results obtained by “taking the insects apart” helped us to resolve previous puzzles about the force estimates in hovering insects, to ellucidate basic mechanisms essential to flapping flight, and to gain insights about the efficieny of flight.

Thrust-producing harmonically oscillating foils are studied through
force and power
measurements, as well as visualization data, to classify the principal
characteristics of
the flow around and in the wake of the foil. Visualization data are obtained
using
digital particle image velocimetry at Reynolds number 1100, and force and
power
data are measured at Reynolds number 40 000. The experimental results are
compared
with theoretical predictions of linear and nonlinear inviscid theory and
it is found
that agreement between theory and experiment is good over a certain parametric
range, when the wake consists of an array of alternating vortices and either
very
weak or no leading-edge vortices form. High propulsive efficiency, as high
as 87%, is
measured experimentally under conditions of optimal wake formation. Visualization
results elucidate the basic mechanisms involved and show that conditions
of high
efficiency are associated with the formation on alternating sides of the
foil of a
moderately strong leading-edge vortex per half-cycle, which is convected
downstream
and interacts with trailing-edge vorticity, resulting eventually in the
formation of a
reverse Kármán street. The phase angle between transverse
oscillation and angular
motion is the critical parameter affecting the interaction of leading-edge
and
trailing-edge vorticity, as well as the efficiency of propulsion.

The vortex streets produced by a flapping foil of span to chord aspect ratio of 4:1 are studied in a hydrodynamic tunnel experiment. In particular, the mechanisms giving rise to the symmetry breaking of the reverse Bénard–von Kármán (BvK) vortex street that characterizes fishlike swimming and forward flapping flight are examined. Two-dimensional particle image velocimetry (PIV) measurements in the midplane perpendicular to the span axis of the foil are used to characterize the different flow regimes. The deflection angle of the mean jet flow with respect to the horizontal observed in the average velocity field is used as a measure of the asymmetry of the vortex street. Time series of the vorticity field are used to calculate the advection velocity of the vortices with respect to the free stream, defined as the phase velocity Uphase, as well as the circulation Γ of each vortex and the spacing ξ between consecutive vortices in the near wake. The observation that the symmetry-breaking results from the formation of a dipolar structure from each couple of counter-rotating vortices shed on each flapping period serves as the starting point to build a model for the symmetry-breaking threshold. A symmetry-breaking criterion based on the relation between the phase velocity of the vortex street and an idealized self-advection velocity of two consecutive counter-rotating vortices in the near wake is established. The predicted threshold for symmetry breaking accounts well for the deflected wake regimes observed in the present experiments and may be useful to explain other experimental and numerical observations of similar deflected propulsive vortex streets reported in the literature.

We study the wake of a cylinder performing rotary oscillations around its axis at moderate Reynolds number. We observe that the structure of the vortex shedding is strongly affected by the forcing parameters. The forced wake is characterized by a ‘lock-in’ region where the vortices are shed at the forcing frequency and a region where the vortices can be reorganized to give a second frequency close to those observed for the unforced wake. We show that these modifications of the wake structure change the dynamic of the fluctuations downstream from the cylinder. We vary the amplitude and the frequency of the oscillations and study the consequences of these modifications on the mean flow and the global drag applied on the cylinder. We then discuss the mechanism responsible for the modification of the fluctuations and the modification of the drag coefficient.

What mechanisms of flow control do animals use to enhance hydrodynamic perfor-mance? Animals are capable of manipulating flow around the body and appendages both passively and actively. Passive mechanisms rely on structural and morphological components of the body (i.e., humpback whale tubercles, riblets). Active flow control mechanisms use appendage or body musculature to directly generate wake flow struc-tures or stiffen fins against external hydrodynamic loads. Fish can actively control fin curvature, displacement, and area. The vortex wake shed by the tail differs between eel-like fishes and fishes with a discrete narrowing of the body in front of the tail, and three-dimensional effects may play a major role in determining wake structure in most fishes.

We study experimentally the vortex streets produced by a flapping foil in a hydrodynamic tunnel, using two-dimensional particle image velocimetry. An analysis in terms of a flapping frequency-amplitude phase space allows the identification of (i) the transition from the well-known Bénard-von Kármán (BvK) wake to the reverse BvK vortex street that characterizes propulsive wakes, and (ii) the symmetry breaking of this reverse BvK pattern giving rise to an asymmetric wake. We also show that the transition from a BvK wake to a reverse BvK wake precedes the actual drag-thrust transition and we discuss the significance of the present results in the analysis of flapping systems in nature.

Dimensionless numbers are important in biomechanics because their constancy can imply dynamic similarity between systems, despite possible differences in medium or scale. A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number, St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U). St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions. Propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < St < 0.4 (refs 3-8). Because natural selection is likely to tune animals for high propulsive efficiency, we expect it to constrain the range of St that animals use. This seems to be true for dolphins, sharks and bony fish, which swim at 0.2 < St < 0.4. Here we show that birds, bats and insects also converge on the same narrow range of St, but only when cruising. Tuning cruise kinematics to optimize St therefore seems to be a general principle of oscillatory lift-based propulsion.

Significant progress has been made in understanding some of the basic mechanisms of force production and flow manipulation in oscillating foils for underwater use. Biomimetic observations, however, show that there is a lot more to be learned, since many of the functions and details of fish fins remain unexplored. This review focuses primarily on experimental studies on some of the, at least partially understood, mechanisms, which include 1) the formation of streets of vortices around and behind two- and three-dimensional propulsive oscillating foils; 2) the formation of vortical structures around and behind two- and three-dimensional foils used for maneuvering, hovering, or fast-starting; 3) the formation of leading-edge vortices in flapping foils, under steady flapping or transient conditions; 4) the interaction of foils with oncoming, externally generated vorticity; multiple foils, or foils operating near a body or wall.

We present a combined numerical (particle vortex method) and experimental (soap film tunnel) study of a symmetric foil undergoing prescribed oscillations in a two-dimensional free stream. We explore pure pitching and pure heaving, and contrast these two generic types of kinematics. We compare measurements and simulations when the foil is forced with pitching oscillations, and we find a close correspondence between flow visualisations using thickness variations in the soap film and the numerically determined vortex structures. Numerically, we determine wake maps spanned by oscillation frequency and amplitude, and we find qualitatively similar maps for pitching and heaving. We determine the drag–thrust transition for both pitching and heaving numerically, and we discuss it in relation to changes in wake structure. For heaving with low oscillation frequency and high amplitude, we find that the drag–thrust transition occurs in a parameter region with wakes in which two vortex pairs are formed per oscillation period, in contrast to the common transition scenario in regions with inverted von Kármán wakes.

When a body oscillates laterally (cross-flow) in a free stream, it can synchronize the vortex formation frequency with the body motion frequency. This fundamental “lock-in” regions is but one in a whole series of synchronization regions, which have been found in the present paper, in an amplitude-wavelength plane (defining the body trajectory) up to amplitudes of five diameters. In the fundamental region, it is shown that the acceleration of the cylinder each half cycle induces the roll-up of the two shear layers close to the body, and thereby the formation of four regions of vorticity each cycle. Below a critical wavelength, each half cycle sees the coalescence of a pair of like-sign vortices and the development of a Karman-type wake. However, beyond this wavelength the like-sign vortices convect away from each other, and each of them pairs with an opposite-sign vortex. The resulting wake comprises a system of vortex pairs which can convect away from the wake centerline. The process of pairing causes the transition between these modes to be sudden, and this explains the sharp change in the character of the cylinder forces observed by Bishop and Hassan, and also the jump in the phase of the lift force relative to body displacement. At precisely the critical wavelength, only two regions of vorticity are formed, and the resulting shed vorticity is more concentrated than at other wavelengths. We interpret this particular case as a condition of “resonant synchronization”, and it corresponds with the peak in the body forces observed in Bishop and Hassan's work.

Significant progress has been made in understanding
some of the basic mechanisms of force production and flow manipulation
in oscillating foils for underwater use. Biomimetic observations,
however, show that there is a lot more to be learned, since
many of the functions and details of fish fins remain unexplored.
This review focuses primarily on experimental studies on some
of the, at least partially understood, mechanisms, which include 1)
the formation of streets of vortices around and behind two- and
three-dimensional propulsive oscillating foils; 2) the formation of
vortical structures around and behind two- and three-dimensional
foils used for maneuvering, hovering, or fast-starting; 3) the formation
of leading-edge vortices in flapping foils, under steady flapping
or transient conditions; 4) the interaction of foils with oncoming,
externally generated vorticity; multiple foils, or foils operating
near a body or wall.

A perching bird is able to rapidly decelerate while maintaining lift and control, but the underlying aerodynamic mechanism is poorly understood. In this work we perform a study on a simultaneously decelerating and pitching aerofoil section to increase our understanding of the unsteady aerodynamics of perching. We first explore the problem analytically, developing expressions for the added-mass and circulatory forces arising from boundary-layer separation on a flat-plate aerofoil. Next, we study the model problem through a detailed series of experiments at Re = 22 000 and two-dimensional simulations at Re = 2000. Simulated vorticity fields agree with particle image velocimetry measurements, showing the same wake features and vorticity magnitudes. Peak lift and drag forces during rapid perching are measured to be more than 10 times the quasi-steady values. The majority of these forces can be attributed to added-mass energy transfer between the fluid and aerofoil, and to energy lost to the fluid by flow separation at the leading and trailing edges. Thus, despite the large angles of attack and decreasing flow velocity, this simple pitch-up manoeuvre provides a means through which a perching bird can maintain high lift and drag simultaneously while slowing to a controlled stop.

An accurate Cartesian-grid treatment for intermediate Reynolds number fluid-solid interaction problems is described. We first identify the inability of existing immersed boundary methods to handle intermediate Reynolds number flows to be the discontinuity of the velocity gradient at the interface. We address this issue by generalizing the Boundary Data Immersion Method (BDIM, Weymouth and Yue (2011)), in which the field equations of each domain are combined analytically, through the addition of a higher order term to the integral formulation. The new method retains the desirable simplicity of direct forcing methods and smoothes the velocity field at the fluid-solid interface while removing its bias. Based on a second-order convolution, it achieves second-order convergence in the L 2 norm, regardless of the Reynolds number. This results in accurate flow predictions and pressure fields without spurious fluctuations, even at high Reynolds number. A treatment for sharp corners is also derived that significantly improves the flow predictions near the trailing edge of thin airfoils. The second-order BDIM is applied to unsteady problems relevant to ocean energy extraction as well as animal and vehicle locomotion for Reynolds numbers up to 10 5 .

Experiments were performed on an oscillating foil to assess its performance in producing large forces for propulsion and effective maneuvering. First, experiments on a harmonically heaving and pitching foil were performed to determine its propulsive efficiency under conditions of significant thrust production, as function of the principal parameters: the heave amplitude, Strouhal number, angle of attack, and phase angle between heave and pitch. Planform area thrust coefficients of 2.4 were recorded for 35° maximum angle of attack and efficiencies of up to 71.5% were recorded for 15° maximum angle of attack. A plateau of good efficiency, in the range of 50–60%, is noted. A phase angle of 90–100° between pitch and heave is found to produce the best thrust performance. Also, the introduction of higher harmonics in the heave motion, so as to ensure a sinusoidal variation in the angle of attack produced much higher thrust coefficient at high Strouhal numbers. Second, experiments on a harmonically oscillating foil with a superposed pitch bias, as well as experiments on impulsively moving foils in still water, were conducted to assess the capability of the foil to produce large lateral forces for maneuvering. Mean side force coefficients of up to 5.5, and instantaneous lift coefficients of up to 15 were recorded, demonstrating an outstanding capability for maneuvering force production.

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 question of which flapping modes, amplitudes, frequencies, and wing shapes produce optimum cruise flight efficiencies. Second, the problem of hovering flight is studied for single flapping wings. Third, aerodynamic phenomena produced by flapping wing interactions are discussed, such as tandem wing configurations, as used by dragonflies, or biplane configurations, as used on the authors' micro air vehicle. Potential flow and viscous flow solutions are presented and the role of vortex shedding, especially from wing leading edges, is discussed. Comparisons with available experimental results are provided.

The average thrust developed by flapping foils seen among the birds and fast swimming marine animals is often estimated experimentally from the time-averaged velocity profile in the wake. We present a parametric investigation of the error due to neglect of the unsteady terms, under the assumption of two-dimensional ideal flow. A numerical simuation algorithm allows comparison of exact average thrust and mean-flow momentum flux in the wake, and two simple models for the wake, based on linear foil theory and the Kármán vortex street, provide further insight into the problem. The main observation is that mean-flow wake surveys tend to overestimate the thrust force, most severely for high reduced frequencies and low values of the proportional feathering parameter. We also find that a thrust estimate based on the Kármán vortex street model yields surprisingly accurate results over the entire range of tested parameters. The linear theory predicts thrust well for all but the highest Strouhal number.

The vortical flow patterns in the wake of a NACA 0012 airfoil pitching
at small amplitudes are studied in a low speed water channel. It is
shown that a great deal of control can be exercised on the structure of
the wake by the control of the frequency, amplitude and also the shape
of the oscillation waveform. An important observation in this study has
been the existence of an axial flow along the cores of the wake
vortices. Estimates of the magnitude of the axial flow suggest a linear
dependence on the oscillation frequency and amplitude.

We present an experimental investigation of the flow structure and vorticity field in the wake of a NACA-0012 airfoil pitching sinusoidally at small amplitude and high reduced frequencies. Molecular tagging velocimetry is used to quantify the characteristics of the vortex array (circulation, peak vorticity, core size, spatial arrangement) and its downstream evolution over the first chord length as a function of reduced frequency. The measured mean and fluctuating velocity fields are used to estimate the mean force on the airfoil and explore the connection between flow structure and thrust generation.
Results show that strong concentrated vortices form very rapidly within the first wavelength of oscillation and exhibit interesting dynamics that depend on oscillation frequency. With increasing reduced frequency the transverse alignment of the vortex array changes from an orientation corresponding to velocity deficit (wake profile) to one with velocity excess (reverse Kármán street with jet profile). It is found, however, that the switch in the vortex array orientation does not coincide with the condition for crossover from drag to thrust. The mean force is estimated from a more complete control volume analysis, which takes into account the streamwise velocity fluctuations and the pressure term. Results clearly show that neglecting these terms can lead to a large overestimation of the mean force in strongly fluctuating velocity fields that are characteristic of airfoils executing highly unsteady motions. Our measurements show a decrease in the peak vorticity, as the vortices convect downstream, by an amount that is more than can be attributed to viscous diffusion. It is found that the presence of small levels of axial velocity gradients within the vortex cores, levels that can be difficult to measure experimentally, can lead to a measurable decrease in the peak vorticity even at the centre of the flow facility in a flow that is expected to be primarily two-dimensional.

We report experimental results of the forced wake of a thin symmetric flat plate, placed parallel to an uniform air stream, in the range of thickness-based Reynolds number 50< Re
e
<200. External wake forcing was introduced by small harmonic oscillations of a moving flap, placed at the trailing-edge of the flat plate. When the flap remains in a fixed horizontal position, the mean velocity profiles obtained by hot wire measurements, for different Reynolds numbers, are self similar. In the presence of harmonic forcing, within a certain range of the forcing frequency, the mean velocity profiles change and coherent structures are formed in the wake. Two independent flow-type resonances were observed: (i) when the inverse of the forcing frequency matches the flight time of the fluid particles along the flap. (ii) when the forcing frequency of the flap equals one half of the vortex shedding frequency of the flat plate and flap system. Implications of the two observed resonances on the wake structure are important. The first resonance (i) is associated to a wide but less intense (energy fluctuations) wake flow and the second resonance (ii) generates a thin but intense resultant wake flow.

Oscillating foils produce thrust through the development of a jet-like average flow. It is found that such jets are convectively unstable with a narrow range of frequencies of maximum amplification, resulting in the formation of a staggered array of vortices with direction opposite to that of the classical Karman street. A stable co-existence of the jet profile and the large-scale patterns is ensured only at the frequency of maximum amplification, hence at this frequency optimal efficiency is obtained, i.e., maximum thrust per unit input energy. The nondimensional frequency of maximum amplification (Strouhal number) is in the range of 0·25 to 0·35. Experiments confirms this results, while the analysis of a large number of data from observations on fish and cetaceans confirm that optimal fish propulsion is achieved within this range of Strouhal number.

A new robust and accurate Cartesian-grid treatment for the immersion of solid bodies within a fluid with general boundary conditions is described. The new approach, the Boundary Data Immersion Method (BDIM), is derived based on a general integration kernel formulation which allows the field equations of each domain and the interfacial conditions to be combined analytically. The resulting governing equation for the complete domain preserves the behavior of the original system in an efficient Cartesian-grid method, including stable and accurate pressure values on the solid boundary. The kernel formulation allows a detailed analysis of the method, and it is demonstrated that BDIM is consistent, obtains second-order convergence relative to the kernel width, and is robust with respect to the grid and boundary alignment. Formulation for no-slip and free slip boundary conditions are derived and numerical results are obtained for the flow past a cylinder and the impact of blunt bodies through a free surface. The BDIM predictions are compared to analytic, experimental and previous numerical results confirming the properties, efficiency and efficacy of this new boundary treatment for Cartesian grid methods.

Ausz. in: Jahrbuch d. Math.-naturwiss. Fak. Göttingen. 1923. - Ersch. auch als Ausz. in: Zeitschrift f. angew. Math. u. Mechan. Bd. 3, H. 4 u. a. T. Göttingen, Math.-naturwiss. Diss., 1923.

General aerodynamic theory-perfect fluids

- T Von Kármán

VON KÁRMÁN, T. 1935 General aerodynamic theory-perfect fluids. Aerodynamic Theory 2, 346-349.