Reverse Von Research Articles

Limiting and optimal Strouhal numbers or tip speed ratios for cruising propulsion by fins, flukes, wings and propellers.

Swimming and flying animals produce thrust with oscillating fins, flukes or wings. The relationship between frequency f, amplitude A and forward velocity U can be described with a Strouhal number St, where St = 2fA/U, where animals are observed to cruise with [Formula: see text]-0.4. Under these conditions, thrust is produced economically and a reverse von Kármán wake is observed. However, propeller-driven craft produce thrust with steadily revolving blades and a helical wake. Here, the simplified aerodynamic geometry of lift-based thrust production is described, applicable to both oscillating and revolving foils. The same geometric principles apply in both cases: if the foil moves too slowly, it cannot produce thrust; if it moves too fast, it produces thrust with excessive power demand. Effective, economic thrust production by animals is not the result of oscillating foils or cyclic vortex shedding; rather, the selection of amplitude and frequency, and wake vortex structure, are corollaries of driving an efficient foil velocity with finite amplitudes. Observed Strouhal numbers for cruising animals appear too low for optimal mechanical efficiency; however, the deviation from optimal efficiency may be small, and there are physical and physiological advantages to relatively low amplitudes and frequencies for swimming and flapping flight.

On plunging using double-peak kinematics

This study investigates the thrust generation mechanism and performance of a plunging airfoil undergoing double-peak periodic motion, specifically utilizing superposition of the regular sinusoidal waveform with another less-weighted faster one. It is revealed that the proposed waveform produces 200% more thrust when compared to sinusoidal plunging with a small efficiency difference. Also, it produces two vortex pairs (2P) per cycle all over the plunging frequency-amplitude space, instead of the reverse von Karman vortices. Using this proposed waveform, the 2P wake is generated through interaction of three types of vortices shed during plunging, opposing the regular 2P pattern found in the literature.

Vortex-shedding modes of a pair of side-by-side thin pitching plates

We have conducted three-dimensional (3D) direct numerical simulations to analyze vortex-shedding modes past vertically arranged, side-by-side 3D pitching plates, considering cases where the plates are in the same, opposite, and different phases. The simulations are performed at a Reynolds number Re=1000, with a dimensionless pitching frequency St=1, and a maximum pitching angle θmax=15°. The vortex-shedding modes reveal horseshoe vortices transforming into distorted hairpin-like structures in the far wake for in-phase plates. Opposite phase plates exhibit helical distortion and core bifurcation, while different phase angles produce meridionally twisted, entangled hairpins. We observe a reverse von Kármán vortex street for in- and different-phase plates, while the opposite shows a reverse Kármán vortex street for the upper plate and a Kármán vortex street for the lower plate. The streakline visualizations reveal wake compression and spoke-like structures symmetrically emanating from the shear layer extremities around the plates' central region. We find leapfrog-like cross-stream vortex interaction when the plates are in phase. The line-time diagram reveals alternating patches of low-intensity cells, switching between negative and positive, alongside continuous bands of both positive and negative cells. Both plates produce similar thrust for the parameters examined in this study.

Hydrodynamics and propulsion of a hydrofoil undergoing leading-edge pitching and traveling wave-based surface undulation

We numerically study the fluid–structure interaction of a free-stream flow across a hydrofoil pitching at its leading edge with superimposed traveling wave-based surface undulations. We utilize an in-house code that employs the sharp interface immersed boundary method and consider a constant pitching amplitude θ0 = 5°, a constant local amplitude-to-thickness ratio AL=0.15, and wave number K = 20 of surface undulation. We compare the effect of surface undulation on a pitching hydrofoil with that of a hydrofoil undergoing pure pitching or experiencing pure surface undulation. The findings reveal that surface undulation on the pitching hydrofoil increases thrust on the hydrofoil. The onset of asymmetry in the vortex street occurs at a lower pitching Strouhal number (St) due to the early formation of a vortex dipole. In addition to the presence of an asymmetric inverse von Kármán vortex street, higher pitching frequencies reveal re-deflection of the asymmetric inverse von Kármán vortices. We quantified dynamics of vortex dipole to explain the occurrence of asymmetric and re-deflected reverse von Kármán vortex street. Furthermore, the analysis reveals an optimum combination of St and phase speed that yields higher propulsive efficiency, as both motions compete in generating thrust. A linearly superimposed scaling analysis for the time-averaged thrust of the combined motion is also presented. The computations and scaling are found to be in good agreement.

Numerical Study of Circular-Cylinder Disturbance Generators with Rigid Splitter Plates

This paper describes the numerical study of oscillating circular cylinders with rigid splitter plates of different lengths. These geometries may be used as disturbance generators for the study of unsteady airfoils and wings operating in highly vortical flowfields. It has been shown that cylinders undergoing forced rotational oscillations at their natural shedding frequency can produce wakes with minimal deviation in cycle-to-cycle vortex strength and position. Adding a splitter plate allows these deviations to be reduced even further. We present cases for oscillating cylinders having splitter-plate lengths up to at a Reynolds number of 7600. Frequencies are maintained at the natural shedding frequency, and a rotational amplitude of 45 deg is used. Numerical simulations are performed using a two-dimensional unsteady Reynolds-averaged Navier–Stokes (RANS) code. Results are presented in the form of vorticity contours and cycle-averaged velocity profiles, as well as the dominant frequencies of cylinder lift force and downstream velocity angles. The results show that splitter-plate lengths shorter than adversely affect the ability to generate a coherent vortex wake due to shear layer roll-up near the trailing edge of the plate. Splitter plates longer than produced a reverse von Kármán wake with consistent cycle-to-cycle vortex shedding.

An unsteady RANS simulation of the performance of an oscillating hydrofoil at a high Reynolds number

The oscillating hydrofoil has been widely used in ocean engineering applications such as ships, underwater vehicles and ocean energy devices. However, Reynolds number of existing research are lower than real ocean applications.This study reports our recent numerical effort in studying a sinusoidally oscillating hydrofoil under unsteady conditions by solving the Reynolds-averaged Navier–Stokes equations. By using experimental test data, the numerical method is well validated. Three wake patterns are identified to fundamentally affect the propulsive performance of the oscillating hydrofoil. Only the wake in the reverse von Kármán vortex street pattern is found to generate thrust for purely oscillating motion. The leading-edge vortex is demonstrated todirectly influence the pressure distribution over the foil’s surface but not the thrust.Furthermore, impact of 3D effect on vortex shedding on the hydrodynamic performance of the hydrofoil is presented. This study is expected to provide guidance on both academics and industries in relevant fields.

Asymmetric wakes in flows past circular cylinders confined in channels

In this paper, we investigate the flow past a circular cylinder confined in a channel at a blockage ratio of $\beta =0.7$ (the ratio of the cylinder diameter and the channel height) for Reynolds numbers between ${\textit {Re}}=300$ and $3900$ using direct numerical simulation (DNS). We show for varying Reynolds numbers a wide range of wake dynamics occur as the spanwise domain length is changed. At a lower Reynolds number of ${\textit {Re}}=300$ , a reverse von Kármán wake alongside either a top- or bottom-biased asymmetry was observed at different spanwise locations. The asymmetry was structurally similar the two-dimensional asymmetry studied by prior investigators, and was found to be a result of the confinement effect. Further, wake-jumping between the two intermittent states was present. For larger Reynolds numbers, ${\textit {Re}}=1000$ and $3900$ , these asymmetric structures were found to become dominant. We also examine the dependence of the asymmetries on the spanwise domain. For small spanwise domains the asymmetries were uniformly orientated across the span. In contrast, for sufficiently large spanwise domains, the asymmetry flips its orientation at different spanwise locations. Comparisons of flow statistics demonstrate good agreement between the different spanwise domains, which suggests the same mechanism maintains the asymmetry in both cases. Further analysis at ${\textit {Re}}=1000$ found the number of times the wake flips is dependent on the initial conditions, with a wake that flips zero (purely asymmetric), two and four times being observed. These structures were also determined to remain stable over time scales of $1000D/U$ .

Effect of aspect ratio on the wake transition behind a thin pitching plate

We study the influence of Aspect Ratio (AR) on three-dimensional wake transition past a thin pitching plate at Reynolds number of 1000 by performing computations for the range 0.54≤AR≤16 at pitching frequencies St=0.5,1 and maximum pitching angles θmax=5°,15°. For all AR, larger θmax and St promote thrust generation. However, higher AR imparts a stabilizing influence in the wake of the drag regime. For the ranges of AR, the drag-producing wake consisting of horseshoe vortices and bridgelets-type vortex structures, whereas twin-jet type bifurcated wakes with entangled vortices are observed for thrust-generating wakes. At higher AR, the wakes show a two-dimensional signature in the drag regime, whereas a spatial wake transition is observed in the thurst regime. The spanwise wake width shows the effect of wake compression for larger St even at θmax=5°, which is also substantiated by particle tracking showing wake compression for the thrust cases up to AR≤12. The near wake oscillations are prevalent for higher AR, although the core region remains unaffected by the aiding influence of spanwise instability for larger AR. The time average streamwise velocity for both drag and thrust regimes resembles an apparent feature of the reverse von Kármán vortex street.

Propulsive performance of a two-dimensional elliptic foil undergoing interlinked pitching and heaving

We computationally study the propulsive performance of a two-dimensional elliptic foil undergoing interlinked pitching-heaving motion. This motion is realized by pitching the foil about an axis on its centerline outside the foil and by varying the distance between the pitching point and the leading edge. A distance of 0 and −∞ corresponds to leading edge pitching and pure heaving. An in-house fluid-structure interaction solver based on the sharp interface immersed boundary method is employed to resolve the flow field around the foil. We conducted simulations for different cases of the location of the pitching axis and pitching frequency at a Reynolds number of 100. The thrust generation is explained by the dynamics of leading-edge and trailing-edge vortices. The wake corresponding to thrust is either reverse von Kármán or a deflected reverse von Kármán vortex street. Analysis revealed the existence of an optimal pitching point for maximum thrust or propulsive efficiency at a given reduced pitching frequency. The optimal regions of the thrust and propulsive efficiency are quantified as a function of reduced pitching frequency and the location of the pitching axis. The pitching point for the maximum thrust and efficiency is found to be different. We discuss the fluid-mechanical reasons for the variation of propulsive performance with the location of the pitching point and the pitching frequency and corroborate our reasoning with the wake signatures.

Route to transition in propulsive performance of oscillating foil.

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.

Propulsive performance of a pitching foil in a side-by-side arrangement with auxiliary pitching foil

A two-dimensional computational study is performed in the present work to simulate two NACA 0012 (National Advisory Committee for Aeronautics) airfoils pitching in a side-by-side configuration at Reynolds Number (Re) 12,000. An auxiliary airfoil with a smaller size is used to enhance the efficiency of the main airfoil. The thrust and lift coefficients of the main airfoil are quantified and compared with an isolated pitching airfoil for a range of reduced frequencies (k=1 to 8) and phase differences (Φ=0°to 180°) between the airfoils. The thrust force increases gradually with a maximum enhancement at an intermediate phase difference (Φ) between the airfoils and decreases further. The simulations cover three main flow regimes depending upon the vortex structure shed by the airfoils, i.e. (i) when both airfoil shed von Kármán (KV) vortex street (ii) auxiliary airfoil sheds von Kármán whereas main airfoil sheds reverse von Kármán (RKV) vortex street (iii) both airfoil sheds reverse von Kármán vortex street. The numerical results reveal that an auxiliary airfoil significantly changes the main airfoil’s hydrodynamic characteristics compared to an isolated airfoil. Maximum thrust enhancement of 23% and maximum efficiency enhancement of 49.2% are obtained compared to an isolated airfoil. The phase difference significantly impacts the propulsive efficiency of auxiliary foil, whereas its influence is marginal on the main foil. The interaction of different wake structures shed by the auxiliary, and the main airfoil leads to the formation of the deflected jet. The jet deflection and wake width are quantified to understand the influence of an auxiliary airfoil in a side-by-side configuration with the main airfoil. The finding of this study is expected to provide new insight for developing and enhancing the performance of hydrodynamic propulsive systems by adding an auxiliary airfoil.

Body-caudal fin fish-inspired self-propulsion study on burst-and-coast and continuous swimming of a hydrofoil model

The present study examines the energy efficiency of self-propelled hydrofoils for various modes and kinematics of swimming adopted by various body-caudal fin fish. In particular, this work considers the intermittent burst-and-coast (B&amp;C) and continuous swimming modes, and examines the effect of the undulating and/or pitching swimming kinematics, adopted by the undulating body of anguilliform fish and pitching caudal fin of carangiform and thunniform fish. Notably, B&amp;C swimming is adopted in nature mostly by the latter class but rarely by the former. This fact forms the basis of our study on the hydrodynamics and propulsion performance for both classes of fish-inspired swimming using a NACA0012 hydrofoil model. This analysis explores a large parameter space covering undulation wavelength, 0.8≤λ*&amp;lt;∞, Reynolds number, 50≤Ref≤1500, and duty cycle (DC), 0.1≤DC≤1, with the DC representing the fraction of time in B&amp;C swimming. The fluid–structure dynamics-based vortex-shedding-process is investigated, where B&amp;C swimming results in either an asymmetric reverse von Karman (RVK) or forward von Karman vortex street, rather than a symmetric RVK vortex street observed during continuous swimming. It is demonstrated that the B&amp;C swimming results in an energy saving, although there is a concomitant increase in the travel time. Moreover, our results show that B&amp;C swimming is effective for carangiform and thunniform tail-like kinematics but not for anguilliform body-like kinematics of the hydrofoil. Thus, the predictions are consistent with the observed swimming behavior adopted by a fish in nature and provide input into the efficient design of unmanned underwater vehicles.

Streamwise and lateral maneuvers of a fish-inspired hydrofoil

Fish are highly maneuverable compared to human-made underwater vehicles. Maneuvers are inherently transient, so they are often studied via observations of fish and fish-like robots, where their dynamics cannot be recorded directly. To study maneuvers in isolation, we designed a new kind of wireless carriage whose air bushings allow a hydrofoil to maneuver semi-autonomously in a water channel. We show that modulating the hydrofoil's frequency, amplitude, pitch bias, and stroke speed ratio (pitching speed of left vs right stroke) produces streamwise and lateral maneuvers with mixed effectiveness. Modulating pitch bias, for example, produces quasi-steady lateral maneuvers with classic reverse von Kármán wakes, whereas modulating the stroke speed ratio produces sudden yaw torques and vortex pairs like those observed behind turning zebrafish. Our findings provide a new framework for considering in-plane maneuvers and streamwise/lateral trajectory corrections in fish and fish-inspired robots.

Spatial wake transition past a thin pitching plate.

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.

Propulsion performance and wake transitions of a customized heaving airfoil

The application of a flapping wing mechanism offers a vast range of development possibilities for unmanned aerial vehicles (UAVs) and autonomous underwater vehicles (AUVs). The influence of wake transitions on flapping wing mechanism’s capabilities is not fully understood particularly at low Reynolds numbers. The numerical investigation of a symmetric airfoil performing sinusoidal heaving oscillations is performed to explore the wake transitions. The influence of heaving parameters on wake transitions when exposed to a constant velocity flow is investigated. The existence of reverse von Karman vortex street, deflected wake and chaotic wake is observed. The wake deflection is found to switch its direction before transforming into a chaotic wake. The coherent structures and its evolution with the flow are presented using proper orthogonal decomposition (POD). The underlying structures and their interactions for different wake situations are identified. Correlations for the nondimensional maximum velocity in the wake in terms of frequency and amplitude is proposed. The wake dynamics is found to depend significantly on the leading edge vortices. The time-varying velocity fluctuations in the flow field are presented and discussed in detail. The velocity fluctuation contours are used to identify the regions of momentum transfer. The transient nature of the flow field is studied using the phase plot. A transition route from the periodic to chaotic regime though a quasi-periodic regime is established using time series analysis. The wake transitions are observed to be more sensitive towards heaving frequency than the heaving amplitude.

Investigations into transient wakes behind a custom airfoil undergoing pitching motion

The wake dynamics behind a sinusoidally pitching custom designed symmetrical airfoil is analyzed computationally. The effect of pitching frequency and pitching amplitude is investigated independently at a constant Reynolds number. The wake patterns are found to exhibit a transition from von Karman to reverse von Karman vortex street and from symmetric reverse von Karman vortex street to asymmetric deflected wake as established. For laminar flow under investigation, the contours of time-varying velocity fluctuations are presented. The mean profiles of velocity and the fluctuations at different locations in the wake show that the thrust wakes are generated at the expense of energy of the surrounding fluid. The temporal behavior of the flow field is analyzed using the phase plot. The time series plots of force coefficients show a periodic loading on the airfoil. Proper orthogonal decomposition (POD) analysis of drag, neutral and thrust wakes displayed the evolution of coherent structures in the flow field. The evolution of these structures portray the transitions in the wake of the airfoil. The merging of coherent structures in an asymmetric deflected wake turns out to be an efficient way of identifying the location of deflection.

Thrust generation by pitching and heaving of an elastic plate at low Reynolds number

We computationally study thrust generation and propulsive characteristics of an elastic plate pitching and/or heaving in free stream laminar flow. The pitching is considered about the leading edge, and the Reynolds number based on the plate length and free stream velocity is 150. An in-house fluid–structure interaction (FSI) solver is employed to simulate the large-scale flow-induced deformation of the structure along with active pitching and heaving in two-dimensional coordinates. The FSI solver utilizes a partitioned approach to strongly couple a sharp-interface immersed boundary method based flow solver with an open-source finite-element structural dynamics solver. We elucidate the mechanism of the thrust generation in the rigid and elastic plate by comparing the time-variation of thrust and work done by the plate, together with the wake signatures in the downstream. The time variation of the thrust is explained using first-order scaling arguments. The computed thrust as a function of pitching frequency for the rigid pitching plate shows a similar trend as compared to the published data of rigid foils, while the elastic plate exhibits a strong influence of the flow-induced deformation of the plate. They both exhibit reverse von Kármán-like vortex shedding in the downstream. We quantify the differences in propulsive characteristics of these two plate types as a function of pitching frequency. We found that there lies an optimum pitching frequency for the elastic plate for efficient propulsion, while the rigid one outperforms the elastic plate at larger pitching frequency. This is due to the fact that the elastic plate locks in to a higher mode of vibration at a larger pitching frequency. Furthermore, the influence of mass ratio, flexural rigidity, pitching amplitude, and Reynolds number on the performance of the elastic plate is also investigated. Finally, we study the combined effect of pitching and heaving on the propulsive performance. The pitching frequency for the maximum efficiency is lesser for the combined heaving and pitching plate as compared to only heaving or only pitching. Our results provide fundamental insights into the propulsive characteristics of the elastic pitching and/or heaving plates, which could help design autonomous underwater vehicles.

Flow Interactions Between Low Aspect Ratio Hydrofoils in In-line and Staggered Arrangements.

Many species of fish gather in dense collectives or schools where there are significant flow interactions from their shed wakes. Commonly, these swimmers shed a classic reverse von Kármán wake, however, schooling eels produce a bifurcated wake topology with two vortex rings shed per oscillation cycle. To examine the schooling interactions of a hydrofoil with a bifurcated wake topology, we present tomographic particle image velocimetry (tomo PIV) measurements of the flow interactions and direct force measurements of the performance of two low-aspect-ratio hydrofoils () in an in-line and a staggered arrangement. Surprisingly, when the leader and follower are interacting in either arrangement there are only minor alterations to the flowfields beyond the superposition of the flowfields produced by the isolated leader and follower. Motivated by this finding, Garrick’s linear theory, a linear unsteady hydrofoil theory based on a potential flow assumption, was adapted to predict the lift and thrust performance of the follower. Here, the follower hydrofoil interacting with the leader’s wake is considered as the superposition of an isolated pitching foil with a time-varying cross-stream velocity derived from the wake flow measurements of the isolated leader. Linear theory predictions accurately capture the time-averaged lift force and some of the major peaks in thrust derived from the follower interacting with the leader’s wake in a staggered arrangement. The thrust peaks that are not predicted by linear theory are likely driven by spatial variations in the flowfield acting on the follower or nonlinear flow interactions; neither of which are accounted for in the simple theory. This suggests that unsteady potential flow theory that does account for spatial variations in the flowfield acting on a hydrofoil can provide a relatively simple framework to understand and model the flow interactions that occur in schooling fish. Additionally, schooling eels can derive thrust and efficiency increases of 63-80% in either a in-line or a staggered arrangement where the follower is between two branched momentum jets or with one momentum jet branch directly impinging on it, respectively.

The Effect of Pitching Frequency on the Hydrodynamics of Oscillating Foils

Abstract The effects of two different pitching frequencies (that is, Strouhal number, St) on the wake structure generated by two foils of aspect ratio 1.0 are examined numerically at a Reynolds number of 10,000. Strouhal numbers of 0.5 and 0.2 were studied, the first corresponding approximately to the peak in efficiency and the second corresponding to the point where the thrust is equal to the drag (the free-swimming condition). The two foils have either a square trailing edge or a convex trailing edge that mimics the shape of the caudal fin exhibited by certain species of fish. In previous works, the convex trailing edge panel was found to have higher thrust and efficiency compared with the square panel trailing edge. Here, these differences are related to their characteristic vortex formation and detachment processes leading to differences in wake coherence and extension. The wake of the square panel at St = 0.2 transitions slowly from a reverse von Kármán street (2S) pattern to a paired (2P) system as the wake develops downstream, whereas at St = 0.5, the wake almost immediately takes on a 2P form with an attendant split in the wake structure. For the convex panel, the transition from a 2S to a 2P structure at St = 0.2 is slower than that seen for the square panel, and for St = 0.5, the wake undergoes an abrupt transition leading to two distinct vortex streets that evolve at a considerably slower rate than seen for the square panel.

Thrust, drag and wake structure in flapping compliant membrane wings

We present a theoretical framework to characterize the steady and unsteady aeroelastic behaviour of compliant membrane wings under different conditions. We develop an analytic model based on thin airfoil theory coupled with a membrane equation. Adopting a numerical solution to the model equations, we study the effects of wing compliance, inertia and flapping kinematics on aerodynamic performance. The effects of added mass and fluid damping on a flapping membrane are quantified using a simple damped oscillator model. As the flapping frequency is increased, membranes go through a transition from thrust to drag around the resonant frequency, and this transition is earlier for more compliant membranes. The wake also undergoes a transition from a reverse von Kármán wake to a traditional von Kármán wake. The wake transition frequency is predicted to be higher than the thrust–drag transition frequency for highly compliant wings.