A simple vortex-loop-based model for unsteady rotating wings

An analytical model is developed for the lift force produced by unsteady rotating wings; this configuration is a simple representation of a flapping wing. Modelling this is important for the aerodynamic and control-system design for bio-inspired drones. Such efforts have often been limited to being two-dimensional, semi-empirical, sometimes computationally expensive, or quasi-steady. The current model is unsteady and three-dimensional, yet simple to implement, requiring knowledge of only the wing kinematics and geometry. Rotating wings produce a vortex loop consisting of the root vortex, leading-edge vortex, tip vortex and trailing-edge vortex, which grows with time. This is modelled as a tilted planar loop, geometrically specified by the wing size, orientation and motion. By equating the angular impulse of the vortex loop to that of the fluid volume driven by the wing, the circulatory lift force is derived. Potential flow theory gives the fluid-inertial lift. Adding these two contributions yields the total lift formula. The model shows good agreement with a range of experimental and computational cases. Also, a steady-state lift model is developed that compares well with previous work for various angles of attack.

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The vortex wake of a ‘hovering’ model hawkmoth

Philosophical Transactions of the Royal Society B Biological Sciences ◽

10.1098/rstb.1997.0023 ◽

1997 ◽

Vol 352 (1351) ◽

pp. 317-328 ◽

Cited By ~ 124

Author(s):

Coen van den Berg ◽

Charles P. Ellington

Keyword(s):

Lift Force ◽

Three Dimensional ◽

Axial Flow ◽

Leading Edge ◽

The Body ◽

Tip Vortex ◽

Vortex Rings ◽

Leading Edge Vortex ◽

Edge Vortex ◽

Visualization Experiments

Visualization experiments with Manduca sexta have revealed the presence of a leading–edge vortex and a highly three–dimensional flow pattern. To further investigate this important discovery, a scaled–up robotic insect was built (the ‘flapper’) which could mimic the complex movements of the wings of a hovering hawkmoth. Smoke released from the leading edge of the flapper wing revealed a small but strong leading–edge vortex on the downstroke. This vortex had a high axial flow velocity and was stable, separating from the wing at approximately 75 % of the wing length. It connected to a large, tangled tip vortex, extending back to a combining stopping and starting vortex from pronation. At the end of the downstroke, the wake could be approximated as one vortex ring per wing. Based on the size and velocity of the vortex rings, the mean lift force during the downstroke was estimated to be about 1.5 times the body weight of a hawkmoth, confirming that the downstroke is the main provider of lift force.

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Three-dimensional vortex wake structure of flapping wings in hovering flight

Journal of The Royal Society Interface ◽

10.1098/rsif.2013.0984 ◽

2014 ◽

Vol 11 (91) ◽

pp. 20130984 ◽

Cited By ~ 18

Author(s):

Bo Cheng ◽

Jesse Roll ◽

Yun Liu ◽

Daniel R. Troolin ◽

Xinyan Deng

Keyword(s):

Experimental Studies ◽

Three Dimensional ◽

Leading Edge ◽

Ring Structure ◽

Tip Vortex ◽

Flapping Wings ◽

Vortex Wake ◽

Wake Structure ◽

Edge Vortex ◽

Vorticity Transport

Flapping wings continuously create and send vortices into their wake, while imparting downward momentum into the surrounding fluid. However, experimental studies concerning the details of the three-dimensional vorticity distribution and evolution in the far wake are limited. In this study, the three-dimensional vortex wake structure in both the near and far field of a dynamically scaled flapping wing was investigated experimentally, using volumetric three-component velocimetry. A single wing, with shape and kinematics similar to those of a fruitfly, was examined. The overall result of the wing action is to create an integrated vortex structure consisting of a tip vortex (TV), trailing-edge shear layer (TESL) and leading-edge vortex. The TESL rolls up into a root vortex (RV) as it is shed from the wing, and together with the TV, contracts radially and stretches tangentially in the downstream wake. The downwash is distributed in an arc-shaped region enclosed by the stretched tangential vorticity of the TVs and the RVs. A closed vortex ring structure is not observed in the current study owing to the lack of well-established starting and stopping vortex structures that smoothly connect the TV and RV. An evaluation of the vorticity transport equation shows that both the TV and the RV undergo vortex stretching while convecting downwards: a three-dimensional phenomenon in rotating flows. It also confirms that convection and secondary tilting and stretching effects dominate the evolution of vorticity.

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Flow structure on a simultaneously pitching and rotating wing

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.458 ◽

2014 ◽

Vol 756 ◽

pp. 354-383 ◽

Cited By ~ 20

Author(s):

M. Bross ◽

D. Rockwell

Keyword(s):

Flow Structure ◽

Large Scale ◽

Three Dimensional ◽

Leading Edge ◽

Tip Vortex ◽

Pure Rotation ◽

Large Scale Vortex ◽

Edge Vortex ◽

Vorticity Flux ◽

Rotating Wing

AbstractA technique of particle image velocimetry is employed to characterize the three-dimensional flow structure on a wing subjected to simultaneous pitch-up and rotational motions. Distinctive vortical structures arise, relative to the well-known patterns on a wing undergoing either pure pitch-up or pure rotation. The features associated with these simultaneous motions include: stabilization of the large-scale vortex generated at the leading edge, which, for pure pitch-up motion, rapidly departs from the leading-edge region; preservation of the coherent vortex system involving both the tip vortex and the leading-edge vortex (LEV), which is severely degraded for pure rotational motion; and rapid relaxation of the flow structure upon termination of the pitch-up component, whereby the relaxed flow converges to a similar state irrespective of the pitch rate. Three-dimensional surfaces of iso-$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{Q}$and helicity are employed in conjunction with sectional representations of spanwise vorticity, velocity and vorticity flux to interpret the flow physics.

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The three–dimensional leading–edge vortex of a ‘hovering’ model hawkmoth

Philosophical Transactions of the Royal Society B Biological Sciences ◽

10.1098/rstb.1997.0024 ◽

1997 ◽

Vol 352 (1351) ◽

pp. 329-340 ◽

Cited By ~ 212

Author(s):

Coen van den Berg ◽

Charles P. Ellington

Keyword(s):

Wing Length ◽

Three Dimensional ◽

Axial Flow ◽

Leading Edge ◽

Tip Vortex ◽

Flow Visualisation ◽

High Lift ◽

Leading Edge Vortex ◽

Edge Vortex ◽

Wing Chord

Recent flow visualisation experiments with the hawkmoth, Manduca sexta , revealed small but clear leading–edge vortex and a pronounced three–dimensional flow. Details of this flow pattern were studied with a scaled–up, robotic insect (‘the flapper’) that accurately mimicked the wing movements of a hovering hawkmoth. Smoke released from the leading edge of the flapper wing confirmed the existence of a small, strong and stable leading–edge vortex, increasing in size from wingbase to wingtip. Between 25 and 75 % of the wing length, its diameter increased approximately from 10 to 50 % of the wing chord. The leading–edge vortex had a strong axial flow veolocity, which stabilized it and reduced its diamater. The vortex separated from the wing at approximately 75 % of the wing length and thus fed vorticity into a large, tangled tip vortex. If the circulation of the leading–edge vortex were fully used for lift generation, it could support up to two–thirds of the hawkmoth's weight during the downstroke. The growth of this circulation with time and spanwise position clearly identify dynamic stall as the unsteady aerodynamic mechanism responsible for high lift production by hovering hawkmoths and possibly also by many other insect species.

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Aspect-ratio effects on rotating wings: circulation and forces

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.44 ◽

2015 ◽

Vol 767 ◽

pp. 497-525 ◽

Cited By ~ 45

Author(s):

Zakery R. Carr ◽

Adam C. DeVoria ◽

Matthew J. Ringuette

Keyword(s):

Aspect Ratio ◽

Lift Force ◽

Slow Growth ◽

Lift Coefficient ◽

Leading Edge ◽

Tip Vortex ◽

Trailing Edge ◽

Data Set ◽

Edge Vortex ◽

Total Circulation

AbstractWe employ experiments to study aspect ratio ($\def\AR{A\mkern-8muR}\AR$) effects on the vortex structure, circulation and lift force for flat-plate wings rotating from rest at 45° angle of attack, which represents a simplified hovering-wing half-stroke. We use the time-varying, volumetric $\AR =2$ data of Carr et al. (Exp. Fluids, vol. 54, 2013, pp. 1–26), reconstructed from phase-locked, phase-averaged stereoscopic digital particle image velocimetry (S-DPIV), and an $\AR =4$ volumetric data set matching the span-based Reynolds number ($\mathit{Re}$) of $\AR =2$. For $\AR =1{-}4$ and $\mathit{Re}_{\mathit{span}}$ of $O$($10^{3}$–$10^{4}$), we directly measure the lift force. The total leading-edge-region circulation for $\AR =2$ and 4 compares best overall using a span-based normalization and for matching rotation angles. The total circulation increases across the span to the tip region, and is larger for $\AR =2$. After the startup, the total circulation for each $\AR$ has a similar slope and a slow growth. The first leading-edge vortex (LEV) and the tip vortex (TV) for $\AR =4$ move past the trailing edge, followed by substantial breakdown. For $\AR =2$ the outboard, aft-tilted LEV merges with the TV and resides over the tip, although breakdown also occurs. Where the LEV is ‘stable’ inboard, its circulation saturates for $\AR =2$ and the growth slows for $\AR =4$. Aft LEV tilting reduces the spanwise LEV circulation for each $\AR$. Both positive and negative axial flow are found in the first LEV for $\AR =2$ and 4, with the positive component being somewhat larger. This yields a generally positive (outboard) average vorticity flux. The average lift coefficient is essentially constant with $\AR$ from 1 to 4 during the slow growth phase, although the large-time behaviour shows a slight decrease in lift coefficient with increasing $\AR$. The S-DPIV data are used to obtain the lift impulse and the spanwise and streamwise components contributing to the lift coefficient. The spanwise contribution is similar for $\AR =2$ and 4, due to similar trailing-edge vortex interactions, LEV saturation behaviour and total circulation slopes. However, for $\AR =2$ the streamwise contribution is much larger, because of the stronger, coherent TV and aft-tilted LEV, which will create a relatively lower-pressure region over the tip.

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Interplay of the leading-edge vortex and the tip vortex of a low-aspect-ratio thin wing

Experiments in Fluids ◽

10.1007/s00348-020-03023-4 ◽

2020 ◽

Vol 61 (9) ◽

Author(s):

Lei Dong ◽

Kwing-So Choi ◽

Xuerui Mao

Keyword(s):

Aspect Ratio ◽

Angle Of Attack ◽

Three Dimensional ◽

Separated Flow ◽

Leading Edge ◽

Tip Vortex ◽

Leading Edge Vortex ◽

Low Aspect Ratio ◽

Thin Wing ◽

Edge Vortex

Abstract Three-dimensional vortical structures and their interaction over a low-aspect-ratio thin wing have been studied via particle image velocimetry at the chord Reynolds number of $$10^5$$ 10 5 . The maximum lift of this thin wing is found at an angle of attack of $$42^\circ$$ 42 ∘ . The flow separates at the leading-edge and reattaches to the wing surface, forming a strong leading-edge vortex which plays an important role on the total lift. The results show that the induced velocity of the tip vortex increases with the angle of attack, which helps reattach the separated flow and maintains the leading-edge vortex. Turbulent mixing indicated by the high Reynolds stress can be observed near the leading-edge due to an intense interaction between the leading-edge vortex and the tip vortex; however, the reattachment point of the leading-edge vortex moves upstream closer to the wing tip. Graphic abstract

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Interactions of a streamwise-oriented vortex with a finite wing

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.51 ◽

2015 ◽

Vol 767 ◽

pp. 782-810 ◽

Cited By ~ 40

Author(s):

D. J. Garmann ◽

M. R. Visbal

Keyword(s):

Separation Bubble ◽

Angle Of Attack ◽

Three Dimensional ◽

Leading Edge ◽

Tip Vortex ◽

Recirculation Region ◽

Rolling Moment ◽

Effective Angle ◽

Finite Wing ◽

Modes Of Interaction

AbstractA canonical study is developed to investigate the unsteady interactions of a streamwise-oriented vortex impinging upon a finite surface using high-fidelity simulation. As a model problem, an analytically defined vortex superimposed on a free stream is convected towards an aspect-ratio-six ($\mathit{AR}=6$) plate oriented at an angle of ${\it\alpha}=4^{\circ }$ and Reynolds number of $\mathit{Re}=20\,000$ in order to characterize the unsteady modes of interaction resulting from different spanwise positions of the incoming vortex. Outboard, tip-aligned and inboard positioning are shown to produce three distinct flow regimes: when the vortex is positioned outboard of, but in close proximity to, the wingtip, it pairs with the tip vortex to form a dipole that propels itself away from the plate through mutual induction, and also leads to an enhancement of the tip vortex. When the incoming vortex is aligned with the wingtip, the tip vortex is initially strengthened by the proximity of the incident vortex, but both structures attenuate into the wake as instabilities arise in the pair’s feeding sheets from the entrainment of opposite-signed vorticity into either structure. Finally, when the incident vortex is positioned inboard of the wingtip, the vortex bifurcates in the time-mean sense with portions convecting above and below the wing, and the tip vortex is mostly suppressed. The time-mean bifurcation is actually a result of an unsteady spiralling instability in the vortex core that reorients the vortex as it impacts the leading edge, pinches off, and alternately attaches to either side of the wing. The increased effective angle of attack inboard of impingement enhances the three-dimensional recirculation region created by the separated boundary layer off the leading edge which draws fluid from the incident vortex inboard and diminishes its impact on the outboard section of the wing. The slight but remaining downwash present outboard of impingement reduces the effective angle of attack in that region, resulting in a small separation bubble on either side of the wing in the time-mean solution, effectively unloading the tip outboard of impingement and suppressing the tip vortex. All incident vortex positions provide substantial increases in the wing’s lift-to-drag ratio; however, significant sustained rolling moments also result. As the vortex is brought inboard, the rolling moment diminishes and eventually switches sign as the reduced outboard loading balances the augmented sectional lift inboard of impingement.

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The leading-edge vortex on a rotating wing changes markedly beyond a certain central body size

Royal Society Open Science ◽

10.1098/rsos.172197 ◽

2018 ◽

Vol 5 (7) ◽

pp. 172197 ◽

Cited By ~ 5

Author(s):

Shantanu S. Bhat ◽

Jisheng Zhao ◽

John Sheridan ◽

Kerry Hourigan ◽

Mark C. Thompson

Keyword(s):

Reynolds Number ◽

Body Size ◽

Central Body ◽

Three Dimensional ◽

Leading Edge ◽

Particle Image Velocimetry Technique ◽

Leading Edge Vortex ◽

Rapid Acquisition ◽

Edge Vortex ◽

Rotating Wing

Stable attachment of a leading-edge vortex (LEV) plays a key role in generating the high lift on rotating wings with a central body. The central body size can affect the LEV structure broadly in two ways. First, an overall change in the size changes the Reynolds number, which is known to have an influence on the LEV structure. Second, it may affect the Coriolis acceleration acting across the wing, depending on the wing-offset from the axis of rotation. To investigate this, the effects of Reynolds number and the wing-offset are independently studied for a rotating wing. The three-dimensional LEV structure is mapped using a scanning particle image velocimetry technique. The rapid acquisition of images and their correlation are carefully validated. The results presented in this paper show that the LEV structure changes mainly with the Reynolds number. The LEV-split is found to be only minimally affected by changing the central body radius in the range of small offsets, which interestingly includes the range for most insects. However, beyond this small offset range, the LEV-split is found to change dramatically.

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