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Experiments in Fluids (2019) 60:12 https://doi.org/10.1007/s00348-018-2657-2

RESEARCH ARTICLE

Volumetric measurement and vorticity dynamics of leading-edge vortex formation on a revolving wing

Long Chen1,2 · Jianghao Wu1 · Bo Cheng2

Received: 3 July 2018 / Revised: 1 October 2018 / Accepted: 26 November 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractAbstract Leading-edge vortex (LEV) is a hallmark of insect flight that forms and remains stably attached on high angle-of-attack (AoA), low aspect ratio (AR) wings undergoing revolving or flapping motion. Despite the efforts on explaining the stability of LEV when it reaches steady state in revolving wings, its formation process remains largely underexplored. Here, we investigate the LEV formation on a revolving wing (AoA = 45°, AR = 4 and Re = 1500), starting with a constant acceleration in the first chord (c) length of travel and then rotating at a constant velocity. The ‘Shake-the-box’ (STB) Lagrangian particle tracking veloci-metry (PTV) system together with a volumetric patching process were performed to reconstruct the entire time-resolved flow field. Results show that LEV reaches steady state after approximately 4c of travel, within which its formation can be separated into three stages. In the first stage, a conical LEV structure begins to form with a tangential downstream convection of (negative) radial shear vorticity. In the second stage, the radial vorticity within the LEV is further convected downwards by the developed downwash, which limits its growth, whereas vorticity stretching increases the LEV strength. In the third stage, vorticity tilting and spanwise convection reduce the LEV strength at its rear end next to the trailing edge and, therefore, preventing it from grow-ing. Our results suggest that insect wings with AR ~ 4, Re ~ 103 and flapping amplitude between 2c and 4c may barely or even not reach the steady state of LEV, indicating an indispensable role of transient LEV dynamics in understanding insect flight.

Graphical abstract

1.5

1

0.5

0432 61 5 70

Chord length of travel at radius of gyration ( )

LEV *

LEV(- r)

Stage 1(

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mechanisms of insect flight, such as clap and fling, absence of stall, rotational lift and wake capture, have subjected to extensive studies [for reviews, refer to Sane (2003), Sun (2005) and Chin and Lentink (2016)]. Arguably, the most prominent and extensively studied mechanism is the absence of stall, which attributes to the stable attachment of leading-edge vortex (LEV) (Ellington et  al. 1996), and its lift augmentation in revolving or flapping wings operating at high AoAs (Usherwood and Ellington 2002; Lentink and Dickinson 2009; Kruyt et al. 2015). Several possible mechanisms that contribute to the LEV stability have been proposed, such as the spanwise flow convection (Ellington et al. 1996; Liu et al. 1998; Wu and Sun 2004; Poelma et al. 2006), tip-vortex-induced downwash flow (Birch and Dickinson 2001; Shyy and Liu 2007; Cheng et al. 2013), Coriolis effect (Lentink and Dickinson 2009; Jardin and David 2015; Jardin 2017), vorticity stretching (Lim et al. 2009; Wong et al. 2013; Wong and Rival 2015) and vorticity annihilation (Wojcik and Buchholz 2014). In addition, the mechanisms for local LEV stability at differ-ent wing spanwise locations (r) depend on local Rossby number Ro (Harbig et al. 2013, 2014; Garmann and Visbal 2014; Carr et al. 2015; Kruyt et al. 2015), and the overall LEV stability and wing force generation depend on the global Ro, which is proportional to the wing aspect ratio (AR) (Lentink and Dickinson 2009; Lee et al. 2016; Smith et al. 2017), and possibly also on the Reynolds number Re (Birch et al. 2004; Shyy and Liu 2007).

Unlike revolving or flapping wings, translating wings operating at high AoAs and Re comparable to insect flight experience stall (Vogel 1967; Dickinson and Gotz 1993; Chen et al. 2010), similar to those in fixed-wing flight at a higher Re (Batchelor 2000). For example, a rapidly-started translating plate with AoA > 13.5° and Re = 192 (compa-rable to that of fruit flies) undergoes a “delayed stall”, as the LEV formation first enhances the lift generation, which then drops after approximately 2–3 chord (c) lengths of travel due to the LEV shedding (Dickinson and Gotz 1993). Notwithstanding the instability of LEV in trans-lating wings, two dimensional (2D) reciprocal translating wings are used previously to model insect flight (Wang et al. 2004; Wang 2004, 2005). This is considered as a rea-sonable approach because the flapping amplitude of most insects are sufficiently small so that the transient behav-ior of LEV, instead of its steady-state behavior (which only exists in continuously revolving wings) occupies the majority of the insect wing stroke. More specifically, the chord length of travel for a wing stroke, also known as the stroke-arc/wing-chord ratio ( � = �Rg∕c , � denotes stroke amplitude) of most hovering insects is within a small range of 2–5, e.g., butterflies and moths � = 1.4–2.6 (Weis-Fogh 1973), wasps and bees �  =  2.8–4.4 (Weis-Fogh 1973; Ellington 1984a, b), flies � = 1.9–4.7 (Weis-Fogh

1973; Ellington 1984a, b; Bomphrey et al. 2017), except for beetles � = 5.1–5.8 (Weis-Fogh 1973). As a result, the transient effects within relatively small � may play a dominant role in the lift generation of insect flight, such as recently shown in hoverfly (Zhu and Sun 2017) and mosquito (Bomphrey et al. 2017). In addition, a study that compares the aerodynamic force and flow structure of a 2D flapping wing model with those measured from 3D wing experiments at Re = 75–115 finds no significant dif-ference between the two, when the wings are within 3–5 chord length of travel before the LEV shedding (Wang et al. 2004). Together, these results suggest that the tran-sient behavior of LEV during its formation period, instead of those at steady state, might play a more significant role in insect flight.

However, while substantial efforts have focused on elucidating the mechanism of LEV stability in the steady state, transient behavior and dynamics of LEV during its formation period in revolving wings receive significantly less attention. This period is highly relevant to insect flight and is of particular significance for those with small � . Here, we examine the LEV formation and transient vortex dynamics on a revolving wing operating at AoA = 45° and Re = 1500. The wing AR is 4 w.r.t the axis of rotation, including a root cut-off of 1c, and the entire travel of the wing is 8c. We only aim at gaining insights into the phys-ics of LEV formation and its transition to the steady state due to unidirectional wing revolving motion, without con-sidering the effects of wing rotation and stroke reversal, which exist in insect flight and may lead to more complex LEV dynamics. Time-resolved 3D velocity field gener-ated by a dynamically-scaled robotic wing revolving in a mineral-oil tank is measured through the ‘Shake-The-Box’ (STB) particle tacking velocimetry (PTV) system. A volu-metric patching of nine individual measurement volumes is conducted to reconstruct the entire flow field. The vor-ticity dynamics of the LEV is analyzed by evaluating the vorticity transport equation in a wing-fixed rotating frame. The evolution of vorticity and velocity are first described in details. The spanwise components of vorticity dynamics terms, e.g. vorticity convection and tilting and stretching, is then calculated and discussed. Finally, the LEV forma-tion towards the steady state is explained.

2 Materials and methods

2.1 Experimental setup

The wing motion was generated by a dynamically-scaled robotic wing mechanism actuated by digital servos (XM450-W350-R, Robotis, resolution 1°/4096). The motion control

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was implemented by Simulink (The MathWorks, Natick, MA, USA) at a rate of 500 Hz using a real-time machine (Performance Real-Time Target Machine, Speedgoat GmbH, Bern, Switzerland). The experiments were con-ducted in an acrylic tank (0.8 m × 0.8 m × 0.8 m) filled with white mineral oil (Tulco, density � = 826 kg m−3, viscosity � = 6 × 10−6 m2 s−1) (Fig. 1a). More details for the tank and motion control can be found in Wu et al. (2018) and Chen et al. (2018).

The wing was a rectangular transparent acrylic plate, with chord c = 0.04 m, span b = 0.12 m, thickness 1.5 × 10−3 m and a root cut-off of 1c. All surfaces of the wing were well polished to reduce diffuse reflections from surfaces and edges while recording. The wing underwent a constant acceleration within the first 1c of rotation at the radius of gyration, Rg (defined in Eq. 1), and then reached a constant angular speed of 2.11 rad s−1 ( Ω0 in Fig. 1b).

The total rotation was 8c at Rg , corresponding to approxi-mately 180°. The angle of attack (AoA) was fixed at 45° throughout the rotation. The Reynolds number (Re) at the constant revolving speed (at Rg ) was 1500, which was defined according to

where � denotes the kinematic viscosity of the oil. Note that the velocity at Rg has been commonly used as the reference velocity in revolving wings (Lua et al. 2014; Lee et al. 2016; Smith et al. 2017). The wing also remained approximately rigid during the experiments with negligible bending.

(1)Rg =

√b+c

∫0

r2cdr/bc.

(2)Re = Ω0Rgc/�,

2.2 Time‑resolved volumetric flow measurement

The three-dimensional flow generated by the revolving wing was measured in a time-resolved fashion using the state-of-the-art 3D Lagrangian particle tracking velocimetry (PTV) system (Shake-the-Box, LaVision Inc., Ypsilanti, MI, USA). Previous results have suggested that, for a vortex flow with high particle density, the ‘Shake-The-Box’ parti-cle tracking algorithm can achieve a higher reconstruction accuracy at much faster processing speed compared with the tomographic particle imaging velocimetry (Tomo-PIV) (Schanz et al. 2014, 2016; Schröder et al. 2015). The Shake-The-Box’ system uses a “MiniShaker” integrated camera system (four digital cameras) and a ‘Flashlight-300’ LED array (pulsed, illumination time 300 µs) for recording and illumination, respectively. A schematic of the complete setup is shown in Fig. 2a. All cameras run at a frame rate of 100 Hz with sync triggers sent by the real-time machine. The LED light, passing through a window on the side wall (Figs. 1a, 2a), produced a 0.8 m × 0.25 m × 0.12 m illumi-nated volume in approximation. The measurement volume was 0.21 m × 0.18 m × 0.1 m, determined by the enabled camera pixels (1023 × 900 pixel) and the dimensions of illuminated region. Polyamide spheres (55 µm, LaVision Inc., Ypsilanti, MI, USA) were used as seeding particles. A 10-min interval was set between individual measurements to ensure a stationary fluid environment, and the settling velocities of the seeding particles were negligible compared with the velocities caused by wing motion (terminal velocity estimated based on Stokes’ law was 0.08 mm s−1).

To measure and reconstruct the entire flow field gener-ated by the revolving wing, a volumetric patching based on total nine individual measurements was performed (Sect. 2.3). The measurement of each individual sub-vol-ume was repeated five times and then averaged to reduce measurement noise. The particle detection, volumetric

Mineral Oil

80cm80cm

74cm10cm

21cm

18cm

Camera

LED

(a)

0

0 1 2 3 4 5 6 7 8(0o) (180o)

(b)

Revolving

Chord lengths of travel at radius of gyration

Ω0

Fig. 1 Schematics of experimental setup and wing kin-ematics. a Experimental setup showing the locations of Min-iShaker camera, LED array, robotic wing, measurement volume

(0.21  m × 0.18  m × 0.1  m) and acrylic tank (0.8  m × 0.8  m × 0.8  m). b Wing angular velocity profile. The flow data are captured over the entire time range of motion (yellow-shaded window)

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self-calibration, particle tracking and velocity interpolation were performed by DaVis 8.4 (LaVision Inc., Ypsilanti, MI, USA). To improve the calibration accuracy, the perspective calibration was modified through multiple iterations of volu-metric self-calibration using the disparity map calculated from particle images. The particle velocities were calcu-lated based on their trajectories, which were reconstructed by the system using an iterative particle reconstruction (IPR) technique in combination with a 4D-PTV algorithm. The velocity vectors of the particles in the measurement volume (1014 × 856 × 470 voxel) were further interrogated by a 40 × 40 × 40 voxel control volume to generate the veloc-ity data on a 51 × 43 × 24 rectangular grid (a 50% overlap between individual control volumes). The spatial resolution (Δ) of the mesh grid was 0.41 cm in all three dimensions.

2.3 Volumetric patching of individual measurement

To capture the flow and reconstruct the velocity field in both the near and far wake of the wing while ensuring suf-ficient spatial resolution, a volumetric patching of nine indi-vidual sub-volumes (0.21 m × 0.18 m × 0.1 m) in a rotated sequence was performed in the post-processing (Fig. 2a, b). The measurement for each sub-volume was done by rotat-ing the starting orientation of the wing counter-clockwisely (increasing �0 in Fig. 2a), whereas the location of the PTV measurement volume and the wing velocity profile were fixed for all measurements of sub-volumes. This was equiva-lent to virtual rotations of measurement volume so that nine individual measurement volumes in a rotated sequence were generated around the wing-rotating axis (from light red to

φ00o

20o

60o

40o

160o

140o

120o

100o80o

Plate

(b)

(c)

X

ZO

Sub-volume #1

#2

#3#4 #5

#6

#7

#8

#9

σpiδ

p

i

LED

Camera

Mineral Oil

(a)

X

Z

Revolving

φ0

(d)

etα

O

Y

X

Z

Revolving

φ

eyer

θ

160o 0o

Fig. 2 Patching total measurement volume from individual sub-vol-umes and definition of coordinates. a Top view of the experimental setup showing the initial wing positions ( �

0 ) of the nine individual

measurements arranged for volumetric patching (Sect. 2.3). For each measurement, the wing rotates clockwise (green arrow), the ini-tial location of which varies from �

0= 0° (light blue) to �

0= 160°

(dark blue) with an interval of 20°. The color-coded blue arrow also shows the change of �

0 . The global frame is defined when the wing

is at zero initial orientation. b Spatial distribution of the sub-volume 1 ( �

0 = 0°) to 9 ( �

0 = 160°) in the global frame. c Spatial averag-

ing of grid data points of individual sub-volumes (dashed grid and blue points) for the grid data points in the total measurement vol-ume (solid grid and red point). � represents the radius that circles the neighborhood for the weighted averaging. �pi represents the distance between a grid point p in the total measurement volume and a grid point i in the individual sub-volume. d Rotating Cartesian frame ( ̂et , êy , êr ), the axes of which vary with the azimuth angle (θ) of the grid nodes, is employed to visualize the tangential and radial components of the flow structure

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dark red in Fig. 2b), corresponding to the sub-volume 1–9. The flow was recorded over the entire time domain for all measurements of sub-volumes (Fig. 1b). Overlap regions were introduced between individual sub-volumes to reduce the error at the volume boundaries. As shown in Fig. 2b, the nine sub-volumes covered a large spatial domain that was expected to capture all primary flow features generated by the revolving wing during the 180° sweep. Note that the pre-cise control of wing motion and PTV measurement timings ensured the repeatability of kinematics and flow features at low Re, and, therefore, the feasibility of the volumetric patching. Similar approach had been used in previous stud-ies (Poelma et al. 2006; Carr et al. 2013; Cheng et al. 2014).

The patching process is further described here in details. First, a Cartesian grid, with the same spatial resolution (Δ) to that of individual sub-volumes, was generated in the global frame for the total measurement volume. As the sub-volumes were spatially distributed in a rotated sequence around the wing-rotating axis, the grid points of total meas-urement volume and each individual sub-volume mostly did not coincide with each other. Therefore, as described below, a spatial searching-averaging algorithm was developed to calculate the velocity on the grid of the total measurement volume. For each spatial layer of data points along Ŷ axis, the velocity ( �� ) on a grid point (p) of the total measure-ment volume was determined by a weighted averaging of the velocity ( �� ) of all sub-volume grid points (i) within a small neighborhood of p (Eq. 3).

The neighborhood was circled by a radius ( � ) of 0.9Δ (Fig. 2c), which was selected after a sensitivity study. The weight is the reverse of the distance ( �i ) between the grid point of total volume (p) and individual sub-volumes (i), i.e., a sub-volume grid point i closer to the total-volume grid point p received a heavier weight.

2.4 Coordinate systems

The origin of the global frame ( X̂ , Ŷ , Ẑ ) was located at the center of the tank with positive Ŷ pointing upward and positive Ẑ from wing tip to root when it is at zero initial orientation (Fig. 2a, b). To more clearly display and ana-lyze the vorticity distribution and dynamics pertaining to the rotational nature of the revolving wing, a set of rotat-ing Cartesian frames ( ̂et , êy , êr ) was introduced (Fig. 2d) (Cheng et al. 2013). The tangential ( ̂et ) and radial ( ̂er ) axes in the rotating Cartesian frames were rotated on the hori-zontal plane according to the azimuth angle (θ) of the grid point being analyzed, while the vertical axis ( ̂ey ) was aligned with the wing rotating axis ( ̂Y ). Note that the rotating Car-tesian frames were independent of wing position. All vector

(3)�� =∑(

��∕�pi)/∑ (

1∕�pi).

quantities in the global frame were then projected into the rotating Cartesian frames using the rotation matrix J(�) . For example, the velocity vectors in the rotating Cartesian frames ( ut , uy , ur ) can be calculated from those in the global frame ( ux , uy , uz ) according to,

2.5 Data analysis

The constructed velocity field in the total measurement volume was analyzed in MATLAB (The MathWorks, Natick, MA, USA) for studying the evolution and dynamics of the LEV generated by the revolving wing. The analysis was based on the vorticity transport equation in wing-fixed rotating frame,

where �∗ = � −� × r and �∗ = � − 2� , represented the rel-ative velocity and vorticity vectors in the wing-fixed rotating frame ( � and � represented the same quantities in the global non-rotating frame); −(�∗ ⋅ ∇)�∗ , (� ⋅ ∇)�∗ and �∇2� repre-sented vorticity convection, tilting/stretching and diffusion, respectively; and �̇� denoted the wing angular acceleration vector. Derivation of the vorticity transport equation in a rotating frame can be found in Batchelor (2000) or Cheng et al. (2013). Equation 5 was further normalized by refer-ence values, i.e., velocity by Ug , length by c and vorticity by Ug

/c , resulting in

where superscripts + indicated the dimensionless values. Note that all quantities shown in the results below were nor-malized by the reference values and the superscripts + was neglected onwards. The radial component ( ̂er ) of Eq. 6, which describes the dynamics of leading-edge and trailing-edge vortices, was given by Eq. 7,

Note that the vorticity diffusion is negligible at Re ~ O(103) compared to vorticity convection and tilting/stretching (Cheng et al. 2013; Garmann and Visbal 2014; Jain et al. 2015), which was, therefore, ignored in the subsequent analysis of LEV dynamics. The radial components of vorticity convection (Eq. 8) and tilting/stretching (Eq. 9) were expanded below to show their compositions,

(4)⎛⎜⎜⎝

utuyur

⎞⎟⎟⎠= J(�)

⎛⎜⎜⎝

uxuyuz

⎞⎟⎟⎠=

⎛⎜⎜⎝

ux sin � − uz cos �

uyux cos � + uz sin �

⎞⎟⎟⎠.

(5)�𝛚∗∕�t* = −(𝐮∗ ⋅ ∇)𝛚∗ + (𝛚 ⋅ ∇)𝐮∗ + �∇2𝛚∗ + �̇�.

(6)�𝛚*+∕�t*+ = −

(𝐮*+ ⋅ ∇

)𝛚*+ +

(𝛚+ ⋅ ∇

)𝐮*+ + 1∕Re∇

2𝛚*+ + �̇�+.

(7)��∗r∕�t* = −(�∗ ⋅ ∇)�∗

r+ (� ⋅ ∇)u∗

r+ 1∕Re∇

2�∗r.

(8)−(u∗ ⋅ ∇)�∗

r= −u∗

t

��∗r

�t⏟⏞⏟⏞⏟

C∗t

−u∗y

��∗r

�y⏟⏞⏟⏞⏟

C∗y

−u∗r

��∗r

�r⏟⏞⏟⏞⏟

C∗r

,

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To calculate the terms in the vorticity transport equation (Eqs. 7, 8 and 9), the velocity gradient tensor in the global frame ∇(x,y,z)�(x, y, z) was first calculated using a central dif-ferential scheme. The corresponding tensor in the rotating Cartesian frames ∇(t,y,r)�(t, y, r) was then calculated using the chain rule (Eq. 10). The vorticity gradient tensor in the rotating Cartesian frames was also obtained following the same formulation.

3 Results and discussion

3.1 Formation of vortical structure

To investigate the formation process of LEV, the evolution of the entire vortical structure (Fig. 3) and the LEV circulation (Fig. 4) are studied here. We use a dimensionless parameter in insect flight, i.e., stroke-arc/wing-chord ratio � = �Rg

/c

(stoke amplitude substituted by revolving angle) or the chord length of travel at Rg , as a measure of the travel distance. Note that the flow data close to wing root is removed due to the reflection from wing fixtures. During the accelera-tion period (1c), rapid generation of shear vorticities from the leading, trailing and tip edges of the wing is observed (Fig. 3d). The leading-edge shear layer rolls into a conical structure above the wing, i.e. the LEV, whereas the trailing-edge shear layer moves downstream and rolls up into a start-ing vortex (SV) in the wake. As the wing continues to rotate at a constant speed, the LEV further grows while keeping the conical structure until 4c of travel (Fig. 3m). In addition, the SV moves further downstream (relative to the wing) and the remaining shear layer at the trailing edge, which is then separated from the SV, forms the trailing edge vortex (TEV). The vorticity shed from wing tip is located along the path of wing tip, connecting the LEV and SV. Note that this vorticity is a mixture of LEV ( −�r ) and TV ( −�t ), which could be a result of the convection of LEV vorticity towards the tip

(9)(� ⋅ ∇)u∗

r= �t

�u∗r

�t+ �y

�u∗r

�y⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

T∗

+�r�u∗

r

�r⏟⏞⏟⏞⏟

S∗

.

(10)

∇(t,y,r)�(t, y, r) =

⎛⎜⎜⎜⎝

�ut

�t

�ut

�y

�ut

�r�uy

�t

�uy

�y

�uy

�r�ur

�t

�ur

�y

�ur

�r

⎞⎟⎟⎟⎠

= J(�)∇(x,y,z)�(x, y, z)JT (�)

= J(�)

⎛⎜⎜⎜⎝

�ux

�x

�ux

�y

�ux

�z�uy

�x

�uy

�y

�uy

�z�uz

�x

�uz

�y

�uz

�z

⎞⎟⎟⎟⎠JT (�).

or a tilting of TV into negative radial direction (parallel to LEV). After approximately the first 4c of travel ( � = 4), the vortical structure generated by the revolving wing remains almost identical, as the flow reaches the steady state. Note that, due to the removal of flow data close to wing root, no root vortex is observed in the current study, although it must exist according to the Helmholtz’s second theorem. This is evidenced by a number of previous experimental and numerical studies on revolving wings, which have reported the formation of a ring-shape vortex structure, composed by the LEV, TV, SV and root vortex (Lua et al. 2010, 2016; Carr et al. 2015).

The arrival at the steady state can be further evidenced by the plateau of LEV circulation and its spanwise distribution. The LEV circulation is calculated at 20 cylindrical slices dis-tributed evenly along the span from the wing root to the tip. Figure 4a shows a representative cylindrical slice at mid-span that cuts through the vortical structure isosurface and Fig. 4b shows the corresponding vorticity contour. The vortical regions including the LEV and trailing-edge shear layer on each slice are identified based on Q value criteria (Jeong and Hussain 1995), i.e., the second invariant of velocity gradient. The LEV circulation on each slice ( Γ∗

LEV(j) , j = 1 − N) is then estimated

by integrating the negative vorticity within the vortical region defined by Q = 0.71 (Eq. 11) and the volume-averaged LEV circulation

(|||Γ̂∗LEV|||) is estimated according Eq. (12):

As shown in Fig. 4c, the magnitude of volume-averaged LEV circulation

(|||Γ̂∗LEV|||) has a steep increase during the first

2c of travel, followed by mild increase until 4–5c of travel, and it levels off (with a fluctuation) afterwards. Likewise, the spanwise local distribution of LEV circulation ( |||Γ∗LEV

||| ) also undergoes a large increase along the entire span within the first 2c of travel (Fig. 4d), during which a conical LEV is quickly formed. For the travels within � = 2–4, the conical LEV is strengthened mostly between the wing root and mid-span (r/c < 2.5) while the circulation distribution close to wing tip undergoes relatively small change. For � ≥ 4, only small increase of LEV circulation is observed between the wing root and mid-span, whereas the LEV circulation at spanwise

(11)Γ∗LEV

(j) =∑

{(t,y)||Q(t,y)≥0.7∩𝜔∗r (t,y)

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location r/c = 3–3.5 experiences a large increase from � = 4 to � = 5, and no further increase is observed onward ( � ≥ 5).

Using the Tomo-PIV technique, Percin and van Oudhues-den (2015) conducted a similar study on the LEV develop-ment of a rectangular revolving plate at Re = 10,000 and AoA = 45°. The wing underwent a linear acceleration and arrived at the constant revolving speed after 1c of travel at 0.75 wing span. A comparison of our results and theirs is shown in Fig. 5. The generation of LEV, tip vortex and start-ing vortex ( � = 1) and the formation of a vortex tube along the footprint of wing tip ( � = 3) are observed in both studies (Fig. 5a). However, the vortex structure observed by Percin and van Oudheusden (2015) is more turbulent, e.g. as illus-trated by the winding substructures in the tip vortex ( � = 1) and the chaotic vortical structures in the LEV ( � = 3), possi-bly due to the higher Re in their study. We further examined the growth of LEV circulation at the 0.75 span as a quantita-tive comparison (Fig. 5b). In summary, there is a reasonable agreement between our and Percin and van Oudheusden’s results in terms of the overall trend of LEV growth, despite that there exist discrepancies for λ > 2, possibly due to the difference of Re and wing geometry.

To further study the LEV formation, the evolution of radial vorticity on a mid-span cylindrical slice is shown in Fig. 6. At � = 2, the LEV is separated into two distinct vortical structures, which are also known as the dual LEV structure (Carr et al. 2013; Garmann and Visbal 2014). How-ever, the two vertical structures merge together as the wing continues to rotate (Fig. 6f–h). The separation and merg-ing of LEV can be explained by the vorticity convection by tangential flow, which will be described in the following section. Note that the merging of LEV may result in the slight increase of circulation at � = 4 (Fig. 4c). In summary, the LEV reaches the steady state at approximately 4–5c of travel in terms of both structure and circulation in the cur-rent study. Here, we consider 4c as the critical distance of travel ( �c ) for the arrival of LEV steady state, the compari-son of which with the previous results is described in the following section.

The growth of the LEV and its arrival at steady state in revolving wings were also reported in Carr et al. (2015) and Poelma et al. (2006). For the revolving rectangular plate with AR = 4 in Carr et al. (2015) (Re ~ 1600 when estimated at Rg ), the LEV circulation experiences a fast growth during

LEV

(a) (c) (d)

(e) (g) (h)

(i) (k) (l)

(m) (o) (p)

=0.15λ (b)

(j)

(f)

(n)

TVSV

LEV

TTEEVV TV-

=0.25λ =0.5λ =0.7λ

=2λ=1.7λ=1.35λ=1λ

=2.5λ

=4.5λ

=3λ

=5λ

=3.5λ

=5.5λ =6λ

=4λ

LEV

TV

SV

TEV

Fig. 3 Formation of vortical structure. The isosurfaces are drawn according to the dimensionless vorticity magnitude at 1.5, and are color-coded with RGB values representing the magnitude of three vorticity components: trailing-edge vorticity ( �r ), red; tip vorticity

( −�t ), green; leading-edge vorticity ( −�r ), blue. The sequence of the isosurfaces is labeled according to the chord-length of travel at the radius of gyration ( � ). A cylindrical slice with r = 1.3c is included in subplots to show the masked region during data post-processing

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� 

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observed above the root-to-tip spanwise flow at the wing tip, indicating that both are induced by TV.

The evolution of radial vorticity �∗r is governed by the

vorticity transport equation (Eq. 7), primarily the vorti-city convection and tilting and stretching terms (Eqs. 8 and 9). Figure 8 shows the vorticity convection isosurface during the LEV formation and its contour on a representa-tive cylindrical slice at mid-span is shown in Fig. 9 (note that a positive value indicates an increase of �∗

r , whereas

a negative value indicates a decrease of �∗r ). First, dur-

ing the wing acceleration period ( � ≤ 1), the convection by tangential flow ( C∗

t ), which dominates the vorticity

transport, is continuously strengthened as the tangential velocity increases (Figs. 8a, 9). An increasing amount of shear vorticity generated at the leading edge (negative)

is transported downstream (from red to blue) to the LEV region (Figs. 8a, 9, � ≤ 2). Since � = 2, two pairs of posi-tive and negative C∗

t regions are observed within the LEV

(Figs. 8a, 9). The positive and negative C∗t regions occur

alternatively (i.e., the ‘LEV reduction’ and ‘LEV enhance-ment’ regions in Fig. 8a at � = 3), corresponding to the generation of the dual LEV structure (Fig. 6d, e). More specifically, the positive-C∗

t around the middle of the dual

LEV structure can transport the negative radial vorticity to the negative-C∗

t region in the primary LEV region, while

the positive-C∗t at the leading edge keeps feeding the nega-

tive shear vorticity into negative-C∗t region in the second-

ary LEV (Fig. 9, � = 2).The convection by vertical flow ( C∗

y ) is shown in Fig. 8b.

A downward convection of LEV vorticity (from red to

(b)

432 61 50λ

LEV * Γ

3

2

1

0

Percin & van Oudheusden, Re=10000 Present study, Re=1500

Percin & van Oudheusden Present study(a)

λ=1

λ=3

Multiple LEV structures

Winding substructures

Chaotic features

Fig. 5 Comparisons with the results from Percin and van Oudhuesden (2015). a Vortical structures outlined by the iso-Q values of 3.125 at two representative time steps ( � = 1 and � = 3) and b LEV circula-tion calculated on the representative slice at 0.75 span as a function of chord length of travel ( � ). Note that the LEV circulation in Percin and

van Oudhuesden (2015) is calculated on a flat plane, instead of on a cylindrical plane, as in our study; in addition, the reference veloc-ity in Percin and van Oudhuesden (2015) is defined at the 0.75 span, instead of at the radius of gyration, as in our study

-6

-3

0

3

6

r * (a) =0.25λ (b) =0.5λ (c) =1λ (d) =2λ

(e) =3λ (f) =4λ (g) =5λ (h) =6λ

LEV

SV

1c

Fig. 6 Radial vorticity on the mid-span cylindrical slice at eight selected time steps

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blue) is observed following the development of downwash (Fig. 7a). Two layers of positive C∗

y (red) are generated

along the span with a layer of negative C∗y (blue) in between

(Fig. 8a, � = 2). Note that the top positive-C∗y region coincide

with the negative-C∗t region ( � = 1–4), suggesting that the

downward convection transports away the negative vorticity that is transported downstream by the tangential flow C∗

t to

the negative-C∗y region. From the C∗

y contour at mid-span

(Fig. 9), the downward convection of LEV is apparent for � ≥ 1. As discussed in previous studies (Birch and Dickinson 2001; Shyy and Liu 2007; Cheng et al. 2013), this downward convection, which can be attributed largely to the existence of enhanced TV in revolving wings, may contribute partly to stabilize the LEV, as the negative vorticity inside the LEV is transported downwards to a region close to the trailing edge, instead of being transported continuously downstream.

The development of spanwise convection is shown in Fig. 8c. During the acceleration period, the spanwise con-vection only occurs in the vicinity of wing tip (Fig. 8c, � = 1). Only after the establishment of spanwise flow at mid-span (Fig. 7b, � = 2), a positive spanwise convection appears along the span above the leeward surface (Fig. 8c, � = 3), and is mostly located at the downstream (rear) end of the LEV (Fig. 9, � = 3). This indicates that, during the LEV formation prior to its arrival of steady state, the spanwise convection can attenuate the LEV strength by transporting the negative vorticity towards wing tip, which supports the role of spanwise flow in LEV stability of a flapping insect wing with finite stroke amplitude (Ellington et al. 1996). Nonetheless, despite that the effect of spanwise convection is not negligible within the LEV development period ( � ≤ 4), its magnitude is reduced after � = 4 when the wing reaches to steady state (not shown here), which suggests that the con-tribution of spanwise convection to LEV stability might still be insignificant for steadily-revolving wings, which agrees

with those shown in previous studies (Shyy and Liu 2007; Cheng et al. 2013).

3.3 Vorticity dynamics: tilting and stretching

The vorticity tilting and stretching can further attenuate the LEV by tilting the radial vorticity into tangential direc-tion (tilting) and compressing the LEV (negative stretch-ing), as previously shown in Lim et al. (2009); Cheng et al. (2013); Wong et al. (2013) and Wong and Rival (2015). Here, we further examine the evolution of vorticity tilt-ing and stretching (Fig.  10). During the acceleration period (1c), the positive tilting and stretching region is only observed at the wing tip region. As the wing rotates forward, another layer of positive tilting and stretching is observed along the tailing edge starting � = 2. As the LEV is transported towards the tailing edge (due to the sequen-tial tangential and downward convection described above), this positive tilting and stretching layer at the tailing edge can reduce the radial vorticity and, therefore, contribute to attenuate the LEV, which is significant starting � = 3. A more evident illustration can be found in Fig. 10b based on the values at mid-span. According to previous studies on the LEV formation (Rival et al. 2014; Widmann and Tro-pea 2015), the wing chord (c) is the characteristic length of LEV shedding, as the Kutta condition at the trailing edge will be broken down when the stagnation point on the upper wing surface moves to the trailing edge during the LEV growth. The existence of the positive vorticity tilting and stretching layer along the trailing edge, therefore, can act as a barrier that keeps the LEV from expanding to the trailing edge and potentially contribute to preventing the breakdown of Kutta condition at the trailing edge, thus preventing the LEV from shedding.

(a)=0.25λ =0.5λ =1λ =2λ =3λ =4λ

Downwash(b)

=0.25λ =0.5λ =1λ =2λ =3λ =4λ

uy=-0.25

uy=0.25

*

*

ur=-0.25

ur=0.25

*

*

Spanwise flowat mid-span

Positive spanwise flow at wing tipof positive spanwise flow

A single region

Fig. 7 Development of a downwash u∗y and b spanwise flow u∗

r

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Unlike the convection terms, the tilting and stretching of radial vorticity ( T∗ and S∗ ) experiences more abrupt behaviors. During the early development of spanwise flow (e.g., � = 2), the LEV strength is enhanced by the tilting and stretching (negative region in Fig. 10b). However, as the spanwise flow further develops, the tilting and stretching

(positive T∗ + S∗ ) quickly switches to remove the LEV vor-ticity close to its rear end above the trailing edge (Fig. 10b). We further looked into the vorticity tilting ( T∗ ) and stretch-ing ( S∗ ) separately at the mid-span to understand this tran-sient behavior (Fig. 11). The temporary LEV enhance-ment at � = 2 results primarily from the negative vorticity

(a) =0.25λ =0.5λ =1λ =2λ =3λ =4λ

(b) =0.25λ =0.5λ =1λ =2λ =3λ =4λ

Ct=-6

Ct=6

*

*

Cy=-6

Cy=6

*

*

(c) =0.25λ =0.5λ =1λ =2λ =3λ =4λCr=-1

Cr=1

*

*

LEV convected downstream

SV convected downstream

LEV reduction LEV

LEV convected downward

LEV convected to tip vortex

enhancement

Fig. 8 Radial component of vorticity convection − (u∗ ⋅ ∇)�∗r (Eq.  8). a Tangential flow convection ( C∗

t ); b Vertical flow convection ( C∗

y ); c

Spanwise flow convection ( C∗r)

-6

-3

0

3

6

Ct *

Cy *

Cr*

1c

-6

-3

0

3

6-6

-3

0

3

6

convectionDownward

toward tipLEV convection

=0.25λ =1λ =2λ =3λ =4λConvection forsecondary LEV Convection for

primary LEV

Fig. 9 Vorticity convection ( C∗t , C∗

y and C∗

r ) on the representative cylindrical slice at mid-span. The vorticity region is outlined by the magnitude

of radial vorticity at 1

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stretching within the LEV region ( S∗1 in Fig. 11b), due to

the existence of a positive spanwise gradient of spanwise velocity ( �u∗

r∕�r ) (vorticity stretching equals to the prod-

uct of �u∗r∕�r and �∗

r ). The contribution of vorticity tilting

( T∗ ) to the LEV enhancement at � = 2 is negligible. During � = 3–4, however, the vorticity stretching abruptly reveres its sign as it switches to compressing within a region at the rear end of the LEV (positive value S∗

2 in Fig. 11b). This

compressing effect limits the growth of LEV but is almost fully compensated by the opposite vorticity tilting within the same region ( T∗

1 in Fig. 11a). In addition, during � = 3–4, a

region of positive vorticity tilting ( T∗2 in Fig. 11a) appears

beneath the negative vorticity tilting region ( T∗1 in Fig. 11a),

thus attenuating the LEV strength as reflected in the total vorticity tilting and stretching ( T∗ + S∗ in Fig. 10).

3.4 LEV formation towards the steady state

The results above suggest that the formation process of the LEV is within the first 4c of travel, which can be separated into three stages (Fig. 12). In the first stage (within approxi-mately 1c), a conical LEV structure begins to form, while only tangential convection of the shear vorticity occurs, i.e., an increasing amount of negative shear vorticity, which is generated from the leading edge, is transported down-stream by the tangential flow. In the second stage (1c to 2–3c), as the downwash is developed after the acceleration period, the negative radial vorticity behind the wing, which is transported from the leading edge by the tangential flow, is further convected downwards by the downwash. Note that the downward convection can be regarded as a stabilizing mechanism as it relocates the negative vorticity inside the LEV towards the trailing edge, preventing the LEV from fur-ther growing and moving downstream. In addition, the radial

gradient of spanwise flow at the second stage (2c) leads to a negative vorticity stretching region within the LEV that enhances its strength ( S*

1 in Fig. 11b and blue region in

Fig. 12). In the third stage (2–3c to 4c), apart from the down-ward convection, additional LEV stabilizing mechanisms further come into play, i.e., spanwise convection (dark red region) and vorticity tilting (light red region), both of which reduce the (negative) radial vorticity at the rear end of LEV next to the trailing edge.

The vorticity dynamics described above during the LEV formation process (stage 3) shares similarity with those described for the steady state in Cheng et al. (2013) (Re = 220 and AoA = 45°). The spanwise convection of LEV at the steady state is found negligible in Cheng et al. (2013) at Re = 220, and small at Re = 1500 in the current study. In addition, considering there exist three stages of LEV forma-tion (Fig. 12), the insect wing flapping strokes, especially those with stroke amplitude less than 3c, might only experi-ence the first two stages of LEV formation, as the stage 3 appears after approximately 2–3c of travel. The spanwise convection and tilting and stretching effects occur in stage 3 here, may not occur within the strokes of those flapping wings.

4 Conclusion

In this study, we investigated the LEV dynamics on a revolv-ing wing (AoA = 45° and Re = 1500) with constant accel-eration in the first 1c of travel and constant angular velocity onwards. We aimed at understanding the vorticity dynamics of the LEV formation during its transition towards the steady state. The 3D flow field was measured in a time-resolved fashion using the ‘Shake-the-Box’ Lagrangian particle

=0.25λ =0.5λ =1λ =2λ =3λ =4λ T +S =-1* *

T +S =1* *

tip vortexLEV tilted to

LEV tiltingand stretching

-6

-3

0

3

6

LEV attenuation

LEV

=0.25λ =1λ =2λ =3λ =4λ

(a)

(b)T +* S* 1c

enhancement

Fig. 10 Radial component of vorticity tilting and stretching (� ⋅ ∇)u∗r ( T∗+S∗ ): a isosurfaces outlined by T∗ + S∗ = ± 1 and b contours on the

representative cylindrical slice at mid-span

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tracking velocimetry (PTV) system. A volumetric patching of nine individual measurement volumes was conducted to reconstruct the entire flow field. Our results showed that a conical LEV was attached to the wing surface during the entire sweep and reached the steady state after approximately 4c of travel. The LEV formation dynamics within the 4c of travel can be separated into three stages. In the first stage (≤ 1c), only tangential convection of shear vorticity occurred, transporting the vorticity originally generated at the leading edge (negative radial vorticity) downstream to the near wake behind the wing. In the second stage (1c to 2–3c), developed downwash transported the radial vorticity from the down-stream wake further downwards towards the trailing edge, thus reducing the strength of LEV. The negative vorticity stretching in stage two leads to a temporary enhancement of LEV. In the third stage (2–3c to 4c), however, an apparent reduction of LEV (negative) vorticity (at its downstream rear end) was observed near the trailing edge due to both spanwise convection and vorticity tilting. Additionally, the vorticity stretching can also reduce the negative radial vorticity inside the LEV at the third stage but is almost entirely compensated by the opposite vorticity tilting within the same region.

The results of the LEV formation dynamics support that, for insect wings with AR ~ 4, Re ~ 103 and flapping

amplitude between 2 and 4c, their pertinent LEVs may barely or never reach the steady state, therefore, the tran-sient characteristics of LEV formation may be more criti-cal than its state-steady characteristics in some insect flights. To further understand the LEV formation dynam-ics, different acceleration/velocity profile of the revolving wing will be examined in future work, while quantifying the relationship between the LEV behaviors and different mechanisms of force generation (e.g., circulatory-based and non-circulatory-based).

Acknowledgements This research was supported by National Science Foundation (NSF CMMI 1554429), Army Research Office DURIP Grant (No. W911NF-16-1-0272), National Natural Science Founda-tion of China (NSFC, Grant No. 11672022), China Scholarship Council (joint Ph.D. program for Long Chen) and the Academic Excellence Foundation of BUAA for Ph.D. Students. We thank Dr. Steve Anderson at LaVision Inc. for his assistance with the ‘Shake-The-Box’ system, and Shih-Jung Hsu for the design of robotic wing mechanism.

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Fig. 11 Vorticity tilting ( T∗ ) and stretching ( S∗ ) on the representative cylindrical slice at mid-span ( � = 2–4). T∗

1 and

S∗1 represents an enhancement of

LEV, whereas T∗2 and S∗

2 repre-

sents an attenuation of LEV

-6

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6=2λ =3λ =4λ

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Affiliations

Long Chen1,2 · Jianghao Wu1 · Bo Cheng2

* Bo Cheng buc10@psu.edu

1 School of Transportation Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China

2 Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 16801, USA

https://doi.org/10.2514/1.J056584https://doi.org/10.2514/1.J056584http://orcid.org/0000-0002-6982-0811

Volumetric measurement and vorticity dynamics of leading-edge vortex formation on a revolving wingAbstractAbstract Graphical abstract

1 Introduction2 Materials and methods2.1 Experimental setup2.2 Time-resolved volumetric flow measurement2.3 Volumetric patching of individual measurement2.4 Coordinate systems2.5 Data analysis

3 Results and discussion3.1 Formation of vortical structure3.2 Vorticity dynamics: convection3.3 Vorticity dynamics: tilting and stretching3.4 LEV formation towards the steady state

4 ConclusionAcknowledgements References