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

Abstract

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.

1. Introduction

When hovering, insects generate lift by imparting downward momentum into the air [1,2], in a way similar to a helicopter with revolving blades [3]. However, instead of revolving the wing continuously, insects flap their wings in a reciprocal motion, using unsteady aerodynamics. Typical helicopter blades operate at low angles of attack (AoA) with a long wing span in order to reduce disc loading and increase aerodynamic efficiency [3]. Insect wings, on the other hand, operate at large AoA in order to exploit augmented lift due to delayed stall [4,5] with a stably attached leading-edge vortex (LEV) [6]. While the literature is still inconclusive on the LEV stability [6–8], it undoubtedly stems from the low-aspect-ratio wing design (or small Rossby number [7]) which gives rise to prominent three-dimensional phenomena [9]. The three-dimensional flow in the near field of flapping/revolving wings has been successfully quantified recently using computational [10–12] and experimental methods [9,13–16].

A complete picture of the vortex wake structure in both the near and far field of flapping wings, however, is not entirely understood experimentally, except for limited theoretical [1,17,18] and preliminary computational studies [19,20]. Much of the knowledge was first brought from the well-understood aerodynamics of helicopters and propellers, which create a helicoid wake with a continuously shed tip vortex (TV). These shed vortices can be modelled as a chain of coaxial vortex rings [3] that, while sufficiently close to each other, confine the induced flow (or downwash) within. The momentum change characterizes the lift generation and the minimum induced power [1,21]. In comparison, flapping wings deposit vorticity into the wake through a prominent TV and starting/stopping vortices, which are formed during revolving and reversal phases of a stroke. Theoretical studies have assumed that these shed vortices also form a distinct vortex ring structure [1,17], which sheds periodically instead of continuously. By considering the mutual interaction of an axial chain of periodically shed circular vortex rings, Rayner [17] described the evolution of the vortex wake structure in terms of several key morphological parameters. Ellington [1] also adopted the vortex ring model when modifying the momentum theory of propellers (Rankine–Froude theory) to evaluate the induced power and efficiency of the wake. Based on this theoretical model, Tytell & Ellington [22] tried to experimentally reconstruct the vortex wake of the hawkmoth using a vortex ring generator, through which they investigated the feasibility of performing measurements of a hovering animal's wake. Recently, using Rayner's vortex model, Wang & Wu [18] further detailed the structure and evolution of the wake and its role on the stroke-averaged aerodynamic lift. They pointed out that the wake has a self-consistent contraction region and an expansion region. The contraction and expansion have a mutual cancelling effect on the lift created, but the induced velocity leads to a contraction of the shed vortices in the near wake, which reduces the lift overall. Recently, Altshuler and co-workers [2] have studied the wake of hovering hummingbirds using two-dimensional flow visualization and discussed the plausible three-dimensional wake topology formed by vortex rings. Nevertheless, without detailed three-dimensional flow visualization, it is still questionable whether such vortex rings exist in the wake considering the complex wing–wake interaction that occurs during the stroke reversals [23,24], which may prevent an integrated vortex structure being formed. Furthermore, revolving/flapping wings at large AoA create substantial drag, which is a critical reservoir for flight manoeuvrability [25,26], and lead to profile power that is comparable to the induced power [27], therefore casting doubt on the feasibility of applying the simplified vortex theory.

For these reasons, it is necessary to resolve the three-dimensional vortex wake structure of flapping wings in both the near and far field with sufficient temporal and spatial resolution. In this paper, a state-of-the-art volumetric three-component velocimetry (V3V) system was used to capture the flow features of a dynamically scaled flapping wing. The objective of this study was to acquire detailed information on the intricate three-dimensional wake structure that is created in the near field as it evolves into the far field and analyse the results in the context of previous theoretical and hypothetical models.

2. Material and methods

2.1. Experimental set-up

A dynamically scaled mechanical wing was designed to rotate with two degrees of freedom (i.e. wing stroke (ϕ) and rotation (ψ); figure 1a). Both degrees of freedom were driven by a bevel-geared shaft system powered using DC motors with encoders (Maxon Motor AG, Sachseln, Switzerland). Wing trajectory tracking was accomplished using a PID feedback control loop in Simulink, Matlab (The Mathworks, Natick, MA, USA; for more detail on the motion control system, see [28]).

Figure 1.

Experimental methods. (a) Schematic showing the flapping wing model and the measurement volume. The total measurement volume (420 × 276 × 354 mm) in the fixed Cartesian frame (X, Y, Z) is constructed by 24 V3V measurement volumes (140 × 140 × 100 mm). (b) Wing kinematics is defined by stroke angle ϕ and rotation angle ψ, which are sinusoidal functions of time. (c) Schematic of the wing kinematics, as viewed along the positive Y-axis. The line denotes the wing chord, with a dot marking the leading edge. Flow measurements are performed at wing positions 1 through 20. (d) The rotating Cartesian frame is defined by vertical (), radial () and tangential () axes. The components vary with the azimuth angles θ of the particles of interest (filled circle). (Online version in colour.)

The wing planform is based on the wing of a fruitfly, Drosophila, representing a typical insect wing profile [29]. The length was 185 mm (from wing tip to centre of rotation) with an aspect ratio of 6 (two times wing length/mean chord length). The wing was made from a transparent polymer sheet with a uniform thickness of 1.52 mm that remained rigid during the experiments. The wing and the gearbox were immersed in the centre of an acrylic tank with transparent walls (dimensions: 610 × 610 × 3050 mm, width × height × length), filled with mineral oil (kinematic viscosity ≈8 cSt at 20°C, density ≈850 kg m⁻³). The wing kinematics were modelled using sinusoidal functions (figure 1b,c) with the stroke amplitude of 130° and maximum wing rotation of 60°. The Reynolds number ((4ΦR²n)/(υAR)), based on the mean chord length, was 2200, where R is the wing length, AR is the wing aspect ratio, n is the wing beat frequency (1/3 Hz) and υ is kinematic viscosity [30]. This Reynolds number is within the range of insect flight [29].

2.2. Volumetric three-component velocimetry process and measurement volume patching

A flow measurement technique known as V3V (TSI Inc., Shoreview, MN, USA), first described by Periera et al. [31], was used to capture the overall three-dimensional flow structure of the dynamically scaled flapping wing. For other applications of the V3V technique, see Kim & Gharib [15], Flammang et al. [32], Chamorro et al. [33] and Cheng et al. [9]. In this experiment, air bubbles pumped out of a porous ceramic filter were used as tracer particles. Experiments were conducted after the large bubbles had risen to the surface leaving behind only small bubbles with an average size of 20–50 μm and a rise velocity less than 0.17 mm s⁻¹ [34]. The fixed coordinate frame (X, Y, Z) was aligned with the measurement volume and had its origin at the centre of the wing rotation (figure 1a). The V3V measurement volume, formed by the intersection of the field of view of the three cameras, was 140 × 140 × 100 mm³ along the X, Y and Z directions.

In order to capture the flow in both the near and far wake while ensuring sufficient spatial resolution, the total measurement volume was patched using 24 individual V3V measurement volumes, which covered a total volume of 420 × 276 × 354 mm³ (figure 1a). Overlap between individual measurement volumes was introduced to reduce the error at the volume boundaries. Relative locations of the individual volumes were changed by adjusting the location of the mechanical wing, while the physical V3V measurement volume remained fixed at the centre of the tank. Note that this approach was based on the fact that the flow was repeatable between different strokes owing to the precise control of the wing kinematics and that similar approaches were applied when reconstructing the three-dimensional velocity field from stereoscopic PIV measurements on two-dimensional planes [13,16,35].

Pairs of sequential images were taken simultaneously by three 4 megapixel digital cameras synchronized with an Nd:YAG pulsed laser illuminating air bubbles inside the measurement volume. The complete flow history within one wing stroke was captured at a laser pulse frequency of 6.67 Hz, yielding 20 images per cycle. After four initial strokes, the wake was conservatively assumed to be fully developed, and the measurements, consisting of a total of nine stroke cycles, were captured. Next, the velocity field at each of the 20 equally spaced time instants (figure 1b,c) was calculated from an ensemble-average of the separate images captured during the nine wing strokes. The particle detection, particle tracking and velocity field interpolation were carried out using InsightV3V software (TSI Inc., Shoreview, MN, USA). The software interpolated (using Gaussian weighting based on vector distance from the grid node) the randomly distributed velocity vectors obtained from the particle tracking algorithm onto a 71 × 47 × 60 rectangular mesh grid (Δx = Δy = Δz = 6 mm).

The spatial uncertainty that results from mean-bias and RMS errors is of the order of 1% for the X and Y velocity components and 4% for the Z component. Higher order quantities such as vorticity and Q-criterion, which rely on velocity gradients, used a central differencing method and have an estimated uncertainty of 8%. For quantification of uncertainty and errors related to V3V measurements, see Pereira & Gharib [36] and Troolin & Longmire [37]. Owing to laser reflections from the shaft and gearbox, parts of the volume close to the wing root were not visualized at certain time instants.

In the current experiments, the wing was flapping vertically in the Earth coordinate frame, and therefore the X-axis (figure 1a) was aligned with the longitudinal (horizontal) axis of the tank. This was done to minimize the ground effect, and since only minimal surface waves were generated, the orientation of the stroke plane did not affect the resulting flow. Additionally, no other wall effects were observed. Throughout the rest of the paper, the coordinates are labelled with respect to the measurement-volume-fixed frame (figure 1a) regardless of the orientation of the stroke plane.

2.3. Data analysis

As shown in Cheng et al. [9], owing to the rotating nature of wing motion, it is advantageous to represent the vector fields (e.g. velocity and vorticity) in a series of rotating Cartesian frames ( figure 1d) rather than a single fixed Cartesian frame. The velocity field in the rotating Cartesian frame u(x, r, t) = (u_x, u_r, u_t) is related to that in the fixed Cartesian frame u(x, y, z) = (u_x, u_y, u_z) by the following relationship:

2.1

where J is the Jacobi rotation matrix which is a function of θ, the azimuthal angle (figure 1d). The same relation applies to all the other vector fields. Additionally, the gradient tensor of a vector field is also written in the rotating Cartesian frame. One can show that the velocity gradient tensor can be expressed as follows:

2.2

where ∇_(x,r,t) and ∇_(x,y,z) represent the gradient operation in rotating and fixed Cartesian frames, respectively. The above relationship also applies to the vorticity gradient ∇ω. The subscripts will be neglected in the rest of the paper for convenience.

The vortex structure was determined by using the second invariant of the velocity gradient ∇u, i.e. the Q-criterion [38]. This criterion has been widely used in identifying the location of vortex cores (for the comparison to other methods when applied in flapping wings, see [14]).

To further understand the evolution of the vortex structure in terms of vorticity dynamics, the spatial derivative terms of the vorticity transport equation in a non-rotating frame [9,39]

2.3

were evaluated (τ denotes time). The individual components of this equation can be written as

2.4

where It can be seen that the temporal change of local vorticity is equal to the sum of the convection  tilt/stretch  and dissipation terms, which are essentially derived from the spatial distribution. It will be proved useful to further expand the convection terms as

2.5

where the terms on the right-hand side represent convection by downwash, radial flow and tangential flow, respectively. Similarly, the tilt/stretch term can be expanded as

2.6

The terms on the right-hand side can be separated into tilting and stretching components; for example, for the tangential component represents the tangential vortex stretching, while the other two terms represent the vortex tilting from the vertical and radial components. The vortex stretching and tilting is absent in two-dimensional flow representations, and therefore represents the three-dimensional properties of the flow.

In the following results section, dimensionless values will be used and were obtained using the following characteristic values: velocity, by mean wing tip velocity (); vorticity, by twice the wing rotation vorticity (, the mean ‘planetary’ vorticity if viewed in the rotating wing frame); time and length, by the period of one wing stroke (1/n) and wing length (R), respectively.

3. Results and discussion

The results are shown primarily as isosurfaces and contour plots on representative slices. Vortices created by consecutive strokes are labelled in order by the subscripts 0, 1 and 2, with 0 representing the ongoing stroke. In order to more fully understand the evolution of the three-dimensional wake produced by the wing, the formation of the near field vortex structures is examined first.

Figure 2 shows the vortex structures (isosurfaces of Q-criterion) at three representative time instants, as well as at two different viewing angles. An animation of the complete sequence is available in the electronic supplementary material, video S1. The isosurfaces are plotted with three different colours (RGB: red, green and blue), the values of which (0–255) are scaled with those of the positive (red) and negative (blue) components of tangential vorticity and the magnitude of radial vorticity (green). A similar method of colour mapping was used in [9]. The advantage of this colouring scheme is that it offers a clear way to visualize both vorticity magnitude and direction within a single isosurface plot. Note that the direction of tangential vorticity was chosen because it distinguishes between the tip and root vortices, which appear to be the dominant components in the wake. The remaining vorticity components (e.g. vertical vorticity) are represented by black colouring. Contour slices of vorticity and velocity distribution are presented in figures 3 and 4, respectively.

Figure 2.

Colour-coded isosurfaces of vortex wake structure defined by Q = 0.25 at T = 0.1 (a), 0.3 (b) and 0.7 (c) viewed at two different angles (i) and (ii). RGB values of the isosurface colour correspond to the vorticity components: root vorticity (), red; tip vorticity (), blue; magnitude of radial vorticity (), green. The TV, RV and LEV are labelled. Subscripts 0, 1, 2 denote vortices created by different strokes with larger numbers representing earlier strokes.

Figure 3.

Contours of vorticity components shown in spanwise slices for T = 0.3. Values of are shown. Slices are from ϕ = −80 to 80° in 10° increments. (a) Tangential vorticity, ω_t with the TV and RV labelled. (b) Radial vorticity, ω_r with the LEV and TEV labelled. (c) Vertical vorticity, ω_x.

Figure 4.

Contours of velocity components shown in horizontal slices for T = 0.3. Values of are shown. Slices are from X = −288 to 36 mm in 36 mm increments. (a) Tangential velocity. (b) Radial velocity (spanwise and reverse spanwise flow). (c) Vertical velocity (downwash and upwash). Arrows indicate the directions of the flow.

3.1. Formation of near wake vortical structure

At the onset of the stroke, LEV₀, TV₀ and the (trailing-edge) starting vortex develop and connect smoothly along the wing edge (figure 2a). While LEV₀ and the starting vortex contain positive and negative radial vorticities, respectively, both are mixed with negative tangential vorticity (cyan). As the wing continues to rotate and the starting vortex has shed, a prominent trailing-edge shear layer structure (TESL₀, yellow, figure 2b) with mixed positive tangential and negative radial vorticity develops and sheds from the trailing edge. In less than a quarter stroke cycle, it quickly rolls up into a root vortex (RV) which travels downwards together with the shed TV (RV₁ and TV₁, figure 2c). Notably, vortex structures based on Q-criterion show that neither TESL₀ nor RV₁ appear to connect with the starting or tip vortices, preventing the formation of a closed vortex ring. In fact, TESL₀/RV₁ and starting/tip vortices contain opposite tangential vorticity; this is shown more clearly in figure 3, where tangential, radial and vertical vorticities are plotted along contour slices at T = 0.3. Note that negative radial vorticity is continuously distributed from the starting vortex to TESL₀ (figure 3b, blue), while it is mixed with negative tangential vorticity in the starting vortex (figure 3a) and positive tangential vorticity in TESL₀.

Owing to strong vortex tilting [9], LEV₀ contains negative tangential vorticity with increasing magnitude from wing root to tip (figure 3a, blue), as it smoothly connects to TV₀. Meanwhile, positive radial vorticity (red, figure 3b) extends from the LEV₀ to the TV₀, which also contains negative radial vorticity from TESL₀ (figure 5a(ii)). Collectively, the wing-attached LEV₀, together with the shed TESL₀ and TV₀ form an integral structure throughout the stroke. In addition, vertical vorticity with comparable magnitude to radial and tangential vorticity is distributed in both TESL₀ and TV₀ (figure 3c). During stroke reversals, as observed in both the current and previous studies [23,35], TV and RV detach from the wing when the LEV is shed; the wing then intercepts the radial vorticity of the shed LEV as the new LEV of the next stroke (with opposite radial vorticity) grows. Consequently, there appears to be no well-established stopping vortex in the wake.

Figure 5.

Contour of vorticity components shown in two spanwise slices. (i) Tangential vorticity, ω_t. (ii) Radial vorticity, ω_r. (iii) Vertical vorticity, ω_x. (a) Spanwise slice at ϕ = 0° (mid-stroke plane) for T = 0.3. (b) Spanwise slice at ϕ = 0° for T = 0.55. (c) Spanwise slice at ϕ = −64° for T = 0.3. (d) Spanwise slice at ϕ = −64° for T = 0.55. Contours of Q = 0.05 are superimposed to show the vortex structure. In c(iii), the area enclosed by the dotted ellipse indicates the location of the newly generated vorticity, in contrast to those from the previous stroke. Arrows represent the velocity vectors projected onto the plane (radial and vertical components).

The results indicate that the net result of the wing action is to generate a shear layer structure (or ideally, a vortex sheet) with continuously distributed radial and tangential vorticity in LEV₀, TV₀ and TESL₀. TV₀ at the wing tip connects smoothly with the wing-attached LEV₀; the TESL₀, shed from the wing trailing edge, rolls up into RV₀ in less than a quarter of a stroke. However, as shown in §3.2, owing to the lack of well-established starting and stopping vortices connecting to the RV₀ and SV₀, the shed vortex structure cannot evolve into a distinct vortex ring.

3.2. Evolution of far wake vortical structure

Figure 5 shows the three vorticity components, tangential (i), radial (ii) and vertical (iii) at two different tangential locations, ϕ = 0° (figure 5a,b) and ϕ = −64° (figure 5c,d) and at two different times, T = 0.3 (figure 5a,c) and T = 0.55 (figure 5b,d). The periodically shed TVs and RVs convect downwards with the induced velocity and contract quickly in the radial direction, while the RV rolls up and moves towards the TV (figure 5a(i)–d(i), electronic supplementary material, video S2 and S3). The trajectories of their spatial locations (identified by the maximum value of Q-criterion) at the mid-stroke plane (ϕ = 0°) are shown in figure 6. It can be seen that both TVs and RVs are moving at a near-constant velocity downwards (figure 6a), while they fully contract radially in less than a half stroke cycle and one-third of the wing length beneath the stroke plane as seen in figure 6b,c (RV before rolling up is not shown). Note that no distinct vortex structure can be identified approximately 75% of the wing length beneath the stroke plane, while the tangential vorticities in TVs (and RVs) have merged together.

Figure 6.

Positions of the shed tip and root vortices as a function of time. (a) Vertical position. (b) Radial position. (c) Vertical versus radial position. The positions of the vortices are located according to the highest Q-value within a planar vortex core on a vertical plane defined by ϕ = 0°. Positions of the stroke plane, wing root and tip are shown. (Online version in colour.)

When contracted radially, the distance between the TV and the RV is about 30% of the wing length. This is significantly less than the ideal wake radius in a helicopter when fully contracted (70.7% of the blade length [3]), where the downwash is confined in a well-defined helical vortex wake. As observed in this study, in addition to contracting radially, TVs and RVs also stretch tangentially to both ends of the stroke and form an arc-shaped distribution. Concurrently, the downwash, first distributed over the wing-swept area in the near field, diverges into the arc-shaped region seen in figure 4c while maintaining significant tangential flow (figure 4a, T = 0.3). Note that the existence of tangential flow in the wake is similar to the swirl in the helicopter's wake, which increases the induced power and therefore may reduce the overall aerodynamic efficiency [3].

In previous studies on revolving wings, positive radial flow (or spanwise flow) induced by the strong TV₀ was observed in a wide region behind the wing in the wake [6,9], which is absent here (spanwise flow can only be observed in the vicinity of the wing tip when the TESL₀ has not rolled up). This is due to the influence of the shed TV₁ in the wake, which induces negative radial flow below TV₀ (figure 5a(i)). Therefore, the shed TVs (and RVs) in the wake form a discontinuous vortex boundary which prevents flow from leaving the confined region of downwash.

Because the magnitude of the downwash varies over the cycle and the vorticity shed at the beginning of the half stroke may travel downward a greater distance than those shed later, the resulting TV and RV have an irregular three-dimensional structure in the wake (e.g. figure 2c). Note that, at the mid-stroke plane (ϕ = 0°), two distinct TVs, shed at two consecutive half strokes, can be observed. However, in the vicinity of stroke reversal (ϕ = –64°), TVs shed at consecutive half strokes are located close to each other and merge together (figure 5c(i),d(i); electronic supplementary material, video S3). In addition, there is only a single RV shed at stroke reversal due to the delayed development of positive tangential vorticity at the beginning of a stroke (as mentioned in §3.1, the starting vortex contains negative tangential vorticity).

The radial vorticity of opposite sign is located between the TVs and the RVs (figure 5a(ii)–d(ii)) and exhibits a V-shaped distribution in spite of the roll-up of the TESL₀ and the contraction of the RV and the TV (figure 4a). Vertical vorticity in the same direction as wing rotation is distributed in the TESL₀ and rolled-up RVs, while that in the opposite direction is distributed in TVs (figures 3 and 5a(iii),b(iii)). At the beginning of a stroke, however, newly generated vertical vorticity is located near the centre portion of the wing (enclosed by a dotted circle, figure 5c(iii)), while those from the previous stroke remain at the wing base and tip (see also figure 3c). This vorticity distribution clearly indicates a prominent wing–wake interaction in which the wing impacts the tangential flow from the previous stroke while generating new tangential flow in the opposite direction (figure 4a, middle slice).

3.3. Vorticity transport

Evaluating terms in the vorticity transport equation shows that convection is the major cause of the local change in vorticity, while the contribution of the tilting/stretching effect is secondary. For instance, the radial vorticity in LEV₀ is convecting downwards with the induced downwash (figure 7b) while its magnitude is reduced by the vortex compression (figure 7c), a phenomenon previously found in revolving wings [9]. Previous studies have hypothesized that spanwise flow was the cause of LEV stability by transporting vorticity into the wake (e.g. [7,8]). However, as found in both revolving [9] and flapping wings herein, the magnitude of vorticity convection due to spanwise flow is significantly lower than that due to the induced downwash and even lower than the vortex compression; therefore, it is unlikely to stabilize the LEV. Note that although the magnitude of the vorticity decays considerably in less than two wing lengths below the wing (figures 3 and 5), the dissipation is about one order lower than the first two terms (not shown).

Figure 7.

Contours of radial vorticity (a), convection (b), and tilt and stretch (c) shown in cylindrical slices located at 50% of the wing span for T = 0.75 (ϕ = 0°). Labels for the LEV correspond to ω_r < 0 and for the TEV correspond to ω_r > 0. Convection by downwash () transports vortices downwards resulting in regions of increasing () or decreasing () vorticity. Vortex tilting and stretching () increases ω_r. (Online version in colour.)

For the tangential component, while the TVs are convecting downwards, it was observed that vortex stretching increases their magnitude (figure 8) by stretching the vortex tube. This is not surprising considering that in spite of wing flapping direction, the gradient of the tangential velocity () is mostly positive and therefore  has invariably the same sign as . Note that, in the vicinity of the wing tip, vortex compression is observed instead due to the decreasing tangential velocity. Similarly, for RVs,  is negative and along the same direction of RVs, which also corresponds to a stretching of the vortex tube (figure 8b). Note that the vortex stretching of TVs and RVs exists in both the near and far field. Interestingly, vortex compression is found over a region in the far wake (figure 8b) where the distribution of positive tangential vorticity from the TV seems to expand radially after the wake is fully contracted (figures 3a and 5a(i)–d(i)). While the cause of such vorticity distribution is currently unknown, it is worth noting that wake expansion after full contraction has been previously shown in simulation results by Wang & Wu [18,40] using the inviscid vortex model first proposed by Rayner [17].

Figure 8.

Stretching and compressing phenomena in the wake. (a) Contour of TV stretching/compressing () shown in spanwise slices for T = 0.4 with a transparent isosurface of ω_t = −1.5. Slices are located at ϕ = −64°, −32°, 0° and 32°. is shown. (b) Slices located at ϕ = 0° shown with lower contour values of  to include more details. (Online version in colour.)

4. Conclusion

The aerodynamics of flapping wings are characterized by the unique system of vortices shed in the wake, the structure and evolution of which reflect the wing morphology and kinematics. While the near wake vortex system directly reflects the action of the wing on the fluid, the far wake vortex pattern is under the influence of vortex interactions and viscous dissipation. It has been shown that the flapping wing model used in this study creates a series of complex vortex structures, which are summarized in the schematics shown in figure 9. The overall effect of the wing motion is to deposit a vortex sheet (TESL₀ combined with TV₀, figure 9a,b) into the wake, which contains vorticity components in all three directions (radial, tangential and vertical). From conventional aerofoil theory, it has been proposed that the radial vorticity in the TESL₀ may change direction as the circulation in the LEV decreases [41] because of the deceleration of the wing angular velocity towards the end of a half stroke. However, as observed in this and previous studies [35], because the wing circulation is primarily contained in the LEV₀, which smoothly connects with the TV₀, it is unlikely that the shed TESL₀ itself reflects the circulation change. Instead, it was observed here that the TESL₀ always contains radial vorticity in the opposite direction of the LEV (figure 9b). The shed TESL rolls up quickly into the RV in less than a quarter stroke cycle (figure 9c). Note that in conventional fixed and rotary wings with large aspect ratios and small AoA, the TESL often rolls up into the TV instead of the RV [42,43]. The TV in flapping wings, however, has been widely found to be a continuation of the LEV, which contains both tangential and radial vorticity. The shed TVs and RVs contract radially while stretching tangentially (figure 9c), which forms a discontinuous vortex boundary that confines the induced flow within an arc-shaped region (figure 4c). The shed radial vorticities in starting and stopping vortices do not connect to the TVs and the RVs, which prevent the formation of a closed-loop structure. Therefore, although the downwash is confined radially, it diverges tangentially. Note that, when applying momentum theory on revolving helicopter or propeller blades [1,3], the downwash flow is assumed to be perfectly confined in a tube formed by the shed TVs. It is therefore apparent that the use of this assumption in the study of flapping wings is invalid owing to the divergence of the downwash tangentially. Finally, we have shown that the shed TVs and RVs undergo vortex stretching in the tangential direction, and together with substantial tangential flow in the wake, plausibly contribute to the stretched arc-shaped vorticity distribution in the wake.

Figure 9.

Schematics of the three-dimensional vortex wake system. (a) Vortex wake system viewed at mid-stroke plane (ϕ = 0°). The arrows in the RVs and the TVs indicate vertical vorticity components, and those in LEV₀ and TESL₀ indicate radial components (mixed with tangential components). (b) Near field vortex system represented by idealized vortex lines. (c) The TESL rolls into the RV, and the vortex system contracts radially but expands tangentially in the wake. Dotted lines indicate a distribution of radial vorticity between the RV and the TV. (Online version in colour.)

In animal flight, the induced flow by the wake vorticity affects both the flight dynamics [44,45] and the biological sensory and heat/mass exchange processes [46,47]. However, owing to the lack of either experimental or computational studies, it is still unclear how such wake patterns may change with Reynolds number, wing profile and kinematics. A recent study on visualizing the wake pattern of a flapping wing with low stroke amplitude and passive wing rotation (also at a lower Reynolds number of 700) [48] shows a similar loss of radial vorticity in the far wake and a dominance of tangential vorticity from tip and root vortices. The downwash in the far wake is confined to a region between two parallel shear layers, instead of the arc-shaped region observed in this study, possibly because of the low stroke amplitude. Additionally, the presence of an animals' body (for example, the tail of a hummingbird) and wing compliance may further change the wake pattern. For example, it has been found that the tip and root vortices of hummingbirds do not contract significantly, while the root vortices remain close to the body (Bret Tobalske 2013, personal communication) [2,49]. Therefore, while this work may serve as a baseline case with an abundance of detail, systematic studies will be desirable in the future. These studies should support the long-term goal of obtaining a vortex wake model pertinent to flapping wings that quantifies force generation, efficiency and power expenditures.