Tip vortices formation and evolution of rotating wings at low Reynolds numbers

Insects and birds inspire novel designs of microaerial vehicles for low-Reynolds-number (Re) flight.^1,2 Unlike conventional fixed-wing and rotary-wing aircrafts, which adopt high aspect ratio wings and operate the wings at a low angle of attack, natural flyers are equipped with low aspect ratio wings and flap their wings at a high angle of attack without stall.³ The formation of a stably attached leading-edge vortex (LEV) has been observed in many species of insects^4,5 and birds,^6,7 and it can provide a flapping wing with additional circulation and enhance its lift production.⁸ 

Previous studies^9,10 have shown that the major vortex structures and lift enhancement mechanisms of rotating wings are similar to those in flapping wings in the middle of the down- and up-strokes. This allows engineers to explore the prominent features of the lift enhancement mechanisms by distilling the complex insect flapping kinematics into canonical rotating motions.¹¹ Usherwood¹² experimentally studied the aerodynamic performance of a dried pigeon wing and a flat card replica in rotating motion. His force measurements showed that rotating wings at a high angle of attack can manifest high-lift mechanisms analogous to those of flapping insects. The maximum lift coefficients of the revolving real and model wings can reach up to 1.64 and 1.44, respectively. These measured values are higher than the bird wings under translating motion (∼1.2) at the same angle of attack in the literature.^13,14 Ringuette’s group^15,16 investigated the vortex structures of rotating flat-plates at different aspect ratios (AR = 2 and 4) using stereoscopic digital particle image velocimetry (S-DPIV) techniques. It was found that, for AR = 2, the aft-tilted LEV merged with the tip vortex. For AR = 4, the LEV and tip vortex can reach the trailing-edge due to the high vortex growth rate. Garmann and Visbal¹⁷ numerically studied the vortex structures and aerodynamic loading of revolving wings. Their simulations illustrated that the pressure gradient force and the centrifugal force together were responsible for maintaining the attachment of the LEV and avoiding the flow separation. Limacher et al.¹⁸ revealed that Coriolis force plays a vital role in determining the spanwise extent of a stable LEV by tilting the LEV core away from the leading-edge toward the wake. Jardin¹⁹ systemically investigated the Coriolis effect on the LEV attachment. It was found that the Coriolis effect is the key element in LEV stability for Re > 200, while viscous effects dominate the LEV stability for Re < 200. In addition to the Coriolis effect, Werner et al.²⁰ reported that planetary vorticity tilting (PVTr) can produce oppositely signed radial vorticity and contribute to the stability of the LEV. Through using a time-resolved volumetric flow measurement, Chen et al.²¹ investigated the LEV formation of rotating wings and revealed that the LEV of insect wings may barely reach a steady state within each flapping stroke due to the limited flapping amplitude. So, it is reasonable to speculate that the formation of transient vortices and evolution could be important in advancing our understanding of the insect flight.

In addition to the LEV formation, other unsteady vortex formations have also been identified in previous studies. Ringuette et al.²² experimentally investigated the role of the tip vortex in the unsteady force generation. Their measurements showed that the tip vortex contributes substantially to the overall force on the plate. Garmann and Visbal¹⁷ revealed that the vortical structures around the wingtip were not stably attached and were less organized due to the significant spanwise flow and vorticity flux from the root to the tip. Near the wingtip, the LEV connects to the tip vortex, extends into the far-wake, and presents unsteady phenomena, especially at high Reynolds numbers.²³ Medina and Jones²⁴ found that the vortex burst around the tip region was correlated with the swirl number, which was defined as the ratio of the axial flux of angular momentum to the axial flux of axial momentum. Through exploring the effect of Rossby number (Ro) on the flow structures of rotating wings, Smith et al.²⁵ found that the higher Ro (>1.2) can cause an LEV burst and expand the leading-edge shear layer to be fed into the tip vortex. Wolfinger and Rockwell²⁶ also observed that the increasing Rossby number can cause the vortex system to become less coherent. Chen et al.^27,28 applied the Brown–Michael vortex model to study the development of the edge vortices for a revolving plate at a wide range of angle of attack. Both the edge vortex circulation and its position were derived analytically. This innovative approach provides insights into the stability analysis of the edge vortex.

The aerodynamic performance of revolving wings has also received attention, especially for low Reynolds number (Re) flight. Bayiz et al.²⁹ compared the hovering efficiency of rectangular wings that underwent revolving and flapping trajectories through a quasi-steady model. Their findings indicated that the flapping configuration performed better in a lower Re regime and can generate higher lift coefficients, while revolving wings were more efficient at a lower lift coefficient for Re above 100. To improve lift augmentation of revolving wings at low Reynolds numbers, Wu and colleagues applied an integrated experimental and computational approach to study a revolving wing with flapping perturbation³⁰ or flapping–pitching perturbations.³¹ Through introduced additional unsteadiness to the revolving wing, their study showed that the flapping and pitching perturbations can lead to substantial lift augmentation and considerable drag reduction. Additionally, Manar et al.³² experimentally studied the tip vortex structure and aerodynamic loading of rotating wings at an angle of attack of 45°. Their measurements suggested that a tip clearance of 5 chord length distance is required to avoid the undesired wall effects. In another study, Jardin and Colonius³³ performed direct numerical simulation (DNS) of rotating wings and found that the lift optimality and vortex structures can be achieved by balancing the effects of wing aspect ratios and Rossby numbers.

Variations of the Reynolds number, angle of attack, and wing aspect ratio are known to generate substantial changes in both vortical structures and correlated aerodynamic forces. However, previous studies mainly focused on the LEV and its associated aerodynamic performance. The coherence changes of the tip vortex have remained unclarified over a range of these key variables. Poelma et al.³⁴ observed a pair of tip vortices around a dynamically scaled fruit fly wing using PIV measurement, which implied that tip vortices have a more complex structure than that is often reported in the literature. Additionally, the vortex formation and force generation of rotating wings over a large travel distance have not been fully addressed yet. Thus, the purpose of the current computational study aims to examine the formation of wingtip vortices and aerodynamic performance of finite-aspect ratio rotating wings in a quiescent flow over a long travel distance via a high-fidelity direct numerical simulation (DNS) in-house solver. An outline of the paper is given below. Section II gives a brief introduction to the numerical methodology and the simulation setup. In addition, the validity of the current solver is demonstrated by comparing current DNS results with the experimental measurements. This is followed by a detailed discussion of the wake topology and aerodynamic performance in Sec. III. Finally, the conclusions are given in Sec. IV.

The numerical simulations are achieved by a second-order, Cartesian-grid-based immersed-boundary method. The non-dimensional equations governing the flow in the numerical solver are the time-dependent, viscous incompressible Navier–Stokes equations, written in an indicial form as Eq. (1),

where u_i (i = 1, 2, 3) are the velocity components in the x-, y-, and z-directions, respectively, p is the pressure, and Re is the Reynolds number.

The above equations are discretized using a second-order central difference scheme on a non-uniform Cartesian mesh, where the velocity and pressure are collocated at the cell centers. The unsteady equations are solved using a fractional step method, which provides second-order accuracy in time. An Adams–Bashforth scheme and an implicit Crank–Nicolson scheme are employed for the discretization of the convective terms and diffusion terms, respectively. Boundary conditions on immersed bodies are imposed through a “ghost-cell” procedure. The value of a variable at the ghost-cell is computed by using a central-difference approximation, and thus, the boundary conditions are prescribed to second-order accuracy. Along with the second-order accurate discretization of the fluid cells, the current numerical treatment leads to both local and global second-order accuracy in the computations. This has been confirmed in our previous study.³⁵ Flow simulations are conducted on stationary non-body-conformal Cartesian grids. This arrangement eliminates the need for the complicated remeshing algorithms that are usually employed by conventional Lagrangian body-conformal methods. In order to visualize the pressure distribution on the membrane wing surface, we first identified the normal direction of the wing surface to differentiate the suction side and pressure side of the wing. Then, the pressure values of the surrounding eight fluid cells on the same side of the wing were extracted from the underlying Cartesian grid and were used to compute the wing surface pressure using a trilinear interpolation. More details of this solver have been described in Ref. 35. Additional validation for this code can be found in the authors’ previous works.^5,36,37 The current flow solver has also been used to simulate canonical flapping plates^38–45 and modeled insect wings.^46–49 

The current study employs a rectangular wing planform with an aspect ratio of AR = 1, 2, 4, and 8, which is defined as (span)²/(area), and the wing thickness is treated as 3% of the wing chord length. The wing surface is represented by a fine unstructured grid with triangular elements. The wing root is extended out at a distance 0.5c away from the rotational axis at a range of angle of attack (α) from 15° to 90°. The rotating wing configuration is shown in Fig. 1(a). The Reynolds number range selected in this paper is from 200 to 1600, which is defined as Re = U_tipR/ν, based on the wingtip velocity and the rotational distance (R) between the wingtip and the rotational center. Due to the focus of the formation of tip vortices in the current study, the selection of the reference velocity and length scales aims to maintain the same tip Reynolds numbers when varying aspect ratios. In addition, using the spanwise distance as the characteristic length can decouple the effects of the Reynolds number and wing aspect ratio on the development of the LEV in revolving wings.⁵⁰ Table I provides a summary of all the parameters involved and their range.

The velocity profile used for the revolving wings consists of a piecewise linear function in three phases: an acceleration phase, a constant speed phase, and a deceleration phase. The acceleration and deceleration were chosen so that the wing reaches the constant angular velocity after 0.125 radians of rotation. The angular velocity profile is given by the following equation:

where $Ω0*$ is the steady state angular velocity, a is the smoothing parameter, τ₁ is the start time, τ₂ marks the end of the acceleration, τ₃ marks the beginning of the deceleration, and τ₄ marks the end of the deceleration.

This expression is a modified form of a function described by Eldredge and Ol,⁵¹ which allows for a continuous motion that is sufficiently differentiable, thereby avoiding discontinuities in the angular acceleration. In current simulations, the value of the smoothing parameter, a, is set to 30. Figure 1(b) shows the smoothed angular velocity profile. The gray shaded region represents the acceleration duration (τ₂–τ₁) and deceleration duration (τ₄–τ₃).The wing is initially at rest in a quiescent flow before accelerating to a constant angular velocity (⁠$Ω0*$⁠), and the total rotational angle is 8π (4 revolving cycles).

The computational domain size of the simulation is 30c × 20c × 30c in terms of wing chord length (c) to get domain independent results. This choice was based on our experience with the simulation of such flows and test simulations on a number of different domain sizes. Such a big domain size can guarantee the required tip clearances to produce data free from wall effects. To maintain consistent grid resolutions in both x- and z-directions, the nominal grid size employed in the current simulations ranges from 185 × 129 × 185 for the smallest aspect ratio wing (AR = 1) to 393 × 137 × 393 for the largest aspect ratio wing (AR = 8). Figure 2(a) shows a typical grid used in the current study for AR = 2. The domain mesh has two refined layers. As can be seen in this figure, very high resolution is provided in a cuboidal region around the plate in all three directions with the smallest resolution of Δx = 0.03c. Around this region, there is a secondary denser layer with Δx = 0.05c. Beyond the secondary denser layer, the grid is stretched rapidly. Boundary conditions along all sides of the computational boundary set as a convective boundary condition. A homogeneous Neumann boundary condition is used for pressure at all these boundaries. Comprehensive studies were conducted to assess the effects of the spatial and temporal grid resolutions on the computed results and to demonstrate that the chosen grids produced accurate results. Using the AR = 2 baseline case as an example [Fig. 2(b)], the spatial grid refinement study was conducted by simultaneously doubling the grid in all three dimensions in the refined zones. The overall grid size for the grid was 305 × 153 × 305 (∼14.2 × 10⁶ grid points). Temporal independence studies were conducted by halving the Δt of the original case while using the same nominal grids. The results shown in Fig. 2(b) demonstrate that the current computational setting can achieve grid independent results.

To quantify the aerodynamic performance, the force coefficients used in the current study are defined as $CL,D=(FL,FD)/0.5ρUtip2S$⁠, in which F_L and F_D are lift and drag, respectively, ρ is the density of the fluid, and S is the area of the wing. The aerodynamic power is defined as $CPW=P/0.5ρUtip3S$⁠, where P is the aerodynamic power consumption. The instantaneous aerodynamic power (P) is defined as the surface integration of the inner product between the pressure and the velocity in each discretized element.⁴⁰ In addition, the normalized pressure shown in the following paper was defined as $CP=2(p−p∞)/(ρUtip2)$⁠.

Rotating wing experiments³² were performed in a 18 in. × 18 in. × 18 in. glass tank filled with mixing water/glycerin. The geometry considered in this study is an aspect ratio two rectangular wing. The root of the wing is extended out at a distance of 0.5c from the rotation axis at a fixed angle of attack of 45°. The wing kinematics profile is similar to the one used in the current study, and the total rotational angle is 720°. The Reynolds number for this case equals to 500. As shown in Fig. 3, the force coefficients experienced a sharp peak associated with the inertial forces, followed shortly by a second peak. In general, our simulation results have a good agreement with the experimental measurements.

In this section, we first present the aerodynamic forces and vortex formation of a rectangular wing with AR = 2 and α = 45° undergoing a rotational motion at a Reynolds number of 200. This is followed by a parametric study to examine the effects of the Reynolds number, angle of attack, and aspect ratio on the vortex structure and associated aerodynamic performance for the range of parameters listed in Table I.

The lift and drag force coefficients are acquired for four revolutions of the rectangular wing with AR = 2, α = 45°, and Re = 200, as shown in Figs. 4(a) and 4(b). The simulation was run for four revolving cycles to ensure that we can observe the vortex formation and force generation of rotating wings over a sufficiently long travel distance. During the acceleration phase of the wing, the peak values of C_L (C_L,max = 1.34) and C_D (C_D,max = 1.47) appear at t/T = 0.09 (after the wingtip travels 0.3 chord length distance). The initial force peak during the acceleration phase is highly dependent on the smoothing parameter (a) in Eq. (2). As the wing transitions to a constant velocity (t/T ≥ 0.15), the lift and drag coefficients drop off quickly and subsequently plateau to an averaged value of C_L = 1.08 and C_D = 1.14 over 2.1 radians of rotation. As the wing enters the second revolution, the lift and drag coefficients drop again and level out to an intermediate averaged value of C_L = 0.74 and C_D = 0.85 during the second revolving cycle. The force reduction between the first revolution and the second revolution in both C_L and C_D is likely due to the rotating wing encounter with its wake from the previous revolution. This finding aligns with the experimental measurements conducted by Venkata and Jones.⁵² After the wing enters the third revolution, both force coefficients slightly increase and level out to another averaged value with C_L = 0.79 and C_D = 0.91, respectively, over the last two revolving cycles before the onset of deceleration. The force coefficients periodically fluctuate during the rest of the revolutions until the deceleration phase is reached. The fluctuating values of force coefficients are due to the unsteadiness of the wake structure, and this phenomenon has been widely observed for both rotating wings⁹ and translational wings⁵³ in the literature.

Figure 4(c) shows the vortex structure near the wing at twelve instants. The shell and core of the vortex structures are visualized using the Q-criterion with two iso-surface values, namely, Q = 10 (in gray) and 50 (in red), respectively. When the wing begins to rotate, the vortex structures form almost immediately, including leading-edge vortex (LEV), trailing-edge vortex (TEV), root vortex (RV), and tip vortex (TV), as shown in Figs. 4(c-i) and 4(c-ii). Slightly later in the wing rotation, the TV starts to detach from the top corner of the tip and form a secondary tip vortex (STV) around the bottom corner of the tip, as shown in Fig. 4(c-iii). As the wing continues to rotate, the wake pattern near the wingtip has a pair of vortex loops: one from the leading-edge tip and the other from the trailing-edge tip [Figs. 4(c-iv)–4(c-vi)]. The general wake pattern maintains the same during the rest of the revolutions [Figs. 4(c-vii)–4(c-xii)]. The typical vortex loops generated by the revolving wing are illustrated in Fig. 5. It appears that a pair of counter-rotating vortex loops were formed near the wingtip. It is also worth noting that such formation of tip vortices could also be produced by a non-rectangular wing shape. Utilizing stereoscopic PIV measurement, Poelma et al.³⁴ also observed a pair of tip vortices around a dynamically scaled fruit fly wing. When the wing travels at a high angle of attack, the TEV will produce a similar stable structure to that of the LEV. Similar dual tip vortex loops have also been reported by Kim⁵⁴ for a rectangular plate. Thus, we believe that this dual tip vortex structure is not attributed to the particular wing shape selected in the current study. To validate our hypothesis, we further simulated three different wing geometries of real insects (i.e., hawkmoth, fruit fly, and dragonfly). The wing rotating kinematics of these insect wings is the same as that of the rectangular wing, but with a slightly higher angle of attack (α = 60°). A slightly higher angle of attack is adopted here because the trailing-edge curvature of real insect wings could potentially reduce the strength of the TEV and prevent the formation of a STV. The correlations between the strength of the TEV and the formation of the STV will be demonstrated in Sec. III C. For all insect wings, the tip Reynolds number is 200, which is the same as the rectangular wing of the baseline case. As shown in Fig. 6, the simulation results indicate that the formation of the counter-rotating dual tip vortices is quite robust, regardless of the change in wing geometry. For this reason, our following sections will only use the rectangular wing to explore the effects of the angle of attack and wing aspect ratio on the tip vortex formation and aerodynamic performance.

To paint a clear picture of the three-dimensional flow structure and associated aerodynamic performance, the revolving wing (AR = 2) has been simulated for a range of Reynolds numbers, 200 ≤ Re ≤ 1600. Figures 7(a) and 7(b) illustrate the effects of the Reynolds number on the lift and drag coefficients at a fixed angle of attack of 45°. It can be seen that all of the curves are characterized by an initial steep increase in the force coefficients, leading to a peak, followed by a sharp drop and subsequent recovery to an intermediate level. The magnitudes of initial force peaks are identical for all the curves, but the force magnitudes after recovery are different. It appears that the higher Reynolds number leads to a higher force magnitude after recovery, but has no significant effect during the acceleration phase when inertial force plays a dominant role. In other words, the effects of the Reynolds number start to impact the aerodynamic forces when the rotating wing reaches the constant angular velocity, as illustrated in the magnified plots on the right side of Figs. 7(a) and 7(b). The mean values of the aerodynamic performance over the last two revolving cycles with the exclusion of the deceleration period are summarized in Table II.

The vortex structures at various Reynolds numbers are shown in Fig. 7(c). The time instant of the vortex structure is labeled by the black dashed line in Figs. 7(a) and 7(b). Two iso-surface values are shown to highlight the inner core (red) and outer shell (gray) of the vortex structure. At Re = 200, the primary vortex structures, such as LEV, RV, and TV, can be identified easily. In addition to the TV located at the top corner of the tip, there is also a STV generated from the bottom corner of the tip. As the Reynolds number increased to 400, all the vortex structures observed in the Re = 200 case are preserved and strengthened as indicated by the thicker red core iso-surface. When both TV and STV become stronger, the tails of these vortex loops start to interact with each other and form a hairpin-like vortex in the far-wake. This interaction becomes much stronger as the Re further increases to 800. The LEV also starts to lift up from the wing surface and bursts when it gets close to the wingtip. The outboard flow of the wing is characterized by smaller scale unsteady vortices. At Re = 1600, the instability of the vortex around the wingtip is further amplified. The vortex structures near the wingtip become less coherent at high Re. The outboard LEV bursts into a non-coherent structure, which indicates the occurrence of an instability. As a consequence, the force fluctuation amplitude increased as Re increased [Figs. 7(a) and 7(b)]. The force fluctuation of revolving wings after a long traveling distance has also been reported for a hawkmoth shaped wing by Usherwood and Ellington.⁹ 

Next, we compared the LEV and TEV formations of the rotating wings with various Reynolds numbers when the aerodynamic forces reached a steady-state value. For each case, multiple 2D slices were taken along the wingspan. Eight slices are shown in Fig. 8 from b/c = 0.25 to b/c = 2.0 along the spanwise direction from the wing root to the wingtip. The size of the LEV continues to grow proportionally along the span. The surface pressure distribution shows the low-pressure region (dark blue) on the top of the wing gradually amplified with the increment in the Reynolds number. For the high Reynolds number cases (Re = 800 and 1600), the LEV becomes more detached from the wing surface around the wingtip, and this leads to a reduction in the low pressure area around the top corner of the tip.

To quantify the strength of the LEV, we first visualized the spanwise vorticity field (ω_span) using contour lines. After the vortex on the 2D slice was manually identified, a closed contour line around this vortex with a specified contour level was selected. Next, the circulation (Γ) of the vortex was calculated by integrating the spanwise vorticity over the area enclosed by the contour level. Although the magnitude of the circulation depends on the chosen contour level, the characteristic behavior of the vortex is not affected by this choice. To make a fair comparison among different cases, in our calculation, the spanwise vorticity was normalized by ω_spanc/U_tip, while the LEV circulation was normalized by U_tip · c. Figure 9 compares the normalized circulation of the LEVs along the wingspan. It shows that the strength of the LEV increases along the wingspan and amplifies ∼1.7 times when the Reynolds number increases from 200 to 1600.

In this section, we examined the effects of the angle of attack on the vortex structures and aerodynamic performance for an AR = 2 wing at Re = 200. Figure 10 shows the evolution of the wingtip vortex with different angles of attack and its associated surface pressure distribution on the suction side of the wing. It can be seen that there is only one vortex loop formed around the wingtip at a lower angle of attack (α = 15° and 30°). As the α increases to 45°, a tiny STV starts to develop from the bottom corner of the wingtip. This might be due to the enhancement of the TEV at a higher angle of attack. The viscous flow around both the leading-edge and the trailing-edge is dominated by the flow separation and produces vorticity. At a lower angle of attack, we can only see the trailing-edge shear layer around the trailing-edge [Figs. 10(a) and 10(b)]. For a revolving wing with a higher angle of attack, however, a strong TEV will be generated and will form a wingtip vortex trail [STV in Figs. 10(c)–10(f)]. With the continuous increase in α, ranging from 45° to 75°, the STV becomes stronger and has the same magnitude as the TV when the α reaches 90°. The proximity of the vortex to the surface of the wing promotes a strong region of suction along the leading-edge. This is indicated by the dark blue region of the pressure coefficient (C_P) on the top wing surface. For the lower α cases [15° and 30°, Figs. 10(a) and 10(b)], the suction region expands along the leading-edge direction and covers the wingtip region. As the α increases, the vortex lifts off the surface into an arch-type structure as it reorients itself along the tip, which reduces the suction near the tip, as shown for α = 45° and 60° [Figs. 10(c) and 10(d)]. The continuous increase in α promotes the LEV detachment, and it leads to a further loss of suction near the tip, such as in α = 75° and 90° cases [Figs. 10(e) and 10(f)]. It appears that the formation of the STV has a strong correlation with the strength of the TEV. Once the strength of the TEV is enhanced, it turns around at the bottom corner of the wingtip and forms its own trail of the vortex loop.

To illustrate the effects of the angle of attack on the LEV strength, Fig. 11 shows the values of LEV circulation at different locations along the wing span. For these rotating wings with the same Reynolds number and aspect ratio, the maximum LEV circulation that they can reach seems bounded within a certain range (normalized circulation ∼0.3 for the current study). The LEV circulation gradually increases before reaching its maximum limit and starts to drop afterward. However, the location where the peak circulation presents along the wing span depends on the angle of attack. For the case with a higher angle of attack (≥60°), the slopes of the curves are high, and the maximum circulations are reached around b/c = 0.5–1.0 (before the mid-span, b/c = 1.0). For the cases with smaller angles of attack (30°, 45°), the LEV circulations increase gradually along the wing span and reach its peak value close to the wingtip, b/c = 1.75. For the case with α = 15°, which has a much less increasing slope, the LEV circulation could not even reach its maximum limit or show any decreasing trend along the spanwise direction.

To present a better visualization of the LEV and TEV evolutions at various angles of attack, Fig. 12 shows the vortex structure on the local coordinates of the wing chord at three different locations along the wingspan, b/c = 0.5, 1.25, and 1.75. As the α gradually increases from 15° to 90°, the LEV starts to lift upward from the suction surface; meanwhile, the trailing-edge shear layer gradually grows up along the pressure surface. When the α reaches 90°, the rotating wing forms a pair of tip vortices with identical strength, as visualized in Fig. 10(f).

Figure 13 shows the temporal variation in the lift and drag coefficients over the entire four revolving cycles. The lift coefficient increases before the angle of attack reaches 45° and then decreases as the angle of attack continuously increases, while the drag coefficients monotonically increase with the angle of attack. The force decrease of rotating wings after the first revolution was previously reported by Venkata and Jones.⁵² In their study, the rotating wing with an angle of attack of 45° experienced a force drop when the wing encountered its wake from the previous revolution. In our study, it is worth noting that the force reductions after the first revolution were not clearly presented for the cases with higher angles of attack (α = 75°, 90°). One possible reason for that is that the high angle of attack of revolving wings will not generate a strong downwash in the vertical direction during the previous rotational cycle. Thus, the wake induced lift drop during the second revolution is not obvious for the higher angle of attack cases (α = 75°, 90°). The mean values of aerodynamic performance including force and power coefficients, lift-to-drag ratio, and lift-to-power ratio are shown in Fig. 14. The maximum lift coefficient appears around $α=45○$⁠, while the maximum aerodynamic efficiency (in terms of lift-to-drag ratio or lift-to-power ratio) is achieved around α = 30°.

In this section, we examined the effects of the wing aspect ratio on the wake topology and aerodynamic performance of the rectangular wing at Re = 200 with a fixed angle of attack of 45°. The effects of wing aspect ratios were investigated by extending the wing span while matching the tip Reynolds number for all cases. The tip Reynolds numbers for all aspect ratios were maintained the same by adjusting the angular velocity of rotating motion. Previous revolving wing studies^33,55 suggested that the Rossby number also needs to be matched in addition to the reference Reynolds number and angle of attack to isolate the effect of aspect ratios. However, matching the Rossby number requires increasing the root offset of the wing configuration, which may not be a suitable option for the vehicle design, and, thus, not adopted in the current study.

Figure 15 shows the vortex structures of rotating wings with different aspect ratios. In Fig. 15, the LEV and TV develop and connect smoothly along the wing span direction. The size of the vortex loop increases with the aspect ratio since the travel distance at the wingtip also scales with the wingspan, but the overall development of the vortex system is similar for each aspect ratio. A detailed analysis of the vortex formation shows that the general wake pattern near the wingtip shifts from a single vortex loop for lower aspect ratio cases (AR = 1) to a pair of counter-rotating vortex loops for higher aspect ratio cases (AR = 2, 4 and 8). Similar to the effect of the angle of attack, the wing aspect ratio can also promote the development of the trailing-edge vortex. The low-pressure area (dark blue) was also enlarged with the increment in the wing aspect ratio due to the intensified LEV. The aforementioned observation is in line with the classic Kutta–Joukowski theory of thin airfoils.⁵⁶ This inviscid theory could achieve a good approximation for the real viscous flow in typical aerodynamic applications. For the revolving wing explored in the current study, a higher strength of leading-edge vorticity will require greater circulation around the trailing-edge for the establishment of the Kutta condition. In other words, if new vorticity is introduced along the leading-edge due to the increase in the angle of attack and wing aspect ratio, then it must be accompanied by equal and opposite vorticities around the trailing-edge.

Figure 16 shows a comparison of normalized LEV circulation along the wing span for various aspect ratio wings. In general, the circulation of the LEV increases approximately linearly up to 70% wing span and then starts to decrease due to the split of the LEV near the wingtip. For high aspect ratio wings, however, the LEV circulation increases again at about 85% spanwise locations. In addition, the LEV of low aspect ratio wings has a relatively higher strength near the wing root.

Figure 17 shows the time course comparison of force coefficients of wings with AR = 1, 2, 4, and 8 in four rotational cycles. During the rotational motion, the instantaneous aerodynamic performances of all aspect ratio cases share the similar decreasing and increasing tendency, and all the force magnitudes gradually arrive at a constant value once the flow reaches a nearly periodical fluctuating state. However, the force drop after the first revolution becomes mild for the wing with a higher aspect ratio (AR > 4). The mean value of forces and power coefficients, lift-to-drag ratio, and lift-to-power ratio are shown in Fig. 18. Although the AR = 1 case owns a relatively higher lift coefficient, the increment in aspect ratios can improve the aerodynamic efficiency in terms of lift-to-drag ratio and lift-to-power ratio. The optimal case tested in the current study is the wing with AR = 3. In addition to the wing aspect ratio, the wing shape could also significantly modulate vortex structures and their associated aerodynamic performance. As suggested by Shahzad et al.,⁵⁷ a desirable wing trailing-edge shape could reduce the power consumption of the outboard wing surface and, thus, improve the power economy by up to 30% with the Reynolds number ranging from 400 to 13 500. Although the effects of wing surface deformation were not considered in the current study, it is worth noting that the wing flexibility could dramatically improve the LEV attachment along the wingspan and, thus, enhance the aerodynamic force generation. For instance, Fu et al.⁵⁸ experimentally investigated the wing flexibility effects on the aerodynamic performance of flapping wings and illustrated that the flexible wing could offer higher lift-to-drag ratios compared to a fully rigid wing. In addition, Shahzad et al.⁵⁹ numerically studied the aerodynamic performance of hovering wings and revealed that the wing flexibility could increase the mean lift generation by up to 39%. It is expected that the aerodynamic performance of the revolving wings explored in the current study could be further improved by adding wing flexibility, especially for the wings with higher wing aspect ratios (AR > 3).

High-fidelity numerical simulations were conducted to examine the vortex structure and aerodynamic loading on revolving wings over a range of Reynolds number, angle of attack, and aspect ratio. The simulations have shown that for the cases with a lower angle of attack or lower aspect ratios, there was only a single TV loop near the top corner of the wingtip. With the increment in the LEV strength due to either higher aspect ratios (AR > 1) or higher angles of attack (α > 30°), the trailing-edge share layer rolls up and develops into a TEV. The TEV turns around at the wingtip to become a STV near the bottom corner of the wingtip. The TV and STV together form a pair of counter-rotating vortex loops around the wingtip, and this vortex structure is quite robust even for different wing geometries.

The instantaneous forces and aerodynamic power coefficients were also evaluated. An initial peak was observed for all simulations right after the acceleration phase due to the impulsive start. Following this initial peak, the forces and power coefficients experience a sharp drop and recover to an intermediate value for the remainder of the first evolution. After the wing enters the second revolution, the force coefficients drop and level out to a lower value, compared with the first revolving cycle. However, the force drop becomes not obvious for the revolving wing with higher angles of attack (α > 75°) and higher aspect ratios (AR > 4).

The authors thank Dr. Anya R. Jones at the University of Maryland, College Park, for sharing her experimental data for our solver validation. The majority of this study was finished at the University of Virginia under the support of the NSF Grant No. CBET-1313217, the AFOSR Grant No. FA9550-12-1-0071, and the ONR MURI Grant No. N00014-14-1-0533 to H.D. This research was also supported by NSF Grant No. CMMI-1554429, Army Research Office DURIP Grant (No. W911NF-16-1-0272) to B.C., 2019 Villanova University Summer Grant Program, and 2019 ORAU Ralph E. Powe Junior Faculty Enhancement Award to C.L.