Unsteady aerodynamics of a pitching-flapping-perturbed revolving wing at low Reynolds number Free

In recent years, as micro air vehicles (MAVs) have become smaller, less expensive, more precise, and nimble, they are becoming a critical technology to the success of a broad range of civil and military operations such as reconnaissance, surveillance, aerial photography, and search and rescue.¹ MAVs typically fly at low speeds and operate at a low Reynolds number (Re) similar to small natural flyers.^2–5 Among all possible layouts of MAVs, those based on unidirectionally revolving wings, such as quadcopters and helicopters, are arguably the most aerodynamically efficient layouts, at least beyond certain critical Re.^6–8 However, due to the adverse viscous effect, their efficiency (e.g., lift-to-drag ratio) is significantly reduced as Re decreases and the maximal lift coefficient achievable is also limited.^9–11 As a result, at smaller sizes, quadcopters and helicopters suffer from short flight duration and low payload capability. Therefore, efforts have been made to look for potentially better solutions from nature, i.e., mimicking insect flights with reciprocating revolving wings (or flapping wings).^12–14 Unlike high Re flights, which predominantly exploit steady (e.g., fix-wing airplanes) or quasi-steady (e.g., helicopters) aerodynamic principles,¹⁵ insects and small birds use flapping wings to exploit unsteady aerodynamic mechanisms in vortex-dominated flow for augmenting force production and aerodynamic efficiency.^16–18 In this regard, designing small-scale MAVs with comparable performance to their natural counterparts should hinge critically on a successful exploitation of the unsteady vortex-dominated aerodynamics at a low Re.

Instead of reversing the revolving motion in flapping flight, an alternative to exploit unsteadiness is to add perturbations to revolving motion, e.g., vertical flapping and pitching. Our previous study¹⁹ has found that in a flapping-perturbed revolving wing (FP-RW) at Re = 1500, merely flapping perturbation can enhance lift and reduce drag, however at the sacrifice of lower power loading.¹⁹ In addition, for FP-RWs at a positive angle of attack (α₀), the leading-edge vortex (LEV) is generated periodically on both upper and lower wing surfaces with a strong asymmetry of strength. This is evidently distinct from a unidirectional revolving wing, where a stable LEV is attached on the upper surface.²⁰ Therefore, the performance improvement of a FP-RW is strongly related to the difference in LEV behaviors between up and down flapping half cycles.

Building upon our previous work¹⁹ on flapping-perturbed revolving wings, the current study further investigates the effects of pitching for a pitching-flapping-perturbed revolving wing (PFP-RW). The pitching motion is expected to provide further benefits for drag and power reduction in revolving wings, which can be informed by previous research on two-dimensional (2D) pitching-plunging airfoils.^21,22 2D pitching-plunging airfoils with freestream velocity have been widely studied in the past decades as an abstracted model for biological propulsion inspired by fish swimming²¹ and bird forward flight²³ or for energy harvesting systems.²⁴ They shed light on our current study because of the similarities of motion despite the lack of spanwise velocity gradient and wing tip effects that exist in the motion of PFP-RW. In 2D pitching-plunging airfoils, the unsteadiness is mainly measured by the Strouhal number (St), while pitching amplitude (α_p) and phase angle (θ) also affect the aerodynamic performance and vortex behaviors. Previous research shows that pitching can simultaneously lower the power consumption and enhance the thrust generation of a 2D pitching-plunging airfoil.^21,22 Triantafyllou et al.²¹ predicts that the optimal thrust efficiency for a pitching-plunging airfoil occurs at St = 0.25-0.35, which is linked to the existence of reverse Karman vortex street. Hover et al.²⁵ and Read et al.²⁶ further pointed out that a sinusoidal pattern of effective angle of attack and a 90° phase offset for pitching is beneficial for propulsion, respectively. Nonetheless, previous studies of 2D pitching-plunging airfoils mainly focus on zero angle of attack, and thus lift production is usually not considered, which however is one of the main performance measures for revolving wings. In addition, compared with 2D pitching-plunging airfoils, the 3D effects in PFP-RW owing to the spanwise velocity gradient generated by flapping and revolving motion as well as wing tip effects may lead to different aerodynamic performance and vortex behaviors and alter their dependencies on the Strouhal number.

Finally, note that the concept of PFP-RW is also related to the flapping rotary wing²⁷ and the “Flotor.”²⁸ The flapping rotary wing undergoes passive rotation (or revolving) due to the thrust generated by forced pitching-flapping motion. At a steady-state, flapping rotary wings simply represent the self-rotating equilibrium states of PFP-RWs. Wu et al.⁶ investigated the aerodynamic performance of a flapping rotary wing and compared it with other layouts, as they concluded that the flapping rotary wing could be a suitable layout for generating high lift coefficient. Recent studies further look into the performance optimization²⁹ and flow control method.³⁰ The “Flotor” is a concept of the MAV layout that actively flaps the wing, while the rotation can be either motor-driven or passive. By switching into active-flapping and active-rotating mode (similar to PFP-RW), a higher lift efficiency over revolving wing can be achieved.²⁸ However, this improvement is not observed in active-flapping and passive-rotating mode (similar to flapping rotary wing). Notably, the pitching motion on the “Flotor” is achieved passively through wing chordwise deformation; therefore, the effects of which cannot be studied systematically. The underlying relation between performance improvement observed in active-flapping and active-rotating mode and pitching motion remains unclear.

Using combined experimental and numerical methods, the goal of this study is to quantify the force production and leading-edge vortex (LEV) behaviors of a pitching-flapping-perturbed revolving wing in hovering flight at Re = 1500, and how they are governed by the Strouhal number (St), pitching amplitude (α_p), nominal angle of attack (α₀), and pitching-flapping phase angle (θ). Here, the wing revolving is considered as a baseline motion and the pitching-flapping motion, which is considered as a form of “perturbation,” is imposed on the baseline revolving motion, thereby giving rise to the term pitching-flapping-perturbed revolving wing (PFP-RW). Details on experimental and numerical methodology are given in Sec. II. We first focus on the effects of symmetrical pitching (i.e., a 90° phase angle in advance with respect to flapping) at α₀ = 0° (Sec. III A) and 20° (Sec. III B). The cycle-averaged performance and LEV behaviors are described in detail. The pitching effect is further discussed in terms of the LEV-mediated pressure component and geometric component. Note that PFP-RW at α₀ = 0° is studied here to further understand the self-rotating equilibrium due to pitching-flapping motion, especially as compared with those at α₀ = 20°. The effects of pitching-flapping phase angle, i.e., asymmetric pitching, are quantified and discussed in Sec. III C. The significance of our findings to applications in MAV is discussed in Sec. III D. Final conclusions are drawn in Sec. IV.

Experiments are performed by operating a dynamically scaled robotic wing mechanism in an acrylic tank (0.8 m × 0.8 m × 0.8 m) filled with white mineral oil (Tulco, density ρ = 826 kg m⁻³, viscosity υ = 6 × 10⁻⁶ m² s⁻¹), as shown in Figs. 1(a) and 1(b). The robotic wing has three degrees-of-freedom (3DOF), i.e., revolving, flapping, and pitching, each of which is actuated by a high-performance digital servo (XM450-W350-R, Robotis) with an angle resolution of 1°/4096. The force and torque acting on the wing are measured by a six-axis load cell (Nano17-IP65, ATI) mounted at the wing root. The sensing ranges for force and torque are ±12 N and ±120 N mm, with resolutions of 1/320 N and 1/64 N mm, respectively. As shown in Fig. 1(c), both robot motion control and force/torque data acquisition are modeled in Simulink (The MathWorks) and then implemented by using a real-time target machine (Performance Real-Time Target Machine, Speedgoat GmbH). The sample rate of motion control and data acquisition system is 500 Hz. More details of the robotic wing system can be found in Ref. 19.

The wing is chosen as a rigid rectangular plate (chord c = 0.06 m, span b = 0.18 m, and thickness 1.6 × 10⁻³ m), a commonly tested geometry in low-Re flight experiments.^10,11 It is made of plastic sheet (PETG) and has a root offset Δr = 0.9c [Fig. 1(b)]. The wing pitch axis is located at mid-chord, which is reinforced by using a carbon fiber rod (diameter 2 × 10⁻³ m) to enhance stiffness. The radius of second and third moments of wing area, given by Eqs. (1) and (2), is 0.154 m and 0.162 m, respectively,

Our definition of the coordinate system for a pitching-flapping-perturbed revolving wing (PFP-RW) is shown in Fig. 1(d). Starting from the inertial frame (OX_iY_iZ_i), the revolving frame (OX_rY_rZ_r), flapping frame (OX_fY_fZ_f), and pitching frame (OX_pY_pZ_p) are introduced successively by rotating the wing with a revolving angle (ψ, defined about Y_i), flapping angle (φ, defined about X_r), and pitching angle (α, defined about Z_f). The wing-fixed frame (OX_wY_wZ_w) is further introduced by shifting the pitching frame to the wing root. The kinematic angles of the wing are thus prescribed according to the following equations:

The wing is revolving at a constant angular velocity (Ω_y = U_g/R₂), and the pitching and flapping perturbations are both sinusoidal. f denotes the frequency of pitching and flapping perturbations, and amplitudes are defined as α_p and φ_f, respectively. α₀ is the nominal angle of attack, and θ is the phase angle between pitching and flapping. Note that a 90° phase angle in advance with respect to flapping is regarded as a symmetrical pitching perturbation, as pitching angle arrives at α₀ at each stroke reversal. The revolving angle is 1440° (four revolutions) and a ramp transit is applied to each angular velocity within the first and last 120° revolution to remove any abrupt change. According to the observation of fully developed wake generated by a FP-RW in our previous study,¹⁹ the data within 1.5–3.5 revolutions are processed.

The parameter space in our study is provided in Table I. Note that, only revolving velocity is considered in our definition of Re. The effective angle of attack (α_e) is defined in Eq. (6). Here, the first term in the RHS is the dynamic angle of attack as a result of the flapping angular velocity, whereas the second term is the geometric angle of attack

As shown in Figs. 2(a)–2(c), a good trajectory tracking is achieved in all three degrees of freedom. The slight phase shift between the reference and the actual trajectories is resulted from the phase lag in the servo motor due to the proportional-integral-derivative (PID) control and motor dynamics. To eliminate the effects of kinematic tracking error, only the actual trajectories measured from the motor encoders are used in the post-processing. To remove the inertial component from raw force data, we repeat our experiments in air. The gravity and buoyancy components are measured at rest and then transformed to each instant along the trajectory. A four-order Butterworth low-pass filter is applied to remove the noise in force measurements, and the cut-off frequency is 5 times of the perturbation frequency in each trial. A sample of raw and filtered data is displayed in Fig. 2(d). Each trial is repeated three times and the force data are separated and averaged according to flapping cycles to eliminate any cycle-to-cycle noise from the motion control and force measurements. Given a trajectory, the uncertainty of cycle-averaged forces is evaluated by calculating the standard deviation among different trials. Within our study, the standard deviation for lift and self-driving torque coefficients [defined in Eqs. (8) and (9)] are below 0.04 and 0.02, separately. In addition, according to our tests of boundary effect (included in the supplementary material), the experimental conditions well approximate an infinite volume.

The aerodynamic loads and angular velocity in the inertial frame can be derived from those measured at wing root through coordinate transformation, the details of which can be found in Ref. 19. The aerodynamic power is calculated by the dot product of aerodynamic torque (T) and wing angular velocity (Ω) in the inertial frame:

The lift coefficient (C_V) is calculated based on vertical force (F_y) [Eq. (8)], and the self-driving torque coefficient (C_T) is derived from the torque acting around the y-axis in the inertial frame (T_y) [Eq. (9)]. The power coefficient (C_P) is defined by Eq. (10)

Here, a negative $C¯T$ (bar represents cycle-averaged value) indicates that an external driving torque is required to rotate the wing, whereas the wing can rotate without external driving torque when $C¯T$ equals to zero. In other words, the wing is at a self-rotating equilibrium state sustained by the pitching-flapping motion. A positive $C¯T$ indicates that the pitching-flapping motion will effectively accelerate the wing to a higher speed and achieve another self-rotating equilibrium (without external driving torque). A dimensionless ratio, power loading (PL), is used to measure the aerodynamic efficiency in lift generation

To quantify the efficiency in the reduction of external driving torque for revolving (i.e., the increase of $C¯T$⁠), a dimensionless ratio (η_DTR) is defined,

where $ΔC¯T$ is the difference of driving torque between a perturbed wing and the steadily revolving wing.

To understand vortex behaviors, the flow field of representative cases is obtained through computational fluid dynamics (CFD) simulation using an in-house numerical solver.^6,31 The solver uses the artificial compressibility method developed by Rogers et al.^32,33 to calculate the velocity and pressure field based on 3D incompressible unsteady Navier-Stokes equations:

where u and p are velocity vector and static pressure, respectively. A reference time (T_ref, defined as c/U_g) is used to transform real time domain (t) into dimensionless time domain (τ, defined as t/T_ref). Another dimensionless time $t^∈$ [0, 1] is introduced to describe wing motion within a flapping cycle, where $t^$ = 0 and 1 denote the start of upstroke and the end of downstroke, respectively.

As shown in Fig. 3(a), the mesh used in our study is an O—H grid with 81 × 81 × 91 nodes in the normal, chordwise, and spanwise direction. The domain size, first grid distance, and time step are 30c, 0.001c, and 400, respectively. Details on numerical method verification and grid validation can be found in Refs. 6 and 31. From our previous study on flapping-perturbed revolving wings (FP-RWs), a good agreement of instantaneous force generation between robotic wing experiments and simulations was achieved.¹⁹ Additional validations between experiments and simulations on pitching-flapping-perturbed revolving wings (PFP-RWs) are provided in the supplementary material.

Based on computed flow data, a robust algorithm, developed according to Graftieaux et al.³⁴ is used to locate vortex core and its boundary. Two scalars, Γ₁ and Γ₂, are derived from the computed velocity field as follows:

A 5 × 5 (total points N = 25) window is swept over the X-Y plane at each spanwise position. q is the center of control window, x is the position vector, u is the velocity vector, n is the unit vector in the z direction, and $ūq$ is the averaged velocity at q. Usually, the vortex core is located at the position of $Γ1$ maximum, whereas the vortex boundary is circled by $Γ2$ = 2/π.³⁴ A sample of vortex detection within mid-span slice of a PFP-RW (St = 0.56, α₀ = 20°, α_p = 8°, and θ = 90°) at mid-downstroke (⁠$t^$ = 0.75) is shown in Figs. 3(b) and 3(c). Here, we only inspect the core location and circulation of leading-edge vortex (LEV). Once the LEV is outlined, the total circulation is calculated by integrating vorticity within the boundary [i.e., at a certain span position r in Eq. (16) and in whole LEV volume in Eq. (17)],

As shown in Fig. 4(a), the generation of (positive) self-driving torque is observed at zero nominal angle of attack (α₀) when a symmetrical pitching perturbation is introduced to a FP-RW. First, compared to a steadily revolving wing, pitching-flapping perturbation produces monotonically strengthening self-driving torque (⁠$C¯T$⁠) as St increases (or equivalently the reduction of external driving torque). Second, $C¯T$ also increases monotonically with the amplitude of symmetric pitching (α_p), despite flapping perturbation solely (i.e., FP-RW) can still result in a positive $C¯T$⁠.¹⁹ However, the increase seems to plateau as pitching amplitude increases. Aerodynamic power coefficient (⁠$C¯P$⁠) also drops monotonically with increasing pitching amplitude [Fig. 4(b)], where no plateauing seems to occur. Consequently, as shown in Fig. 4(c), the efficiency of the external driving torque reduction (η_DTR) undergoes an approximately exponential decay with St, at higher rates for higher α_p.

Note that, the propulsive efficiency of 2D pitching-plunging wings at zero angle of attack, which reaches maximum with St = 0.2 ∼ 0.4,²¹ is akin to the η_DTR defined here. The propulsive efficiency of 2D pitching-plunging wings measures the efficiency in forward propelling, whereas η_DTR measures the efficiency in rotating the wing. Within the range of St tested in the study of PFP-RWs, no peak of η_DTR is captured [Fig. 4(c)]. In other words, the efficiency peak may only occur at infinitesimal St, which has less practical significance as the reduction of external driving torque at infinitesimal St is negligible. This discrepancy between 2D and 3D cases can be possibly attributed to the tip vortex effects in PFP-RW. The tip vortex effect and the resulting induced drag are an inherent phenomenon in finite-span airfoil aerodynamics.¹⁵ The existence of tip-vortex-induced downwash flow can tilt the force backward, leading to a lower lift but a higher drag. It is further shown that the above-mentioned tip vortex effect also occurs in low Re flapping wings,³⁵ which can reduce force generation. For PFP-RWs, the tip vortex might similarly tilt the pressure force backward, reducing its horizontal component (thrust) but enhancing the vertical component in the wing frame. Therefore, compared with 2D cases, for a perturbation with similar St and pitching amplitude, the PFP-RW could consume more power but gain less thrust. Thus, the generation of self-driving torque of PFP-RW might not dominate the increase of additional power at small St, which could possibly explain the lack of an optimum St. Another possible 3D effect in PFP-RW is the spanwise gradient of velocity. Although its existence cannot significantly enhance force generation on 2D translating plates,³⁶ its effect on the force generation in PFP-RWs remains unclear, which demands future study.

Since the shear force (friction) is relatively low, the lift and self-driving torque of a PFP-RW depend primarily on the normal pressure force and its geometric projection in the vertical and horizontal direction (in the wing frame), both of which vary with pitching amplitude. The pressure force, however, is largely determined by the behaviors of leading-edge vortex (LEV) and its circulation. Generally, for a PFP-RW at α₀ = 0°, the addition of pitching leads to mutually opposing effects of pressure force and geometric projection on the generation of self-driving torque: pitching reduces the pressure force but increases the horizontal projection (with a proper phase of pitch, see Sec. III C). As a result, the increase of $C¯T$ becomes stronger as pitching amplitude increases; however, plateaus due to the significantly weakened pressure force as pitching amplitude becomes large. In the following, we first explain how increasing pitching amplitude leads a lowering pressure force in terms of wing surface pressure distribution and LEV circulation. Then, the increase of $C¯T$ as a result of the geometric projection of the LEV-mediated pressure force is described. Without losing generality, the effects of pitching are explained only using the example of St = 0.56, where the increase of $C¯T$ is most prominent.

The addition of pitching first reduces the normal pressure force compared with that of a FP-RW. Due to the symmetric addition of pitching at α₀ = 0°, both the reduction of pressure force [Fig. 5(a)] and the variation of the pressure distribution [Fig. 5(c)] are almost symmetrical between upstroke and downstroke. As α_p increases, the low-pressure region (LPR1) on the lower surface at mid-upstroke (⁠$t^$ = 0.25) and the LPR1 on the upper surface at mid-downstroke are both reduced (⁠$t^$ = 0.75). At each stroke reversal (⁠$t^$ = 0 and 0.5), the LEV generated in previous stroke can impinge to the wing surface, resulting in another low-pressure region (LPR2).¹⁹ As α_p increases, the LPR2 is also attenuated and shifted toward wing tip. This symmetrical variation of pressure distribution indicates that the pitching effect on the flow is symmetrical. Further analysis shows that the attenuation of low-pressure regions (LPR1 and LPR2) is mainly resulted from a symmetrical reduction of LEV circulation.

Based on a quantitative analysis of FP-RWs and PFP-RWs at α₀ = 0°, the local circulation distribution [Fig. 6(a)] is still symmetrical between upstroke and downstroke, except for the wing tip region, where a strong tip vortex effect and LEV tilting occur.²⁰ As α_p increases, the LEV is attenuated monotonously in terms of size and strength [$t^$ = 0.25 and 0.75, Fig. 6(a)]. The conical structure of LEV even becomes absent in the inboard region at mid-stroke (⁠$t^$ = 0.25 and 0.75) when α_p = 24°, as no obvious increase of local LEV circulation along the span is observed [Fig. 6(a)]. The introduction of pitching is expected to reduce both the geometric and effective angle of attack, which in turn weakens the strength of LEV. To quantify the relationship between the LEV circulation and the effective angle of attack (α_e) as pitching amplitude (α_p) increases, we integrate the LEV vorticity globally over the entire volume of LEV [Fig. 6(b)], which is then plotted against α_e for different pitching amplitudes. As α_p increases, both the volume-averaged LEV circulation and the α_e decrease, whereas the former decreases linearly over the latter [Fig. 6(b)]. Therefore, this suggests that α_e predominately controls the strength of the LEV for α₀ = 0° cases, and therefore also the reduction of pressure force described above. On the contrary, as shown in Figs. 6(c) and 6(d), when pitching is introduced, no significant variation of the LEV core position occurs along the span.

Although the weakened LEV circulation (due to the pitching induced reduction of effective angle of attack) reduces the total pressure force acting normal on the wing surfaces, pitching also increases the horizontal projection of dimensionless pressure force (C_Fh) in the wing-frame [Fig. 5(b)]. Here, projection angle (β) is defined as the angle between pressure force F_n and its vertical component F_v. A positive C_Fh indicates that the wing is generating instantaneous thrust and therefore driving torque to rotate itself (i.e., a positive C_T). The combined effects of the reduced pressure force and strengthened horizontal projection determine the increase of $C¯T$ [Fig. 5(c)]. However, as α_p further increases, the increase of projection is compromised by the decrease of pressure force, and therefore no further increase of $C¯T$ can be achieved [Fig. 4(a)].

At a positive nominal angle of attack (α₀ = 20°), the pitching effects on both cycle-averaged lift and self-driving torque are further investigated for the PFP-RW. As shown in Fig. 7(a), the cycle-averaged lift (⁠$C¯V$⁠) can be augmented with only flapping (as in a FP-RW), and the augmentation increases monotonically with St.¹⁹ The addition of pitching (in PFP-RWs) further increases the lift augmentation, but the increase seems to plateau at large pitching amplitude (α_p), as the difference of $C¯V$ between α_p = 16° and 24° cases is almost negligible [Fig. 7(a)]. This is similar to the increase of $C¯T$ at α₀ = 0°. In a FR-RW with $α0$ = 20°, where pitching is absent, the external driving torque for revolving (negative $C¯T$⁠) is enhanced as St increases [Fig. 7(b)]. However, in PFP-RWs, the addition of pitching can reduce the external driving torque and further leads to a self-rotating equilibrium [Fig. 7(b)], which strengthens as either St or α_p increases. Notably, at St = 0.56 and α_p = 24°, the self-rotating equilibrium can be achieved for the PFP-RW at α₀ = 20°.

Aerodynamic power, as measured by $C¯P$⁠, increases with St and is higher than that of steadily revolving wings [Fig. 7(c)]. The addition of symmetric pitching apparently reduces power consumption and enhances the aerodynamic efficiency [Figs. 7(c) and 7(d)]. The power loading (PL) decreases monotonically with St regardless of pitching amplitude (α_p), which indicates that the lift generation efficiency deceases with St for both FP-RW and PFP-RW. However, pitching with larger amplitude can mitigate the decrease of PL [Fig. 7(d)]. In general, the PL of FP-RW and PFP-RW is lower than that of the revolving wing, indicating that the lift augmentation is at the cost of lower aerodynamic efficiency. However, there exists a notable exception for the largest pitching amplitude tested, i.e., at α_p = 24°, that an enhancement of PL over a revolving wing can be achieved when St is below 0.22 [the blue segment in Fig. 7(d)]. Remarkably, this implies that a perturbation of small St (with either small flapping velocity or high revolving speed) and large pitching amplitude can simultaneously enhance lift magnitude and efficiency of a revolving wing. Also note that, at St ∼ 0.1 and α₀ = 10°-20°, the “Flotor” with active rotating and flapping can generate a higher lift and Figure of Merit (FoM) than pure revolving wings.²⁸ The critical St and α₀ corresponding to the performance improvement of the “Flotor” shows a good agreement with our findings, despite no details on pitching amplitude is reported in their study as the pitch motion is passively achieved by wing flexibility.

The high lift generation efficiency of the PFP-RW at small St and large α_p can be explained by the variation of effective angle of attack (α_e), as shown in Fig. 7(e). According to a previous study on the efficiency of a collection of 2D flapping kinematics,³⁷ a two-stroke motion (vertical flapping combined with forward speed) can reach a lift generation efficiency close to (but slightly lower than) that of the optimal steady flight, when downstroke is a gliding flight and upstroke has an angle of attack near the optimal value of the steady flight. For the PFP-RW at α₀ = 20° and St = 0.11, as pitching amplitude increases, the downstroke is approaching a gliding flight, as the geometric angle of attack is reduced. The α_e in upstroke, on the other hand, is increased. An α_e larger than 20° nominal angle of attack is reached in upstroke when pitching amplitude is 24° [Fig. 7(e)]. Note that, the downwash due to revolving motion can reduce α_e which is not considered in its definition [Eq. (6)]. Thus, for a PFP-RW with α_p = 24° and St = 0.11, its averaged effective angle of attack during upstroke is closer to the optimal one of a steady revolving wing (15°–20°). Therefore, according to Wang’s study,³⁷ at α₀ = 20° and St = 0.11, the PFP-RW is operating close to the most efficient status in lift generation. Though the peak efficiency of PFP-RW surpasses its steady counterpart (i.e., revolving wing) at α₀ = 20° [Fig. 7(d)], it is still slightly lower than the global optimal efficiency¹⁹ (power loading of the revolving wing peaks about 2.75 at α₀ = 15°), which further agrees with Wang’s conclusion.

Similar to α₀ = 0° cases, the pitching effects on the lift and self-driving torque of PFP-RWs at α₀ = 20° depend both on the attenuation of pressure force and the variation of geometric projection. As shown in Fig. 8(a), the normal pressure force and its vertical component only have minor differences throughout a flapping cycle as pitching amplitude changes, indicating that the change of projection due to pitching has negligible effects on lift generation. Therefore, the variation of lift mainly depends on the asymmetry of pressure force during upstroke and downstroke. Specifically, the drop of pressure force due to pitching is more substantial during upstroke, as there is an apparent shrinkage of LPR1 at $t^$ = 0.25 when pitching is introduced [Fig. 8(c)]. However, the changes of LPR1 at mid-downstroke (⁠$t^$ = 0.75) are less significant, suggesting an asymmetric modulation of pressure force and LEV circulations due to pitching, which is categorically different from that in the case of α₀ = 0°.

Evidence further shows that the asymmetrical variation of pressure distribution due to pitching is strongly related to the asymmetrical LEV behaviors. Given a pitching amplitude (α_p) of 8°, as shown in Fig. 8(c), the LEV is significantly attenuated in strength during upstroke (⁠$t^$ = 0.25) but roughly retains its intensity during downstroke (⁠$t^$ = 0.75). As α_p increases to 24°, the conical LEV structure during upstroke is mostly replaced by a shear layer of positive vorticity (⁠$t^$ = 0.25), but less variation is observed during downstroke. Quantitative analysis of LEV shows that, as α_p increases, a significant decrease of LEV circulation is observed along the span at mid-upstroke, whereas no notable variation occurs at mid-downstroke [Fig. 9(a)]. Note that an attenuated LEV at mid-upstroke can directly predict the shrinkage of LPR1 (⁠$t^$ = 0.25) and LPR2 (⁠$t^$ = 0.5, caused by LEV impingement). Our analysis further shows that although the decrease of effective angle of attack (α_e) is identical between upstroke and downstroke due to pitching, its effect on the LEV circulation is asymmetrical [Fig. 9(b)], i.e., a quick attenuation of LEV circulation at mid-upstroke but an almost constant LEV circulation at mid-downstroke. This asymmetrical behavior of LEV circulation over effective angle of attack (α_e) is obviously distinct from α₀ = 0° cases. Previous study³⁸ on the unsteady flow of pitching-plunging airfoils revealed the importance of α_e on flow evaluation. Here, a positive nominal angle of attack can shift the positive α_e to where LEV circulation is almost saturated, resulting in the asymmetrical trend. However, it is not convinced that the LEV circulation is solely manipulated by α_e, as the attenuation of circulation at mid-upstroke is faster than a linear trend [the LEV circulation at α_e = −18° is almost zero in Fig. 9(b)]. This further implies that introducing a nominal angle of attack can fundamentally tune the rule of LEV circulation over α_e, in addition to shifting α_e itself. In addition, as α_p increases, the significant reduction of LEV circulation at mid-upstroke can also shift its core toward the leading edge [Figs. 9(c) and 9(d)].

In addition to the lift enhancement, the role of LEV and geometric projection in the reduction of external driving torque (or the generation of self-driving torque) is explained here. As mentioned above, the pitching can lead to an attenuation of LEV and pressure force mainly during upstroke. However, similar to the α₀ = 0° cases, the geometric effect of pitching can otherwise mitigate the drop of instantaneous thrust (C_Fh) in the wing frame by enlarging the projection in the horizontal direction [Fig. 8(b)]. Thus, the drop of C_Fh is smaller than that of pressure force, especially at a lower pitching amplitude. More importantly, the pressure force and the backward horizontal projection are simultaneously decreased due to pitching during downstroke (though the drop of pressure force is tiny), both of which are beneficial for the increase of $C¯T$⁠. This is obviously distinct from α₀ = 0° cases, where the LEV-mediated effect and geometric effect of pitching are mutually opposing. Thus, the geometric effect of pitching is critical in the increase of $C¯T$⁠, though the lift enhancement is mainly caused by the LEV-mediated effect.

As seen from above, the generation of lift and self-driving torque of a PFP-RW depends on the temporal coupling of pressure force and geometric projection, which can be modulated by the pitching amplitude and the nominal angle of attack. Here we further discuss the effects of pitching phase relative to flapping (θ), which was assumed to be 90° (or symmetric) in previous results. For α₀ = 0° and θ = 90° cases, where a significant self-driving torque (i.e., a positive $C¯T$⁠) is resulted from a favorable projection of pressure force due to the nose-down wing in downstroke (also nose-up wing in upstroke), the phase angle is of great importance. Similar to 2D cases, apart from the influence in flow evaluation, an improper phase angle can significantly change the geometric projection and may even lead to a drag generation.²⁷ Details on this can be found in Refs. 26, 39, and 40.

The effect of phase angle on the lift, self-driving torque, and efficiency of PFP-RWs at α₀ = 20° are further studied in detail to make clear its potential in performance improvement. Based on a baseline case (θ = 90°, α_p = 16°), advanced pitching (θ < 90°) and delayed pitching (θ > 90°) are introduced. As shown in Fig. 10(a), results show that cycle-averaged lift decreases slightly with θ when a delayed pitching is introduced to a strong flapping perturbation (i.e., high St). However, the lift generated with advanced pitching is close to that produced by a symmetrical one. On the other hand, more significant phase angle effect is found in the self-driving torque [$C¯T$ shown in Fig. 10(b)]. As St increases, a steep increase of $C¯T$ over θ is observed when a delayed pitching is applied. The self-driving torque of a PFP-RW with θ = 130° is almost identical to that of the revolving counterpart [Fig. 10(b)]. On the contrary, a further increase of $C¯T$ compared with the symmetric pitching case is resulted from an advanced pitching at high St, but plateaus when θ reaches 60°. Considering the lift generation efficiency, generally, the power loading (PL) peaks with a symmetrical pitching at all St tested [Fig. 10(c)]. For a PFP-RW with delayed pitching, the lift loss and augmentation in external driving torque for revolving can both lower the efficiency. On the other hand, notwithstanding the improved reduction in external driving torque of a PFP-RW with advanced pitching, the power consumption is still higher than its symmetrical counterpart (not shown here for brevity), resulting in a lower power loading.

A further examination of instantaneous force for three representative cases (St = 0.56, θ = 50°, 90°, and 130°) is shown in Fig. 11. Generally, the variations of pressure force and its vertical component have negligible differences [Fig. 11(a)], both of which follow the variations of effective angle of attack (α_e) [Fig. 11(c)]. This again illustrates the importance of LEV behavior in the variation of instantaneous lift. Both advanced and delayed pitching result in an increased negative peak of lift, while a delayed pitching slightly lowers the positive lift peak during downstroke, resulting in a lower cycle-averaged lift [Fig. 11(a)]. For an advanced pitching, the cycle-averaged lift is almost unchanged, as a slight increase of lift is observed in the downstroke [Fig. 11(a)].

In addition to the variation of pressure force, the pitching phase angle also significantly affects the instantaneous self-driving torque by altering the horizontal projection of pressure force. As shown in Figs. 11(b) and 11(d), although the pressure force is almost identical for all cases at the onset of downstroke, a delayed pitching can tilt the pressure force backward (a higher β), leading to a significantly increased drag in the wing frame at $t^$ = 0.65. On the contrary, the horizontal projection of pressure force is reduced by an advanced pitching and thus a decrease of instantaneous drag occurs between $t^$ = 0.5 and $t^$ = 0.7 [Fig. 11(b)].

Our results show that pitching-flapping perturbation could serve as a novel mechanism to improve the aerodynamic performance of rotary wing micro air vehicles (MAVs) in multiple ways. First, a perturbation consisting of slight flapping (low St) but large symmetrical pitching can improve the power loading of a revolving wing [blue in Fig. 7(d)], indicating that a rotary MAV can utilize this perturbation to improve its payload capacity using a fixed power source, or to expand its functioning time by saving power. One extra design consideration however is to add compliant structures to the wing hinges with low structural loss, thereby to conserve the additional inertial power introduced due to the pitching-flapping perturbation.

In addition, at high St, although the power loading of a PFP-RW is lower than its revolving wing counterpart, significant gain can be obtained in lift coefficients through the pitching-flapping perturbation. Consider that a rotary wing MAV can enhance its lift production by increasing the rotational speed of its wings/blades or by increasing the lift coefficient at fixed rotational speed, this result has practical significance because motors of a rotary wing MAV cannot increase their speed indefinitely. Due to the motor speed-torque characteristics,⁴¹ to improve efficiency, a motor is commonly geared up to operate at a preferred range of rated speed, which is commonly 70%-80% of its maximal speed.⁴² If a rotary MAV was to double its lift generation, it can theoretically increase its wing rotational speed by approximately 40% (assuming negligible Re effects). However, assuming that the motors are running at their rated speed during steady flight, an increase of speed by 40% would lead to 98%–112% of the maximal speed, which is not a sustainable operating state without causing significant mechanical noise, heat, and wear. Moreover, for MAVs, small-sized brushed motors, which are more prone to mechanical wear, noise, and heat generation, are more commonly used than the brushless motor due to the size constraint,^13,14 further preventing a large increase of rotational speed in a sustained fashion. Alternatively, based on our findings, a rotary wing MAV can easily double its lift generation without changing the motor rotational speed, but with a pure flapping perturbation (FP-RW) at St ∼ 0.55 or with a pitching-flapping perturbation (PFP-RW) at St ∼ 0.45 (Fig. 7). The desired pitching-flapping perturbation can be generated either passively through designing a 2-DOF compliant joint at the wing hinge or through introducing additional low-weight and low-power actuation mechanism, e.g., electromagnetic actuators.^42,43

The unsteadiness in low Re fluids fundamentally module the aerodynamic performance in both biological propulsions, i.e., insect and fish, and micro bio-mimicking system. Here, as a follow-up of our previous study on a flapping-perturbed revolving wing (FP-RW, i.e., introducing unsteadiness via vertical flapping), we look into the combined effects of pitch motion to FP-RWs (i.e., pitching-flapping-perturbed revolving wings, PFP-RWs) at Re = 1500 via experiments and simulations. The aerodynamic performance is measured experimentally considering pitching amplitude (α_p), Strouhal number (St), angle of attack (α₀), and phase angle (θ), whereas further flow analysis is based on the velocity field from computational fluid dynamics (CFD) simulations.

An enhanced generation of self-driving torque is observed at α₀ = 0° when symmetrical pitching is applied to a FP-RW. The enhancement, which can be plateau as α_p keeps increasing, is resulted from a mutually opposing effect of pressure force and geometric projection: pitching reduces the pressure force but increases the horizontal projection. Further analysis on flow shows that pitching motion can lead to a symmetrical attenuation of LEV circulation between upstroke and downstroke, which is approximately linear over the reduction of effective angle of attack. At α₀ = 20°, symmetrical pitching can lead to a lift enhancement as well as a self-rotating equilibrium on PFP-RWs. The lift enhancement is mainly resulted from an asymmetrical reduction of pressure force, while the reduction of external driving torque for revolving is further related to the variation of projection. The lift generation efficiency of PFP-RWs is mostly lower than revolving wings, except for those with a perturbation of large α_p but small St. This agrees with previous suggestion for the most efficient two-stroke motion. Analysis shows that the LEV circulation experiences an asymmetrical behavior at mid-strokes when pitching is introduced: a quick reduction at mid-upstroke but almost constant at mid-downstroke, which directly predict the asymmetrical reduction of pressure force. A further study on the effect of phase angle for PFP-RWs at α₀ = 20° suggests that delayed pitching is inferior to a symmetrical or advanced pitching because of a loss of lift and an increase of external driving torque for revolving. An advanced pitching can further reduce the external driving torque for revolving but shows less impact on lift, which is explained by considering the variation of pressure force and its projection.

Our findings reveal the significance of effective angle of attack (α_e) in the flow evaluation (i.e., LEV circulation) and pressure force generation of PFP-RWs. However, based on our results between α₀ = 0° and 20°, the LEV circulation could be asymmetrically affected by α_e between upstroke and downstroke depending on the nominal angle of attack. In addition to uncovering the underlying physics, our study emphasizes the possibility that pitching-flapping perturbation serves as a novel mechanism to improve the aerodynamic performance of rotary wing micro air vehicles (MAVs).

See supplementary material for the comparison between computational and experimental aerodynamic forces and the tests of boundary effect of our experimental setup.

This research was supported by the National Science Foundation (Grant No. NSF CMMI 1554429), Army Research Office DURIP Grant (No. W911NF-16-1-0272), National Natural Science Foundation of China (NSFC, Grant No. 11672022), and China Scholarship Council (joint Ph.D. program for Long Chen). The authors gratefully acknowledge Ya$ǧ$iz Efe Bayiz and Pan Liu for their help in the experimental setup.