Scaling Bioinspired Mars Flight Vehicles for Hover

Abstract

With the resurgent interest in landing humans on Mars, it is critical that our understanding of the Martian environment is complete and accurate. One way to improve our model of the red planet is through aerial surveillance, which provides information that augments the observations made by ground-based exploration and satellite imagery. Although the ultra-low-density Mars environment has previously stymied designs for achieving flight on Mars, bioinspired solutions for flapping wing flight can utilize the same high lift producing mechanisms employed by insects on Earth. Motivated by the current technologies for terrestrial flapping wing aerial vehicles on Earth, we seek solutions for a 5 gram bioinspired flapping wing aerial vehicle for flight on Mars. A zeroth-order method is proposed to determine approximate wing and kinematic values that generate bioinspired hover solutions. We demonstrate that a family of solutions exists for designs that are O(10¹) g, which are verified using a 3D Navier-Stokes solver. Our results show that unsteady lift enhancement mechanisms, such as delayed stall and rotational lift, are present in the bioinspired solution for a 5 g flapping wing vehicle hovering in Mars conditions, verifying that the zeroth-order method is a useful design tool. As a result, it is possible to design a family of bioinspired flapping wing robots for Mars by augmenting the adverse effects of the ultra-low density with large wings that exploit the advantages of unsteady lift enhancement mechanisms used by insects on Earth.

II. Introduction

Improved information on the Martian environment will reduce uncertainties and their associated risks in future Mars missions, including human missions. Aerial vehicles can perform sensing and acting tasks that fill the gaps in the information gathering capabilities of Mars orbiters and land-based rovers. Flight vehicles capable of near surface hovering are attractive candidates for such information gathering missions. These missions may include surveying remote locations, retrieving samples, efficiently generating topographic models of the surrounding terrain, providing near-surface weather data, and assisting rovers in path planning. Swarming flight vehicles offer additional design options and possibly improved efficiencies for these missions. Previous efforts in designing Mars aerial vehicles have yet to yield a realizable solution for accomplishing flight, which is mainly due to the ultra-low-density environment inherent to the Martian atmosphere. Although NASA plans to send a helicopter to Mars during the 2020 rover mission, a rotorcraft solution results in a much larger (over 850 gram vehicle with a 1.21 m rotor diameter) [1] and potentially less efficient [2] platform.

We have previously shown [3] that bioinspired flapping wing solutions can overcome the difficulties associated with flight in the ultra-low-density Martian atmosphere. This bioinspired solution is dynamically similar to terrestrial insect-scale flapping wing motion in that the flapping wing motion on Mars can also take advantage of unsteady, high lift coefficient producing mechanisms. To maintain dynamic similarity with insects on Earth, the relevant set of dimensionless parameters for our study, i.e. aspect ratio AR, Reynolds number Re, reduced frequency k, Mach number M, and angle of attack AoA, were kept in the insect flight regime. Starting from the morphological parameters corresponding to a bumblebee, the wings were uniformly scaled to approximately the size of cicada wings. The flapping frequency was also reduced to approximately 43% that of a bumblebee. Generating such a solution through biomimicry not only provides a means of producing sufficient lift, but it also results in a flight vehicle that has the potential to benefit from the attractive qualities of insect flight, such as hovering, high maneuverability, long range/endurance flight, and high payload capacity. To exploit the advantages of insect-inspired flight, we extended the study to quantify the payload margin of the bumblebee-inspired flapping wing micro air vehicle (FWMAV) and determined that it can carry a payload on the order of its own body weight [4]. It was determined that the payload capabilities are limited by maintaining dynamically similar motion.

The present work seeks to continue in the exploration of solutions for bioinspired flapping wing hovering flight on Mars. This study is motivated by the fact that current flapping wing robots on Earth have a mass O(10¹) grams such as Chiba University’s bio-inspired MAV (6 grams) [5], the DelFly II (16 grams) [6], and the Nano Hummingbird (19 grams) [7]. This contrasts with our previous solution at the insect scale [4] which had a mass of 0.2 g. Whereas insect size micro-air vehicles can employ stealth, larger vehicles with increased payload capability can be more beneficial for Mars exploration. To move towards the design of a bioinspired flapping wing flyer for Mars of various sizes and weight, we provide a method for generating dynamically similar hovering solutions for flapping wing robots with total masses on the order of the current Earth FWMAVs.

The approach to finding such a bioinspired solution is similar to our past studies [3,4]. We first determine a dynamically similar solution based on the desired vehicle mass by using a zeroth order method that combines the relevant dimensionless parameters for maintaining dynamic similarity with the lift-weight balance given by

$$2L=W=mg=\frac{1}{2}\rho {U_{ref}}^{2}S{C}_L.$$ (1)

In Eq. (1), L is the aerodynamic lift force of one wing, W is the vehicle weight on Mars, m is the vehicle mass, g and ρ are the gravitational acceleration and atmospheric density on Mars, U_ref (defined in Section III.A) is the reference velocity of a flapping wing, S is the planform area of a single wing, and C_L is the lift coefficient. We then use a well-validated [8,9], three dimensional Navier-Stokes (NS) equation solver to determine the forces for hovering flight of our insect-inspired flapping wing flyer. Finally, the power required for the flapping motion is calculated based on the resulting kinematics and morphological parameters.

We only consider the aerodynamics of the flyer and make no consideration for the actuator dynamics or their imposed mass on the system. Additionally, we only simulate the case of rigid wings, which has been shown to elucidate a multitude of phenomena for flapping wing simulations [9–11]. After a baseline solution is found, the mass of the body is scaled to simulate additional payload. Hover solutions are determined from the NS equation solver, and the resulting wing kinematics and power required to sustain the given payload are analyzed.

In the remainder of the paper, we describe the methodology, which includes the details of the governing parameters and the bioinspired dynamic scaling using the zeroth-order method. Additionally, the governing equations for the wing kinematics and aerodynamic modeling are included in the methodology section. We then report the results for the 5 g flapping wing robots on Mars based on the NS solver. A design approach for a family of large flapping wing robots on Mars that are O(1) g is reported and discussed as well.

III. Methodology

A. Bioinspired scaling to achieve dynamically similar motion to insects on Earth

The proposed method seeks to ensure that the resulting body parameters and motion are dynamically similar to insects on Earth [3,12] and can therefore benefit from the high lift mechanisms typical of unsteady insect motion. We are motivated by the fact that insect lift coefficients with rigid wings are governed by five dimensionless parameters such that C_L=C_L(AR,k,Re,AoA,M) [10,13]. These parameters are defined by combinations of morphological and kinematic parameters as shown in Table 1.

Table 1.

Dimensionless parameters, their definitions, and typical values which govern the high lift coefficients of insects.

Dimensionless Quantity	Symbol	Definition	Typical Value for Insects [10]
Aspect ratio	AR	$\frac{R^2}{S}$	2 ≤ AR ≤ 10
Reduced frequency (hover)	k	$\frac{\pi f_C}{U_{ref}}\to \frac{1}{2Z\cdot AR}$	0.1 ≤ k ≤ 0.4
Reynolds number	Re	$\frac{U_{ref}{L}_{ref}}{v}\to \frac{2\pi \rho fZR{\widehat{r}}_{2}c}{\mathrm{\mu }}$	O(102–104)
Angle of attack	AoA	$\frac{\pi }{2}-|\alpha |$	~ 45°
Wing tip Mach number	Mtip	$\frac{U_{ref}}{a}\to \frac{2\pi fZR}{a}$	Mtip ≤ 0.1
Lift coefficient	C L	$\frac{L}{\frac{1}{2}\rho U_{ref}^{2}S}$	O(101)

In Table 1, R is the span of one wing, S is the planform area of one wing, f is the flapping frequency, c is the mean chord length, Z is the half peak-to-peak flapping amplitude, U_ref is the reference velocity taken at the second moment of area location along the span ${\widehat{r}}_2$ where$U_{ref}=2\pi fZR{\widehat{r}}_2$, L_ref is a reference length, μ is the fluid dynamic viscosity, and a is the speed of sound. Using the ranges provided in Table 1 and the physical parameters defined in Fig. 1 for the FWMAV, a range of solutions with resulting dimensionless parameters in the insect regime can be determined for a given vehicle mass.

Fig. 1.

Schematic illustration of the key parameters for (a) the wing with pitch axis location x_pa at the leading edge (as used by some FWMAVs [5,14]), (b) the flapping motion, and (c) the pitching motion. The wing parameters are determined by considering a semi-elliptic wing, similar to the wing planforms used in practical applications of FWMAVs [5].

B. Zeroth-order method for dynamically similar solutions

To generate an informed solution for dynamically similar motion on Mars given a vehicle mass, we provide a method that combines the lift-weight balance equation (Eq. (1)) with the typical ranges of relevant dimensionless parameters that govern the high lift coefficients of unsteady insect motion in Table 1.

Obtaining a dynamically similar solution is non-trivial as there are multiple dimensionless parameters to consider in the design of a flapping wing robot. Also, a small change in one parameter can quickly drive another parameter out of the insect-inspired regime. The proposed zeroth-order method will result in dynamically similar solutions for the flapping frequency f and amplitude Z, as well as the wing span R and mean chord c required to balance the weight of the desired vehicle W on Mars.

The largest constraint imposed by dynamically similar motion on Mars is the effect of the Mach number M on the reference velocity U_ref. The reduced speed of sound on Mars (a_Mars=0.72a_Earth) results in more rapidly approaching the speed of sound for a given velocity, as compared to that same velocity on Earth. We can use the information that insects typically operate at M<0.1, such that M_max=0.1, to determine the maximum of the reference velocity, while maintaining an appropriate Mach number. The maximum wing velocity is defined as the velocity at the wing tip and is constrained by the Mach number and speed of sound as

$$U_{tip}=2\pi fZR=M_{max}{a}_{Mars}.$$ (2)

Since the reference velocity is taken at the ${\widehat{r}}_2$ location, we can express U_ref in terms of U_tip as

$$U_{ref}={\widehat{r}}_2{U}_{tip}.$$ (3)

The wing area of each wing required to sustain the vehicle with weight W, while operating at a flapping speed of U_ref, can be determined by rearranging Eq. (1) in terms of S, such that

$$S=\frac{W}{\rho U_{ref}^2\overline{C_L}}$$ (4)

As a result, the wing span R and mean chord c can be expressed in terms of the desired wing aspect ratio AR as

$$R=\sqrt{S\cdot AR}\text{and}c=R/(AR)\text{.}$$ (5)

Expressing R and c in this manner gives us the advantage of solving for the wing morphological and kinematic parameters explicitly in terms of two dimensionless parameters, M and AR. As such, we can guarantee that the resulting solution is dynamically and geometrically similar with respect to two (M and AR) of the five relevant dimensionless parameters (Table 1). To factor in a third dimensionless parameter to guide our solution, we express the flapping amplitude Z in terms of the reduced frequency in hover k as

$$\text{Z}=\frac{1}{2k\cdot AR}.$$ (6)

Lastly, the flapping frequency f can be expressed in terms of the reference velocity (Eqs. (2) and (3)), wing span (Eq. (5)), and flapping amplitude (Eq. (6)) as

$$f=\frac{U_{ref}}{2\pi \text{Z}R{\widehat{r}}_2},$$ (7)

which ensures that the flapping frequency will result in dynamically similar motion with respect to M, AR, and k.

Note that only three of the five dimensionless parameters are used in the above approach for determining c, R, f, and Z. This is because the angle of attack AoA is governed by the instantaneous pitch angle α (see Section III.C.1) and is therefore independent from the wing geometry or flapping motion. Additionally, the Reynolds number Re is left to vary as a function of the previously defined wing parameters. This is because Re is directly proportional to the atmospheric density ρ (see Table 1), which is extremely low on Mars. As a result, Re is generally low for a given combination of kinematic and morphological parameters on Mars and will therefore remain in the desired insect-inspired regime.

C. Governing equations

Flapping wing flight is characterized by unsteady flow with large vortical structures. To properly account for the unsteady lift enhancing mechanisms, the generation of large scale vortices [15] and their nonlinear interaction with the wing [16,17] must be properly resolved by solving the Navier-Stokes equations. The computational framework and flapping wing motion employed is based on our recent work [18] which explored a design space of 3D flapping wing motions.

A fully validated NS equation solver is used to calculate the velocity and pressure field around the flapping wing vehicle. NS equations are solved using an in-house three-dimensional, structured, pressure-based finite volume solver [8,13,19]. The computational setup as well as grid and time step sensitivity studies can be found in the Appendix.

1. Wing kinematics

The wing kinematics are described using bioinspired relations [20,21]. The flapping motion with respect to the wing root is described as the sinusoidal motion

$$\zeta (t/T)=Z\cos (2\pi t/T)$$ (8)

where Z is the flapping amplitude, and the time t is normalized by the period T=1/f. The pitching motion is described by the piecewise function

$$\alpha (t/T)=\{\begin{array}{lll}{\alpha }_d+\frac{\text{A}}{\mathrm{\Delta }{\tau }_{\text{r}}}\left\{\left(\frac{t}{T}-{\tau }_1\right)-\frac{\mathrm{\Delta }{\tau }_{\text{r}}}{2\pi }\sin \left[\frac{2\pi }{\mathrm{\Delta }{\tau }_{\text{r}}}\left(\frac{t}{T}-{\tau }_1\right)\right]\right\}\hfill & \text{for}\hfill & \frac{t}{T}\le {\tau }_1\hfill \\ {\alpha }_d-\frac{\text{A}}{\mathrm{\Delta }{\tau }_{\text{r}}}\left\{\left(\frac{t}{T}-{\tau }_2\right)-\frac{\mathrm{\Delta }{\tau }_{\text{r}}}{2\pi }\sin \left[\frac{2\pi }{\mathrm{\Delta }{\tau }_{\text{r}}}\left(\frac{t}{T}-{\tau }_2\right)\right]\right\}\hfill & \text{for}\hfill & {\tau }_2\le \frac{t}{T}\le {\tau }_2+\mathrm{\Delta }{\tau }_{\text{r}}\hfill \\ {\alpha }_u+\frac{\text{A}}{\mathrm{\Delta }{\tau }_{\text{r}}}\left\{\left(\frac{t}{T}-{\tau }_3\right)-\frac{\mathrm{\Delta }{\tau }_{\text{r}}}{2\pi }\sin \left[\frac{2\pi }{\mathrm{\Delta }{\tau }_{\text{r}}}\left(\frac{t}{T}-{\tau }_3\right)\right]\right\}\hfill & \text{for}\hfill & \frac{t}{T}\ge {\tau }_3\hfill \end{array}$$ (9)

where A is the pitch amplitude, and α_d and α_u are the forward and backward pitch amplitudes, respectively. In this study, we consider symmetric flapping/pitching motion such that A=α_d=α_u. Note that because the pitching angle is a function of time, the angle of attack is also a function of time. When referring to a single AoA value, (i.e., Table 1), we are referring to the bulk angle of attack that occurs during the translation phase of the motion. The parameter Δτ_r is a non-dimensional number that determines the duration of the pitch rotation in terms of the flapping period. When the pitching duration is short, the pitching is confined to the beginning and end of each stroke. For example, when the pitching duration is Δτ_r=0.5, the pitching motion is slow and near a perfect sinusoid. This results in the pitching motion being active during the entire forward and backward strokes. Alternatively, as Δτ_r→0, the pitching motion approaches a square wave, resulting in rapid pitch rates. The parameters τ₁, τ₂, and τ₃ govern the timing of the pitch rotation. Since only symmetric pitching is considered in this study, τ₁=Δτ_r/2, τ₂= τ₁+Δτ_r/2, and τ₃= ½(1-Δτ_r). The pitching motion takes place about the pitch axis location x_pa=0, which is coincident with the leading edge of the wing. This pitch axis location is considered to most directly model the pitch axis locations of current flapping wing air vehicles [5,14].

2. Aerodynamic modeling

We consider hover and assume any aerodynamic forces generated by the body can be neglected [22]. We only simulate a single, rigid wing, assuming left-right symmetry of the system with respect to the longitudinal plane. We directly solve the three-dimensional incompressible NS equations

$$\begin{gathered} \mathbf{\nabla }*\cdot \mathbf{V}*=0, \\ \frac{k}{\pi }\frac{\partial \mathbf{V}*}{\partial \tau }+\left(\mathbf{V}*\cdot \mathbf{\nabla }*\right)\mathbf{V}*=-\mathbf{\nabla }*p*+\frac{1}{Re}\mathbf{\nabla }{*}^2\mathbf{V}*, \end{gathered}$$ (10)

to determine the pressure and shear stress distributions on the wing. The velocity field V is normalized with the reference velocity U, or V*=V/U. Time is normalized by the flapping period (1/f), T=ft. Lengths are normalized by the mean wing chord c, and pressure is normalized per p*=p/ρU². These equations are solved using a well-validated structured, finite-volume, pressure-based incompressible NS equation solver used extensively in flapping wing studies by Shyy and coworkers [8,9]. For all motions considered in this study, the resulting Reynolds number is in the range of Re<1000. In this Reynolds number regime, the fluid flow can be considered as laminar and the computational accuracy of the NS equation solver employed in this study is satisfactory [23].

IV. Results and discussion

A. Family of solutions for Mars flapping wing robots of various masses

The zeroth-order method can be used to determine a family of solutions for dynamically similar flapping wing motion on Mars for vehicles of various masses. The approach described in Section III.A is applied for a range of masses m, allowing the relationships between the aspect ratio, Reynolds number, and kinematics to be investigated. The results from this analysis are presented in Fig. 2.

Figure 2.

The resulting Reynolds numbers a) and kinematics b) for a family of dynamically similar flapping wing robots on Mars for various masses at aspect ratios between 2 and 10. The results are obtained using the zeroth-order approach, which assumes C_L=1.5 (similar to hummingbirds in hover [27]) and k=0.2 (a reduced frequency that results in high flapping amplitudes for a given aspect ratio). Since the flapping amplitude is only a function of the aspect ratio and the reduced frequency, it remains invariant with respect to the changes in vehicle mass for a given value of k. Note that the red curve is for the case of m=5 g, which is the focus of the present work.

Figure 2(a) shows a trend of increased Reynolds number as the vehicle mass increases for a given aspect ratio. This is because heavier vehicles require a larger wing planform area (Eq. (4)). Since the Reynolds number is directly proportional to the chord length, Re will increase accordingly. Note that as AR increases, the area distribution of the wing is distributed through the span length more than the chord length, which explains the reduced Re with increased AR for a given vehicle mass. Most important is the observation that Re stays within the insect-inspired regime (Table 1) for a broad range of vehicle mass and aspect ratios. This is again because the fluid density ρ of Mars’ atmosphere is extremely low, and Re directly scales with ρ.

Figure 2(b) relates the aspect ratio to the flapping frequency and amplitude across a range of vehicles masses. Note that flapping amplitude remains constant with respect to vehicle mass. This is due to the trend exhibited by Eq. (6), where the flapping amplitude is a function of AR and k only. Within the proposed zeroth-order method, the flapping amplitude is calculated first with respect to aspect ratio and reduced frequency. Then the flapping frequency is successively calculated, thus incorporating the effects of aspect ratio, reduced frequency, and the resulting flapping amplitude into the flapping frequency required for sufficient lift generation.

Figure 2(b) also demonstrates that the flapping frequency increases with respect to aspect ratio for a given vehicle mass. This increase is required to account for the decreasing flapping amplitude as a function of aspect ratio since the lift scales directly with both flapping frequency and amplitude (Eqs. (1–3)).

Lastly, Fig. 2(b) illustrates that for a given aspect ratio and flapping amplitude, the flapping frequency reduces with respect to increased vehicle mass, which is less intuitive than the previously mentioned trends. The lighter flapping wing vehicles have smaller wings compared to the heavier vehicles, as demonstrated in Fig. 2(a) and Eq. 4. As a result, they generate significantly less lift for a given set of kinematics, since lift scales with wing size to the third order, as seen implicitly in Eqs. (1–3) and discussed in more detail in our previous work [4]. Since the flapping amplitude is fixed for a given aspect ratio, the lift required must be achieved through a high flapping frequency for lighter vehicles with relatively small wings. This is a trend found in nature where fruit flies and mosquitos, on the lower end of the weight spectrum, flap at frequencies of 800 Hz [24] and 200 Hz [25], respectively, and butterflies and hummingbirds, on the upper end of the weight spectrum, flap at frequencies of 10 Hz [26] and 20 Hz [27], respectively.

B. Solution for a 5 g flapping wing robot on Mars in hovering flight

To systematically determine a solution for the kinematic and morphological parameters for a 5 g flapping wing robot with motion dynamically similar to the motion of insects on Earth, we begin with the zeroth-order approach described in Section II.A. We use the zeroth-order results to define the input parameters into the NS solver. We then analyze the NS results to determine whether or not the zeroth-order method produces wing geometry and kinematics that generate sufficient lifting force to balance the weight of the vehicle in the Martian conditions.

1. Zeroth-order solution

We apply the solution the procedure outlined in Section III.A to a 5 g vehicle to determine the morphological and kinematic parameters to supply to the NS solver. In order to find a solution, some assumptions must be made about the flight characteristics. First, since the average lift coefficient C_L can only be determined a posteriori, we must assume a value to solve Eq. (4) for the wing planform area S. Since insects are capable of producing lift coefficients in excess of 1 [10,28] and our previous study revealed that the average lift coefficient for symmetrically flapping 3D wings is near 1.5 [18], we use assume a value of C_L=1.5 in the zeroth-order method. The validity of this assumption will be verified once the NS solutions are obtained (Section IV.B.2). Additionally, the reduced frequency in hover k governs the flap amplitude required to maintain dynamic similarity (Eq. (6)). In light of our previous results showing that increasing the flapping amplitude and reducing the flapping frequency to achieve a desired lift reduces the inertial flap power required (the dominant power component for flapping wing motion on Mars) [4], we set k=0.2 such that it remains in the insect regime, while having the effect of increasing the flapping amplitude.

2. Hover solution from 3D NS solver

To achieve a hover solution on Mars, we provide the results from the zeroth-order model into the NS solver to determine the resulting lift force. As the aspect ratio is one of the main parameters driving the lift coefficient in bioinspired flight, we choose discrete values of AR=2, 4, and 6 to cover the aspect ratios in a range from fruit flies (AR=2.4 [25]) to bumblebees (AR=6.6 [29]). Many studies have investigated the effects of aspect ratio on the lift coefficient of flapping wings [30,31]. The goal of the present work is not to perform a parametric investigation of aspect ratio effects. Instead, we include various aspect ratios to demonstrate the ability of the zeroth-order model to properly inform the dynamically similar solutions of flapping wing vehicles with varying dimensionless parameters (aspect ratio included). The results of the zeroth-order informed 3D NS simulations for a range of dimensionless parameters in the insect-inspired regime can be found in Table 2.

Table 2.

Nondimensional and dimensional parameters for medium-size (5 g) bioinspired flapping wing air vehicles on Mars compared to fruit fly, bumblebee, and hummingbird parameters on Earth.

		fruit fly on Earth [25]	medium-size bioinspired flight on Mars			bumblebee on Earth [29]	hummingbird on Earth [27,32]
dimensionless parameters	AR	2.4	2	4	6	3.3	4.1	---
	k	0.17	0.2	0.2	0.2	0.16	0.10	---
	Re	153	586	415	339	1326	11000	---
	α	~ 45	45	45	45	~ 45	~ 20	°
	Mtip	0.001	0.1	0.1	0.1	0.038	0.049	---
	$\overline{C_L}$	1.36	1.87	1.87	1.77	1.25	1.46	---
dimensional parameters	m	0.00096	5	5	5	0.175	8.4	g
	W	9.418×10−6	0.0164	0.0164	0.0164	0.00172	0.0824	N
	f	218	26.6	37.6	46.1	155	23	Hz
	Z	70	71.6	35.8	23.9	58	75	°
	R	0.0239	11.8	16.7	20.4	1.32	8.5	cm
	S	2.38×10−4	70	70	70	0.264	8.81	cm2

It is worth noting first in Table 2 that the lift coefficients from the NS solutions, generated for the Mars solutions are in excess of the assumed lift coefficient in the zeroth-order model (i.e. C_L>1.5). This result is a clear confirmation that the zeroth-order method can be used as an appropriate tool to scale the bioinspired solutions for hovering flight on Mars. Since the lift force scales with C_L, any increase in the lift coefficient from the assumed value is an increase in the vehicle mass for the given wing geometry and kinematics.

The results for bioinspired flight on Mars in Table 2 are couched between representative bioinspired values at the upper and lower ends of the relevant nondimensional parameters. It is evident that all of the dimensionless parameters for the medium-size bioinspired flight vehicle on Mars are well within the typical values common to biological flapping wing flight. The Mars solutions appear to have dimensional and nondimensional parameters that lie close to those of a hummingbird, which is not surprising given the 5 g mass of the bioinspired flapping wing vehicles, similar to the 8 g mass of a hummingbird. However, the most notable differentiation between the Mars solutions and the representative solutions on Earth is the wing size required for sufficient lift on Mars, noted by the relatively large values of S compared to the biological flappers on Earth. Augmenting the low density on Mars with large wings is the same approach used in our previous study [3] and is built into the zeroth-order method to ensure appropriate lift generation along with dynamically similar motion. Additionally, the Reynolds number is nearly two orders of magnitude lower than hummingbirds and one order of magnitude lower than bumblebees, which is mostly due to the ultra-low density of the Martian atmosphere.

3. Exploitation of bioinspired lift-enhancement mechanisms for hovering on Mars

It is critical to understand the behavior of the aerodynamics that generate sufficient lift to hover in the ultra-low density Mars atmosphere. In fact, the zeroth-order method proposed in the present work hinges on the ability of dynamically similar motion to generate forces that benefit from the unsteady lift-enhancing mechanisms of insects on Earth. To test if this is the case, we simulate the motion resulting from the zeroth-order method for the 5 g and analyze the resulting aerodynamic mechanisms present in the solutions, which can be found in Fig. 3.

Figure 3.

The resulting (a) 3D NS lift coefficient and (b) kinematics time history for the dynamically similar flapping wing motions generated by the zeroth-order method for 5 g flapping wing vehicles on Mars with various aspect ratio. (c) Contour plots of the iso-surfaces of Q-criterion (Q=15) indicative of vorticity for AR=4. Contours are indicative of vorticity and demonstrate the presence of unsteady lift enhancement mechanisms such as (t/T=0.05) combination of wake capture and rotational lift effects, (t/T=0.10) developing leading-edge (LEV) and trailing-edge (TEV) vortices, (t/T=0.25) translational lift due to delayed stall prior to shedding of tip-/trailing-edge vortex, and (t/T=0.40) increase in lift due to rotational lift mechanism. Note that 3D NS solutions are for one wing only. The second wing and the body are included for visualization purposes only and are not to scale.

It should first be noted that the full-cycle-averaged lift coefficients $\overline{C_L}$ (Fig. 3(a)) are sufficiently high to balance the weight of the vehicles in hover. Additionally, a constant $\overline{C_L}$ for 2<AR<4, then a reducing $\overline{C_L}$ for AR>4 is consistent with aspect ratio effects on $\overline{C_L}$ as described by other researchers [30]. Additionally, the $\overline{C_L}$ curve in Fig. 3(a) reveals lift peaks similar to those produced by insects on Earth. Each half-stroke is characterized by two dominant lift peaks near the beginning and end, which bookend the translational lift section in the middle of each half-stroke. These dominant lift peaks on the ends of each half-stroke are mainly due to the rotational lift mechanism due in part to the wing forcing the surrounding flow down and generating a positive reaction force in the direction of lift. The translational lift section can be found between 0.15<t/T<0.35 in Fig. 3(a), during which the wing maintains a constant angle of attack of ±45° (Fig. 3(b)).

During the translational phase, a strong leading-edge vortex (LEV) remains attached until the mid-stroke, which is near the time when it sheds and the lift reduces significantly. Figure 3(c) contains snap shots of contour iso-surfaces of the Q-criterion which are indicative of regions of high vorticity in the flow. The presence and locations of these contour iso-surfaces can be correlated to unsteady lift enhancement mechanisms utilized by biological fliers. For instance, the attached LEV can be seen at t/T=0.10 and t/T=0.25 in Fig. 3(c), in addition to the shed TEV at t/T=0.25. A break down and separation of the tip vortex (TV) can be seen during the rotational lift period at the end of the down stroke in Fig. 3(c) at t/T=0.40. Since the lift histories and cycle-averaged lift coefficients of AR=2 and 6 are similar to the lift histories of AR=4, it is assumed that they benefit from the same unsteady lift mechanisms seen in the case of AR=4.

V. Concluding remarks

This study builds on our prior work of achieving hovering flight on Mars using bioinspired flapping wing motion based on a bumblebee with enlarged wings. Using a zeroth-order method, approximate wing size properties and kinematics are determined such that sufficient lift is generated to balance the weight of a 5 gram flapping wing flyer on Mars. Despite using wings that are larger than typical insect wings, the dimensionless parameters governing the high lift coefficient of insects are maintained in the zeroth-order solution method.

The trends revealed through the zeroth-order method are useful in understanding the limits of achieving insect-inspired flapping wing solutions for aerial vehicles on Mars as well as determining the vehicle parameters to supply to the NS solver. Based on a given flapping wing vehicle configuration, we can computationally determine the aerodynamic characteristics before designing a physical solution. This will allow for a more informed and efficient design process of a flapping wing flyer capable of flight on Mars.

Future work will include parametric studies of factors such as wing kinematics, morphological parameters, and dimensionless parameters to optimize the aerodynamics and power requirements. Additionally, the 3D NS solver will be coupled with our existing flight dynamics solver to determine the controls required for trimmed, hovering flight of a 6 degree-of-freedom flapping wing air vehicle.

Nomenclature

A

pitch amplitude

a

speed of sound

αd

downstroke pitch amplitude

αu

upstroke pitch amplitude

AoA

bulk angle of attack

AR

aspect ratio of a single wing

c

mean chord length

C_L

instantaneous coefficient of lift

$\overline{C_L}$

mean coefficient of lift (full-cycle-averaged)

Δτ_r

pitch duration

f

wing flapping frequency

g

gravitational acceleration

k

reduced frequency

L

lift

L_ref

reference length

m

total vehicle mass

M_tip

wing tip Mach number

n

kinematic viscosity of the fluid

p

pressure

p*

nondimensional pressure

R

Span of a single wing

Re

Reynolds number

${\widehat{r}}_2$ =

spanwise location of the center of the second moment of wing area

ρ

density of the fluid

S

planform area of one wing

t

time

T

flapping period

τ

pitch rotation timing

U_ref

reference velocity

V*

fluid velocity vector

W

weight

x_pa

chordwise pitch axis location normalized to chord

Ζ

stroke amplitude (half peak-to-peak)

ζ

Instantaneous flapping angle of the wing

Appendix

Time and spatial sensitivity analyses were performed for each of the three aspect ratios considered in this study. For each aspect ratio (AR=2,4,6) 5 grids were generated to find a solution that was independent of the grid size for each geometry. The details of the converged grid used for AR=4 can be found in Fig. A1. Each simulation in the grid sensitivity analysis was run with 480 time steps per period and simulated for 3 periods. Additionally, a time sensitivity analysis was run at three different numbers of steps per period for the converged grid based on the spatial sensitivity analysis. These studies were conducted for Re=2000, k=0.35, U_ref=1, ρ=1, c=1, Δτ_r=0.3, x_pa=0, α_d=α_u=45°. The second finest grids (34×68×136) with 2091874 cells and 480 time steps per period yields sufficiently converged solution for lift for each aspect ratio as shown in Fig. A2 and Table A1.

Figure A1.

Illustrations of the converged grid for AR=4 in (a) an isometric view, (b) a planar slice of the computational mesh for both near and far field locations, and (c) a detailed view of the mesh around the wing. Similar trends are present in the converged grids for the other aspect ratios considered.

Figure A2.

Time history of the lift coefficient for the (a,c,e) spatial and (b,d,f) time sensitivity studies during the third flapping period for AR=2, 4, and 6, respectively. Curves suggest convergence for the 34×68×136 grid at 480 time steps per period for each aspect ratio considered (red curves in figure). Note, all NS results in the present work are from the third flapping period.

Table A1.

Spatial and temporal sensitivity for 3D grids. Five different meshes and three timestep sizes were included in the spatial and temporal sensitivity study for each aspect ratio. The second finest grids (34×68×136) with 2091874 cells were chosen with 480 timesteps/period for the NS simulations.

		Cells	Timesteps/Period	Total Cells	<CL>	L1-norm	L2-norm
AR = 2	spatial	10×20×40	480	46930	1.96	0.134	0.180
		15×30×60	480	168200	1.98	0.155	0.196
		23×46×92	480	631800	1.93	0.0997	0.124
		34×68×136	480	2091874	1.86	0.0634	0.0823
		51×102×204	480	7181504	1.83	---	---
	temporal	23×46×92	240	2091874	1.99	0.0314	0.113
		23×46×92	480	2091874	1.86	0.0223	0.0558
		23×46×92	960	2091874	1.81	---	---
AR = 4	spatial	10×20×40	480	46930	1.95	0.272	0.323
		15×30×60	480	168200	1.93	0.229	0.274
		23×46×92	480	631800	1.89	0.159	0.219
		34×68×136	480	2091874	1.88	0.0705	0.0981
		51×102×204	480	7181504	1.90	---	---
	temporal	23×46×92	240	2091874	1.95	0.0370	0.147
		23×46×92	480	2091874	1.88	0.0279	0.0764
		23×46×92	960	2091874	1.86	---	---
AR = 6	spatial	10×20×40	480	46930	1.95	0.418	0.504
		15×30×60	480	168200	1.89	0.276	0.330
		23×46×92	480	631800	1.85	0.138	0.169
		34×68×136	480	2091874	1.84	0.135	0.173
		51×102×204	480	7181504	1.80	---	---
	temporal	23×46×92	240	2091874	1.89	0.0499	0.189
		23×46×92	480	2091874	1.84	0.0368	0.0941
		23×46×92	960	2091874	1.82	---	---