On the quasi-steady aerodynamics of normal hovering flight part II: model implementation and evaluation

Abstract

This paper introduces a generic, transparent and compact model for the evaluation of the aerodynamic performance of insect-like flapping wings in hovering flight. The model is generic in that it can be applied to wings of arbitrary morphology and kinematics without the use of experimental data, is transparent in that the aerodynamic components of the model are linked directly to morphology and kinematics via physical relationships and is compact in the sense that it can be efficiently evaluated for use within a design optimization environment. An important aspect of the model is the method by which translational force coefficients for the aerodynamic model are obtained from first principles; however important insights are also provided for the morphological and kinematic treatments that improve the clarity and efficiency of the overall model. A thorough analysis of the leading-edge suction analogy model is provided and comparison of the aerodynamic model with results from application of the leading-edge suction analogy shows good agreement. The full model is evaluated against experimental data for revolving wings and good agreement is obtained for lift and drag up to 90° incidence. Comparison of the model output with data from computational fluid dynamics studies on a range of different insect species also shows good agreement with predicted weight support ratio and specific power. The validated model is used to evaluate the relative impact of different contributors to the induced power factor for the hoverfly and fruitfly. It is shown that the assumption of an ideal induced power factor (k = 1) for a normal hovering hoverfly leads to a 23% overestimation of the generated force owing to flapping.

1. Introduction

Our aim is to develop and validate a generic, transparent and compact modelling treatment for representing the aerodynamics of insect-like flapping wings in hover. The main intended use of the resulting model is in the preliminary engineering design of insect scale flapping wing vehicles; however, the model can also be used to support quantitative studies of insect physiology.

There have been several attempts to construct sophisticated analytical models for the aerodynamics of insect flight, for example those developed by Minotti [1] and Ansari et al. [2]. Another class of aerodynamic models representing a medium cost, medium fidelity treatment is based on the unsteady vortex lattice method as developed by Fritz & Long [3] and Roccia et al. [4]. However, the simpler so-called ‘semi-empirical’ quasi-steady models [5] are generally more widely used because they are relatively fast, offer insight into the generated forces and allow the comparison between different types of wing morphologies and kinematics. These models require the use of experimental data for the flapping translational force coefficients within the model. Hence, their applicability is dependent on the availability of appropriate experimental data, such as those presented by Dickinson et al. [6] and by Usherwood & Ellington [7,8].

Examples of semi-empirical quasi-steady models are provided by Walker & Westneat [9], Sane & Dickinson [10], Berman & Wang [11], Whitney & Wood [12] and Khan & Agrawal [13]. The main physical assumption in these models is that the instantaneous aerodynamic forces on a flapping wing are equal to the forces during steady motion of the wing at an identical instantaneous velocity and angle of attack. These models start with a definition of wing kinematics from which the angle of attack and the incident velocity in the wing frame of reference are obtained. The lift and drag forces acting on the wing owing to its flapping translation are then calculated using the available experimental data on flapping translational force coefficients. Finally, force components owing to wing rotation as well as the non-circulatory added mass effects are usually added. The success of a quasi-steady aerodynamic model is based primarily on the availability of appropriate flapping translational force coefficients from experimental data. However, quality experimental data are limited to a few specific geometries and test cases, and force coefficients can show considerable change with variations in the wing shape [14].

In contrast to analytical models, computational fluid dynamics (CFD) models have the benefit of providing detailed information on both the generated aerodynamic forces and the structure of the wake and surrounding fluid. However, from an interpretation point of view, it is generally difficult to separate the contributions of the various fluid dynamic mechanisms to force generation [5], and as a result these models may not provide insight appropriate to engineering design or insect physiological analysis. CFD models are also computationally expensive meaning that they are of limited use as part of optimization studies. Example CFD models include those developed by Liu et al. [15] to study a hawkmoth wing in hover, Ramamurti & Sandberg [16] for a fruitfly wing as well as the comprehensive CFD studies of different insect wings developed by Sun et al. [17–19].

In this paper, a generic analytical methodology for evaluating the steady translational force coefficients of flapping wings in normal hover and its validation against available experimental data are presented. The coefficient expressions are then implemented within a quasi-steady blade element model for the analysis of several hovering insects. Generated aerodynamic forces and consumed power are calculated and validated against other existing aerodynamic modelling methodologies, including CFD simulations. A fundamental aspect of the model is its ability to be generic through its ability to account for different aerodynamic effects related to flapping flight including, for example, the influence of wing shape and kinematics on the aerodynamic characteristics. It allows wider application of semi-empirical quasi-steady models as it removes constraints imposed by the availability of experimental data, allowing flexible analysis, design and optimization of hovering flapping wings.

2. Flapping wing analytical model

2.1. Wing morphology

The modelling process begins with definition of the wing shape in terms of chord distribution as a function of span. In this work, we use the procedure proposed by Ellington [20] to define the chord distribution through a beta function representation. This representation provides a compact analytical description of wing shape using just three variables: wing length, mean chord and the non-dimensional radial location of the wing centre of area. Details of this method are provided in Part I of this work and hence are not considered further here.

2.2. Kinematics

The wing kinematics are defined using the axis systems shown in figure 1. The reference axis system is (x₀, y₀, z₀) with the x₀-axis taken parallel to the Earth's surface. Kinematics of the wing are defined by Euler rotations relative to the reference axis system. These rotations govern the stroke-plane angle, σ (figure 1a), wing-flapping angle, ϕ (figure 1b) and wing pitching (varying incidence) angle, θ (figure 1c).

Figure 1.

Axis systems. (a) Stroke-plane axes (for clarity, x₁ and x₂ are aligned with the body axis), (b) flapping axes, (c) pitching axes and (d) lift–drag frame.

In normal hovering, most insects use symmetrical half-strokes and horizontal stroke plane (σ of −90°) [5,14]. Maximum angular deviation from the stroke plane is typically small (less than 15°, see fig. 1 in [21] and fig. 6 in [22]) and it is common to assume that the motion is planar [5,13,14,23]. Therefore, the required kinematic angles to be defined are the flapping angle ϕ and the pitching angle θ. The angles ϕ(t) and θ(t) are defined here using representations similar in concept to those given by Berman & Wang [11]. However, we reduce their parametric expressions to simpler expressions more compatible with the current work as follows:

2.1

and

2.2

where ϕ_max is the flapping angle amplitude, θ_max is the pitching angle amplitude and f is the flapping frequency. The parameters C_ϕ and C_θ are used to control the shape of flapping and pitching cycles, respectively, where 0 < C_ϕ < 1 and 0 < C_θ < ∞. In the limit where C_ϕ → 0, ϕ(t) becomes sinusoidal, and when C_ϕ approaches 1, ϕ(t) becomes a triangular waveform as shown in figure 2a. On the other hand, as C_θ approaches 0, θ(t)becomes sinusoidal while as C_θ tends to ∞, θ(t) becomes rectangular as shown in figure 2b. The phase lag angle, δ, controls the pitching timing through the flapping cycle and is 90° for the symmetric case.

Figure 2.

Kinematics model for symmetric normal hovering. (a) Flapping angle, ϕ(t) and (b) Pitching angle, θ(t). The parameters C_ϕ and C_θ are used to control the shape of flapping and pitching cycles. (Online version in colour.)

Optimizing kinematics for minimum power for a given wing shape and lift constraint leads to a C_ϕ that tends to unity while C_θ tends to be as large as possible. This means a triangular waveform variation for the flapping angle (i.e. constant flapping velocity) and a rectangular waveform variation for the pitching angle (i.e. fixed pitch angle in each half-stroke). These waveforms are consistent with the optimal hovering kinematic variations discussed by Taha et al. [14] and are similar to the kinematics proposed by Schenato et al. [24]. Moreover, they are compatible with optimal hovering rotor aerodynamics where, in the absence of unsteady effects, a rotor is usually operated at a constant angular speed and a constant optimal angle of attack [25].

Once the kinematics of the wing are defined, the instantaneous angular velocity in the wing frame is derived. The linear velocity vector, V, is then obtained by cross-multiplication of the angular velocity vector and the position vector in the wing frame [12]. The wing angle of attack, α, defined as the angle between the zero lift line and the instantaneous velocity vector (figure 1d) in the x–z plane, is obtained as

2.3

The instantaneous lift and drag force components on each wing strip are expressed as

2.4

where ρ is the density, c is the chord, while C_L and C_D are the flapping wing translational lift and drag coefficients, respectively, which will be discussed comprehensively in the next section.

2.3. Aerodynamics

2.3.1. Modelling principles

Previous experimental work conducted on flapping wings has included experiments on model insect wings in parallel translation motion as well as revolving and flapping translations. These three possible wing motions are shown schematically in figure 3 with an idealized conception of their associated vortex structures. An objective of this diagram is to show that while the three wing motions lead to quite different wake structures, the fundamental building blocks of the wake are similar, and hence it should be anticipated that an aerodynamic theory for the flapping case can be built from modification of existing components developed for translating and revolving flight. It has been observed in experimental studies that at small angles of attack the wing lift coefficients are almost the same for all three wing motions (see fig. 12 in [7], fig. 1 in [26] and fig. 7 in [27]). However, once the wing enters the high angles of attack region, the lift coefficients of parallel translating wings drop significantly owing to wing stall. On the other hand, revolving and flapping wings do not exhibit classical abrupt stall characteristics and the lift coefficient tends to increase up to the maximum value at around 45°. The reason for this is typically attributed to the formation of a leading-edge vortex (LEV) on the top surface of the wing [7,28,29]. This LEV has stable characteristics and is often continuously attached during the flapping cycle.

Figure 3.

Illustration of an idealized conception of the vortex wakes from the three principal wing motions used for practical lift generation: (a) parallel translating, (b) revolving and (c) flapping. For flapping motion, it is assumed that wing rotation at the beginning and end of each half-stroke occurs instantaneously and the angular velocity is constant through each half-stroke such that the starting and stopping vortices are coincident. In practice, the flapping wake structure is significantly less stable than the wakes for translating and rotating motion owing to self-induced velocities and thus represents the most idealized case of the three wakes shown. (Online version in colour.)

2.3.2. Translational aerodynamic force coefficients

This study considers a very simple aerodynamic model for the translational aerodynamic force on a flapping wing based on a normal force as a function of angle of attack [23]:

2.5

where is the magnitude of the normal force coefficient at 90° angle of attack, which will depend primarily on Reynolds number and wing shape [11,30]. Resolving the normal force in the lift and drag directions gives

2.6

This model is based on the following assumptions:

• (1) Absence of classical wing stall; i.e. the wing undergoes a three-dimensional flapping translation where the LEV is stable and does not grow with time. Because no new vorticity is generated at the leading edge, there is no additional vorticity generated at the trailing edge and the wing satisfies the Kutta condition at angles beyond which classical stall would occur for parallel translating wings [31]. This means that the lift is a continuous function of angle of attack.

• (2) The wing is an infinitesimally thin flat plate, and hence there is no chordwise component to the integrated surface pressure force.

• (3) The chordwise tangential force owing to skin friction is negligible compared with the integrated surface pressure force acting normal to the wing chord.

• (4) The magnitude of the normal pressure force is proportional to the projected wing chord perpendicular to the flow direction [32].

It is understood that assumption 1 (absence of classical wing stall) will become invalid at angles of attack approaching 90° where, from a symmetry argument, there must be a separation at both the leading and trailing edges, and hence the Kutta condition cannot be satisfied.

Experiments on model insect wings and CFD simulations [32,33] have shown that the above model provides a close approximation of the measured translational steady lift coefficient and is widely used for modelling lift for insect physiology and engineering studies [11,12,30]. Within this community, it is customary to present the model as

2.7

where C_T is referred to as the translational lift constant and is equal to half the peak normal force coefficient.

A value for translational lift constant loosely based on a method used by Hewes [34] is obtained as follows. Taking the limit of equation (2.7) in the vicinity of small angles of attack gives

2.8

hence,

2.9

where C_Lα is the three-dimensional wing lift curve slope at small angles of attack. For a given wing shape, the wing lift curve slope can be obtained using an appropriate wing theory and hence an expression for C_T can be obtained. A suitable way for the evaluation of the three-dimensional wing lift curve slope is to use Prandtl lifting line theory [35,36]:

2.10

The above lifting line expression gives good results for aspect ratios above 3 [37], hence can be applied to most insect wings, which have aspect ratios ranging between 3 and 5 [20]. In cases where the wing aspect ratio is smaller than this range, the extended lifting line theory results can be applied instead [37]. An important aspect of the above relation is that it accounts for the influence of the vorticity in the wake on the wing lift curve slope [29,38,39]. An expression for the steady lift coefficient owing to translation is thus obtained by substituting equations (2.9) and (2.10) into equation (2.7), giving

2.11

The two-dimensional aerofoil lift curve C_Lα,2d has a theoretical value of 2π rad⁻¹ (0.11 deg⁻¹) for a flat plate. However, Spedding & McArthur [40] showed experimentally that this value reduces at the low Reynolds number at which insects operate. Okamoto et al. [41] have shown that for a flat plate wing at typical insects Reynolds numbers, the lift curve slope takes a value of 0.09 deg⁻¹; hence this value will be used in this study. The parameter E is the edge correction proposed by Jones [42] for the lifting line theory and is evaluated as the quotient of the wing semi-perimeter to its length [42,43]. The aspect ratio, AR, is based on the span of a single wing, [2,11,12,18] on the assumption that the lift and hence bound circulation drops to zero at the inboard edge of the wing and there is no carry-over of lift to the opposite wing. The parameter k is the so-called ‘k-factor’ included to correct for the difference in efficiency between the assumed ideal uniform downwash distribution and real downwash distribution [35,40,44]. For this work, the k-factor required within equation (2.11) will be estimated using the induced power factor expression of hovering actuator disc models [45]. The induced power factor of normal hovering flight was discussed comprehensively in Part I of this work and was analytically expressed in terms of three contributors accounting for the non-uniform downwash velocity distribution, tip losses and effective flapping disc area.

Once the lift coefficient is obtained, the steady translational drag coefficient can be obtained from equation (2.6) using trigonometry, with the assumption that the tangential friction force is zero:

2.12

The above relation will underestimate the drag coefficient at very low angles of attack where skin friction becomes the main component of drag. However, during hover, flapping wings typically operate at relatively higher angles of attack (between 25° and 45°) where the model accuracy is good [7].

2.3.3. Non-translational coefficients

At the end of every half-stroke, the flapping wing pitches about its spanwise axis and there has been an argument that rotational forces exist as a function of this instantaneous rotation rate [6,10]. These forces are usually modelled using the quasi-steady treatment for the case of small-amplitude flutter on thin rigid wings [10] and it is therefore necessary to make an assumption that the theory holds true for large angles of attack. Given this, the rotational component of the total aerodynamic force is then defined using the Kutta–Joukowski equation and the instantaneous circulation owing to wing rotation as follows:

2.13

and

2.14

where Ω is the wing angular velocity around a spanwise axis at and the rotational coefficient, C_rot, is given by

2.15

where varies from 0 to 1 and is usually taken as 0.25 [46]. It should be noted that for symmetric hovering half-strokes (which is the main concern of this work), the forces owing to rotation effects sum to zero and therefore can be ignored [10,12]. Nevertheless, rotational effects are believed to have an important role in control and manoeuvrability [7].

The final class of aerodynamic force considered is non-circulatory and forces in this class are referred to as ‘added mass’ forces. These are the forces that result from accelerating or decelerating the neighbouring air mass surrounding the wing owing to the flapping motion. They are usually modelled as [9–11]

2.16

The outer bracket in the above equation is the mass of air surrounding a wing element, while is the first derivative of the normal velocity component of the chord relative to air. However, once again, for symmetrical half-strokes, the net added mass force is zero [5,10,30,32].

3. Results and discussion

3.1. Comparison with the leading-edge suction analogy

Here, we will show that the developed formula for the steady lift coefficient of a hovering wing has strong similarity with the model derived from the leading-edge suction analogy, which was originally proposed by Polhamus for delta wings [47]. Owing to the similarities in the flow structure between delta and flapping wings, the leading-edge suction analogy has usually been considered as a possible aerodynamic treatment for the flapping problem (see reviews by Sane [31], Ansari et al. [5] and Taha et al. [14]) and the model has been adopted by a number of researchers [48–50] to tackle the flapping wing problem. In this section, the Polhamus model will be thoroughly analysed in the light of its application to flapping wing aerodynamics modelling.

The leading-edge suction analogy is based on an assumption, substantiated from experiments, that the flow external to the LEV passes around the vortex and reattaches to the wing upper surface. Hence, it assumes that the total lift is composed of two parts. The first is the potential flow lift with zero leading-edge suction. The second is a vortex lift equal to the force required to maintain the equilibrium of the potential-type flow around the vortex. Thus, the total lift coefficient is given by [47]

3.1

where K_P is the wing lift curve slope at small angles of attack, Λ is the wing sweep angle and K_i is the derivative of the induced drag coefficient with respect to the square of the lift coefficient. Assuming a non-swept wing, which is consistent with insect wings, the above relation has almost the same shape of variation with angle of attack as the sin 2α relation. This is shown in figure 4a where each of the two relation plots is normalized by its maximum (amplitude) value. Next, in order to compare the amplitudes, we rearranged equation (3.1) and wrote it as

3.2

Note that the first bracket in the above expression is exactly the same formula we are using to calculate the steady lift coefficient (equations (2.7) and (2.9)). The second bracket represents an additional term multiplied to the proposed lift coefficient relation, which we name K_Polhamus. Figure 4b illustrates the variation of this term with the wing aspect ratio. In this illustration and without losing generality, the parameter k was assigned a unity value and the angle of attack is taken as 45° to represent the condition of maximum C_L value. The wing lift curve slope is calculated here using the extended lifting line expression [37] such that it is also valid at low aspect ratios (AR < 3).

Figure 4.

(a) Comparison of the shape of variation of the Polhamus and the sin 2α relation. Each plot is normalized by its amplitude value. (b) K_Polhamus variation with the wing aspect ratio for a unity k-factor at 45° angle of attack using the extended lifting line model. (Online version in colour.)

The implications of the results in figure 4b are as follows. First, the value of the K_Polhamus term is of the order of unity. From this, we can conclude that the Polhamus model is matching quantitatively with our proposed model. Second, the term K_Polhamus has a slightly higher value than unity. This is to be expected as Polhamus did not take into account the effect of the vortex flow on the attached flow [49], hence would overpredict the lift. Third, the term K_Polhamus increases with the increase in aspect ratio. Once again, this is expected as the Polhamus model is known to increasingly overpredict the wing lift coefficient as the wing aspect ratio increases. Polhamus obtained very good agreement for his model with delta wing experimental data of aspect ratios up to 1.5; however, for an aspect ratio of two, lower experimental lift coefficient values were evident (see discussion of Polhamus [47]). Therefore, for the range of insect wing aspect ratios, the Polhamus model is expected to overpredict the lift. It should be noted that the values of angle of attack and k used for the result in figure 4b lead to maximum values of K_Polhamus; hence, the maximum deviation case is considered. Furthermore, the values of K_Polhamus are only weakly sensitive to the angle of attack and/or k values; hence, the conclusions derived here may be considered as general results.

3.2. Comparison with experimental results of revolving wings

The force coefficients modelled in this work are three-dimensional steady coefficients, which account for the downwash effect on the aerodynamic characteristics. In this section, equations (2.11) and (2.12) are used to calculate the variation of the lift and drag coefficients with angle of attack for hawkmoth and bumblebee wings and results are compared with the measured steady force coefficients from revolving wing experiments of Usherwood & Ellington [7,8]. These experiments were performed at Reynolds numbers similar to those experienced by the actual insect with (single) wing aspect ratios of 2.83 and 3.16 for the hawkmoth and the bumblebee, respectively. In our model, these aspect ratio values are used as well as the revolving wing k values (i.e. k = k_indk_tip) evaluated in Part I giving a k value of 1.27 for the hawkmoth and 1.29 for the bumblebee. Also lift and drag coefficients are evaluated for the case of fruitfly wing using a revolving wing k value of 1.37 (table 2). The results are compared with the experimental data of Lentink & Dickinson [51] who provided revolving wing data at similar Reynolds number experienced by the actual insect (Re = 110).

Table 2.

Contributions to the induced power factor, k.

insect	kind	ktip	kflap	kind ktip	kind ktip kflap
fruitfly	1.21	1.14	1.10	1.37	1.51
bumblebee	1.17	1.10	1.25	1.29	1.61
hawkmoth	1.14	1.12	1.22	1.27	1.55
honeybee	1.18	1.10	1.17	1.30	1.52
cranefly	1.24	1.09	1.21	1.35	1.63
hoverfly	1.20	1.10	1.41	1.33	1.88
dronefly	1.16	1.06	1.29	1.22	1.57
ladybird	1.15	1.12	1.01	1.29	1.30

Figure 5 compares the calculated lift coefficient variation for the entire range of geometric angles of attack as well as the calculated drag polars against the experimental data. There are two aspects to the fit between theory and experiment that need to be considered: firstly, the degree to which the form of the data fits the model, and secondly the agreement between predicted and measured amplitude. The agreement between the model and experiments with respect to either the fit or the amplitude is good, though the agreement is better for lift than for drag. Given that the flow topology is different at 90° angle of attack for the reasons discussed in §2.3.2, some sort of discrepancy at very high angles of attack is not unexpected. In terms of the impact of the model discrepancy, it should be noted that most insect wings operate at mid-stroke angle of attack range between 25° and 45° (table 1), and in this range there is very good fit between the model and data. The fruitfly data have a drag offset, and a possible remedy, as proposed by Dickson & Dickinson [32], is to add a constant representing the drag coefficient at zero lift. However, it is noteworthy to mention that Lentink & Dickinson [51] measurements at higher Reynolds number (Re = 1400 and 14 000) did not show any existence of this drag offset.

Figure 5.

Lift coefficient variation with (i) angle of attack and (ii) drag coefficient. Results evaluated using the current model are compared to available experimental measurements for (a) hawkmoth; experimental data digitized from fig. 6 of [7], (b) bumblebee; experimental data digitized from fig. 7 of [8] and (c) fruitfly; experimental data digitized from fig. 7 of [51]. (Online version in colour.)

Table 1.

Weight, morphological and kinematic parameters of hovering insects [18].

insect	mass (mg)	R (mm)	(mm)		f (Hz)	ϕmax (deg)	αmid (deg)
fruitfly	0.72	2.02	0.67	0.55	254	75	46
bumblebee	175	13.2	4.02	0.49	155	58	28
hawkmoth	1648	51.9	18.26	0.46	26.3	60.5	32
honeybee	101.9	9.8	3.08	0.5	197	65.5	25
cranefly	11.4	12.7	2.38	0.56	45.5	61.5	30
hoverfly	27.3	9.3	2.2	0.52	160	45	29
dronefly	68.4	11.4	3.19	0.48	157	54.5	26
ladybird	34.4	11.2	3.23	0.47	54	88.5	43

3.3. Comparison with previous flapping wing aerodynamic models

Table 1 presents morphological and kinematic data of eight hovering insects taken from Sun & Du [18] against which we are going to validate our results. In our model, a wing is divided into 50 evenly spaced wing strips in the spanwise direction and a wing-flapping period is divided into 500 evenly spaced time steps. Aerodynamic forces on each strip are integrated along the wing length and averaged over the flapping period. For the evaluation of the aerodynamic power consumed during the flapping cycle, we assume that the energetic cost to the hovering insect is given by the time-averaged mechanical power output, where power can be positive or negative. This approach is consistent with that used by Sun & Du [18]. A specific power, P^*, is then obtained as power divided by the total mass. Accounting for negative power assumes that mechanical energy can be stored and released when the wing does positive work. This method of accounting also means that inertial power cancels out and thus can be ignored (see fig. 10 of [18]).

Table 2 presents the values of k_ind, k_tip and k_flap accounting for the non-uniform downwash velocity distribution, tip losses and effective flapping disc area, respectively. They are calculated using the procedure presented in Part I for eight insect species based on their data provided in table 1. These values are repeated here, as they will be used within the analysis of the obtained results.

Results from the quasi-steady blade element implementation are now compared with CFD results from Sun & Du [18] as well as the quasi-steady blade element model of Berman & Wang [11]. Both simulations used a horizontal stroke plane, symmetrical half-strokes, a sinusoidal variation of flapping angle (corresponding to C_ϕ → 0) and a trapezoidal-like variation of the pitching angle with a time interval over which rotation lasts of about 25% of the flapping cycle duration. This corresponds to a value of 5 for C_θ in our model. The mid-stroke angles of attack, α_mid, used by Sun and Du in their calculations are given in table 1. Vertical force to weight ratio as well as specific power results from our model compared to other models are given in table 3. The vertical force is defined as the magnitude of the z-component of the total force produced by a pair of wings in the reference frame (x₀, y₀, z₀) and is the force used for weight support.

Table 3.

Comparison between current model and that of CFD by Sun & Du [18] and quasi-steady model of Berman & Wang [11].

insect	current model		CFD of Sun & Du [18]		quasi-steady model of Berman & Wang [11]
	Fz0/W	P* (W kg−1)	Fz0/W	P* (W kg−1)	Fz0/W
fruitfly	0.98	26	1	30	1.003
bumblebee	1.06	43	1	42	0.95
hawkmoth	0.92	30	1	33	1.15
honeybee	1.1	43	1	41	—
cranefly	1.3	18	1	16	—
hoverfly	1	25	1	27	—
dronefly	1.2	37	1	32	—
ladybird	0.96	30	1	28	—

The results in table 3 show very good agreement with CFD results for weight support and power consumption for the eight insects. It should be remembered that Sun and Du developed their simulation model to obtain the value of α_mid (supplied in table 1) that would enable weight support. Hence, all their F_z0/W ratios are 1. In our model we used their α_mid values and calculated F_z0/W. The present model correctly predicts near-unity values of F_z0/W, but with a small underestimation in the hawkmoth and an overestimation in the cranefly and dronefly. The above results demonstrate that the proposed model is able to obtain usefully accurate results for a wide range of insects with different wing shapes and operating conditions. Table 3 also presents results from the quasi-steady blade element model developed by Berman and Wang, which was limited to the analysis of only three insects for which experimental data are present. Additionally, Berman and Wang represented all wing planform shapes as a half ellipse; the model implemented in this work represents the wing planform shape using the more convenient beta representation.

An important feature of the proposed model is its transparency, in that it provides greater insight into how the different problem parameters affect the solution; hence, allowing improved understanding of the flapping problem. As an example, results shown in figure 6 allow an assessment of how each of the three effects included within k affects the calculated aerodynamics (in terms of the weight support ratio). The fruitfly represents a case where k_ind and k_tip are the main contributors to k, while the hoverfly represents a case where the k_flap effect is the most significant. Figure 6 also shows the importance of accounting for k in aerodynamic calculations, as assuming an ideal case (k = 1) can lead to significant overestimation of the generated force (e.g. 23% for the hoverfly case).

Figure 6.

Effect of accounting different contributors to k on the weight support calculation for (a) fruitfly and (b) hoverfly. In both figures, Δ represents the percentage difference in value between calculated F_z0/W and unity. (Online version in colour.)

4. Conclusion

A generic, transparent and compact model for the design and/or analysis of rigid flapping wings in hover has been presented. The model is generic in that it can be applied to wings of arbitrary planform geometry following arbitrary kinematic cycles, and is transparent in that the model parameters are clearly linked to attributes of the flow physics. The model is compact in the sense that relatively modest computational effort is required for solution compared to higher order models such as those based on CFD, and hence the model is suitable for use as part of preliminary engineering design and optimization of flapping wings.

The modelling capability provides an improvement in the state of the art in that relevant aerodynamic model parameters are obtained by analytical means from geometry and kinematic information alone; aerodynamic data from experiments or higher order models are not required. The model is implemented using a quasi-steady blade element framework with parametric control of both wing chord distribution and wing kinematics.

The developed model has been validated against analytical, numerical and experimental results from the literature with the following outcomes:

• (1) Comparison of the model with analytical results from the Polhamus leading-edge suction analogy shows that there is good agreement with respect to the magnitude and the shape of variation of lift over the full range of incidence up to 90° angle of attack. Moreover, the proposed model avoids the overestimation in the lift coefficient values by the Polhamus analogy for typical insect wing aspect ratios.

• (2) Comparison of the calculated steady force coefficients with available experimental data for revolving hawkmoth, bumblebee and fruitfly model wings shows good agreement with respect to both the shape of variation of the force coefficients with incidence and the magnitude.

• (3) Comparison of the calculated aerodynamic forces and consumed power with available numerical CFD simulations for eight insect cases shows good agreement.

The validated model is used to evaluate the relative impact of different contributors to the induced power factor for the hoverfly and fruitfly. It is shown that the assumption of an ideal induced power factor (k = 1) for a hoverfly leads to a 23% overestimation of the generated force owing to flapping.