Aerodynamic modelling of insect-like flapping flight for micro air vehicles

Transcription

1 ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) Aerodynamic modelling of insect-like flapping flight for micro air vehicles S.A. Ansari,R.Żbikowski, K. Knowles Department of Aerospace, Power and Sensors, Cranfield University, Defence Academy of the United Kingdom, Shrivenham, SN6 8LA, England Available online 1 September 2006 Abstract Insect-like flapping flight offers a power-efficient and highly manoeuvrable basis for a micro air vehicle capable of indoor flight. The development of such a vehicle requires a careful wing aerodynamic design. This is particularly true since the flapping wings will be responsible for lift, propulsion and manoeuvres, all at the same time. It sets the requirement for an aerodynamic tool that will enable study of the parametric design space and converge on one (or more) preferred configurations. In this respect, aerodynamic modelling approaches are the most attractive, principally due to their ability to iterate rapidly through various design configurations. In this article, we review the main approaches found in the literature, categorising them into steady-state, quasi-steady, semi-empirical and fully unsteady methods. The unsteady aerodynamic model of Ansari et al. seems to be the most satisfactory to date and is considered in some detail. Finally, avenues for further research in this field are suggested. r 2006 Elsevier Ltd. All rights reserved. Keywords: Insect flight; Aerodynamic modelling; Micro air vehicles; Wing aerodynamic design; Unsteady aerodynamics; Low Reynolds number flow Contents 1. Introduction Insect flight Wing kinematics Insect flight aerodynamics Aerodynamic phenomena Wagner effect Wake capture Apparent mass effects Kramer effect Leading-edge vortex CFD studies Corresponding author. Tel.: ; fax: address: s.a.ansari@cranfield.ac.uk (S.A. Ansari) /$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi: /j.paerosci

2 130 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Aerodynamic modelling Important features to model Aerodynamic modelling methodologies found in the literature Steady-state methods Quasi-steady methods Semi-empirical methods Unsteady methods Other methods The method of Ansari et al Description of the method Methodology Analysis Implementation Validation Limitations Conclusions Acknowledgements References Introduction The origins of the micro air vehicle (or MAV) date back to about 1997 when DARPA launched a pilot study into the design of hand-held (150 mm) flying vehicles [1]. While being readily portable, such MAVs would also show particular promise in socalled D 3 dull, dirty and dangerous environments. Current surveillance assets (e.g. satellites, UAVs), however, possess virtually no capabilities of information-gathering inside buildings whereas a suitable MAV would provide such a facility. Agile flight inside buildings, caves and tunnels is of significant military and civilian value [2 4]. The focus on indoor flight leads to the requirement of a distinct flight envelope, including small size, low speed, hovering capability, high manoeuvrability at low speeds and small acoustic signature, amongst other things. In addition, autonomy is required to enable mission completion without the assistance of a human telepilot; this requires precise flight control. Although scaled-down versions of conventional designs may be the obvious approach to MAV design and are being pursued for outdoor applications, such flight platforms are unattractive for indoor flight for a number of reasons. Fixed-wing aircraft, for example, do not have the required agility (due to bank-to-turn manoeuvres) necessary for obstacle-avoidance in indoor flight and are incapable of hovering. Although rotorcraft offer good agility and VTOL 1 capability, they suffer from 1 Vertical take-off and landing. wall-proximity effects and are too noisy and inefficient (particularly at small scale). A plausible alternative, therefore, is flapping-wing flight, if only due to its abundance in nature. More significantly, however, flapping flight offers an important advantage it is more efficient in terms of specific power requirement at low speeds than either fixed- or rotary-wing flight [5]. Since power on a small flying platform is in limited supply, this is a very important constraint. There are two classes of aerial flapping flight birdlike and insect-like. Birds have an endoskeleton so that muscles attached to bones along the wing are used for flight and manoeuvring. This, however, makes them heavy and relatively less efficient (in terms of specific power). They can compensate for this by spending a significant time of their flight gliding. The flapping motion of their wings exploits the propulsive nature of plunging and pitching wings the Knoller Betz effect (after Knoller [6], Betz [7]) while their forward speed provides the necessary lift. Birds consequently spend most of their time in forward flight and are, therefore, too fast to be useful for indoor applications. 2 Reproducing their endoskeleton and muscles by engineering means poses a considerable challenge as it requires essentially the development of a sophisticated prosthetic device. Insects, on the other hand, possess an exoskeleton: all actuation is carried out at the wing root and, 2 The hummingbird is a notable exception and is described, from an aerodynamics viewpoint, as a notional insect. It has small wings with few elastic bones and flaps like insects, and is also capable of sustained hover.

3 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) consequently, the wing structure is very light, generally accounting for 1% of the insect s weight [8]. This makes insect flight very attractive as a model, while also satisfying all the other requirements of the flight envelope identified above (especially hover) for an indoor MAV. It provides the most viable and proven approach for a MAV [2 4]. Insects have been around ever since life moved out of the oceans and on to land about 350 million years ago [9 11]. Over this time, flying insects have evolved and perfected their flight, making them the most agile and manoeuvrable creatures for their size today. It is this characteristic of insect flight that we seek to mimic. Insects are by far the most abundant species around. Yet, several aspects of their livelihood have not been well understood, owing principally to their small size and the concomitant experimental difficulties. One such aspect is their flight. They rely on flapping motion, whereby they move their wings in a more or less reciprocating manner to generate sufficient forces (and moments) for flight. Although most choose not to do so, insects are also generally capable of hover. This feature makes them particularly attractive for indoor flight. The development of an efficient MAV requires a careful wing aerodynamic design study. This is particularly true for an insect-like flapping-wing MAV (or FMAV) as the wings are the source of lift, propulsion and manoeuvres, all at the same time. Whereas insect flapping wings offer a proven platform and are abundant in nature (there are over 170,000 species of flying insects), little is known about the optimality of their wing design. Unlike for fixed or rotary wings, the parametric space associated with flapping wings is yet unexplored. A study that covers the effects of both wing kinematics and wing geometry (amongst others) on their aerodynamic performance is required, thus highlighting the need for a suitable aerodynamic tool. For the aerodynamic design, a tool that can be used iteratively to converge on a preferred design is required. Ellington [12] proposed design guidelines based on scaling from nature but this does not give physical insight or allow design optimisation. Conventional analytical and semi-empirical methods are incapable of handling this unexplored flight regime while CFD methods are only now beginning to become competitive. Aerodynamic modelling techniques, therefore, offer the best compromise and are the theme of this article. In the rest of this section, we address the basics of insect flight, particularly their kinematics. We also detail the concept of aerodynamic modelling after discussing briefly some CFD studies in the field. In Section 2, we discuss the aerodynamic phenomena that make insect-like flapping flight possible and identify the key elements that need to be included in a representative aerodynamic model. Section 3 is dedicated to a review of the various aerodynamic models proposed for this flight regime. Of these, the model of Ansari et al. ([13,14], for full details, see Ansari [15]) seems the most satisfactory to date and is discussed in more detail in Section 4. Finally, we present our main findings and suggest avenues for future research in the concluding section (Section 5) Insect flight Insect flight has fascinated man over the ages, especially since the beginning of the 20th century and the development of manned flight. Historically, it was envisaged that men would fly by flapping artificial wings like birds and, in fact, the Wright brothers studied birds in flight when designing their first gliders ([16], p. 89). The recent interest in MAVs has rekindled interest in flapping flight, especially insect flapping flight, and more so because of recent advances in the knowledge base. Several reviews of insect-flight aerodynamics have been given, for example, Sane [17], Rozhdestvensky and Ryzhov [18], Lehmann [19] and Wang [20]. Ho et al. [21] have also reviewed flapping-wing flyers in the context of MAVs while Lian et al. [22] have addressed the flexibility of insect-like wings at these Reynolds numbers. The most distinguishing characteristic in insect flight is the wing kinematics. These kinematics are not found in other flying creatures or machines (except for hummingbirds; see Footnote 2, Section 1). Due to their much smaller scale, insects differ fundamentally from birds in that all actuation is carried out at the wing root. By contrast, birds have internal skeletons to which muscles are attached, permitting more localised actuation along the wing, e.g. wing warping, although much bird wing deflection may be passive. As a result of these kinematics, the aerodynamics associated with insect flight are also very different from those encountered in conventional fixed- and rotary-wing or even bird flight. Therefore, for the development of an engineering model that faithfully represents insect

4 132 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) flight, it is necessary first to understand these kinematics Wing kinematics Ever since the advent of cameras, insect flight has been photographed [23] but it has only been since the relatively recent availability of high-speed photography that good descriptions of their kinematics have emerged [24 28]. In terms of flight, insects can be broadly classified into two groups those with one pair of wings and those with two pairs. We restrict ourselves to the former and, in particular, Diptera which are two-winged flies, e.g. the fruit fly Drosophila, as they are easier to analyse and emulate, and are excellent flyers. The description that follows is for such an insect. Insects use a reciprocating wing motion for flight. The overall flapping motion can be compared to the sculling motion of the oars on a rowing boat, consisting essentially of three-component motions sweeping (fore and aft movement), heaving or plunging (up and down movement) and pitching (varying incidence). Typical flapping frequencies are in the range of Hz, generally decreasing with increasing insect size and weight [29]. Through a cycle, the motion of the flapping wing itself can be divided broadly into translational and rotational phases. The translational phase consists of two half-strokes the downstroke and the upstroke (see Fig. 1). The downstroke refers to the motion of the wing from its rearmost position (relative to the body) to its foremost position. The upstroke describes the return cycle. At either ends of the half-strokes, the rotational phases come into play stroke reversal occurs, whereby the wing rotates rapidly and reverses direction for the subsequent half-stroke. During this process, the morphological lower surface becomes the upper surface and the leading edge always leads (Fig. 1). Pronation and supination are the terms used to describe the stroke reversals preceding the downstroke and upstroke, respectively. The path traced out by the wing tip (relative to the body) during the wing stroke is similar to a figure-of-eight on a spherical surface since the wing semi-span is constant (see Fig. 2). The wings flap back and forth about a plane called the stroke plane, which is analogous to the tip-path-plane described by the tips of the blades of a rotorcraft, and may be inclined to the horizontal at the strokeplane angle b (Fig. 2). Downstroke Upstroke Flapping Cycle Downstroke Upstroke Fig. 1. Half-strokes during an insect flapping cycle. Note that the leading edge (thick line) always leads.

5 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) horizontal H T β inclined stroke plane reference downstroke upstroke H :: head T :: thorax A :: abdomen A (a) Isometric view (b) Detailed side view Fig. 2. Insect flapping figure-of-eight kinematics. During a half-stroke, the wing accelerates to a roughly constant speed around the middle of the half-stroke, before slowing down to rest at the end of it. The velocity during the wing-beat cycle is, therefore, non-uniform and for hover in particular, the motion of the wing tip does not differ dramatically from a pure sinusoid [25]. Wing pitch also changes during the half-stroke, increasing gradually as the half-stroke proceeds. On average, the angle of attack during a half-stroke is about 35 at the 70%-span position [25] and typical stroke lengths are of the order of 3 5 wing chords [30]. In normal hovering flight, most insects use symmetrical half-strokes and horizontal stroke planes (b ¼ 0 ). Forward flight is made possible through asymmetry between up- and downstrokes downstrokes are generally longer than upstrokes and an inclined stroke plane (ba0 ). Flapping asymmetries between starboard and port wings are used for turning manoeuvres (rolling or yawing) whilst pitching motions seem to be the result of aerodynamic moments generated by the wings with some assistance from the insect body. In the review that follows, frequent reference is made to various kinematics parameters of insect flight. Sweep angle and sweep rate are denoted by f and f, _ respectively, while a and _a refer to angle of attack and pitch rate, respectively. Wing length is R, chord length is c and the arc swept by the flapping wings through a half-stroke is the stroke amplitude F (see also Fig. 17(a)). 2. Insect flight aerodynamics The aerodynamics of insect flapping flight is too vast a subject to be addressed in great detail here. We restrict ourselves, therefore, to the most pertinent flow characteristics in this flight regime. The aim of this section is to provide a logical progression towards the description of useful aerodynamic models for these flows. It had been a long-held belief that insects should be unable to fly on the basis of conventional aerodynamics: their wing area and weight corresponded to very high wing loadings and, hence, mean lift coefficients that were too large. In fact, Magnan declared that insect flight was impossible: Tout d abord poussé par ce qui fait en aviation, j ai applique aux insectes les lois de la resistance de l air, et je suis arrive avec M. Sainte-Lague a` cette conclusion que leur vol est impossible. [31]. However, as knowledge in this field advanced through more detailed observations, a much clearer picture of the aerodynamic phenomena has emerged. In this section, we address the aerodynamic modelling of insect flight. Before such a model can be devised, the main aerodynamic phenomena characterising insect flight must be identified. This is addressed in Section 2.1. A number of existing CFD studies are reviewed in Section 2.2 followed by a discourse on the concept of aerodynamic modelling, with particular reference to insect-like flapping flight (Section 2.3). Finally, the key elements that need to be included in any representative

6 134 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) aerodynamic model for insect flight are identified (Section 2.4) Aerodynamic phenomena The flow associated with insect flapping flight is incompressible, laminar and unsteady, and occurs at low Reynolds numbers. Despite their short stroke lengths, insect wings can generate forces much higher than their quasi-steady equivalents due to the presence of a number of unsteady and vortical aerodynamic effects. Under steady-state conditions, the high angles of attack at which flapping insect wings operate would normally stall the wings and give deteriorated aerodynamic performance. In practice, however, these wings continue to produce favourable forces even in these extreme conditions. We start, therefore, by discussing relevant aerodynamic phenomena Wagner effect The main identifying feature of an insect s flapping cycle is the wing s repeated acceleration (starting), deceleration (stopping) and reversal. This start stop reversal behaviour is fundamental to the aerodynamics that make this flight possible. Each time the wing starts, it sheds starting vortices in accordance with Kelvin s law (see e.g. [32]). These vortices are of the opposite sense to the bound circulation around the wing (see Fig. 3) and, therefore, have an inhibitory effect on lift the socalled Wagner effect (after Wagner [33]). Increasing angle of attack also has a similar effect. When the wing decelerates towards the end of a half-stroke, stopping vortices are shed which resist the drop in lift on the wing. However, as stroke reversal occurs and the wing begins moving in the opposite direction, a new set of starting vortices are shed. These are of the same sign as the stopping vortices from before stroke reversal (which are still in the vicinity; see Fig. 3), thus giving rise to a more acute form of the Wagner effect [34] Wake capture As the flapping motion proceeds, the fluid around the wing is no longer quiescent (especially during hover or low-speed flight) and the flapping wing repeatedly moves into its own wake. Dickinson proposed the term wake capture for this mechanism of unsteady lift generation on the basis of his experiments [35,27,36], although wake passage [37] is an attractive alternative. The wake behind a flying object contains energy imparted to the surrounding fluid in the form of momentum and heat. Wing passage through the wake could, therefore, be a method to recover some of this lost energy and utilise it usefully for flight. The importance of wake capture was also suggested by Grodnitsky and Morozov [38] who proposed that insects and birds have special mechanisms whereby they extract (a) (b) (c) Fig. 3. Wagner effect and wake capture (clockwise vortices are shown in grey). As the wing decelerates at the end of a half-stroke, a stopping vortex is shed (a); during stroke reversal, a vortex of opposite sign arising from the sudden pitching rotation is shed (b); at the start of the next half-stroke, a starting vortex (of the same sign as the previous stopping vortex) is shed (c). The previous leading-edge vortex is also of wrong sign for the current half-stroke, further impeding lift.

7 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) energy back from their near vortex wake. A similar view was expressed by Ennos [39] who speculated that in two-winged flies the kinematics were helped by the aerodynamics Apparent mass effects As an object moves through an inviscid fluid at constant velocity, the fluid ahead of it moves aside and closes up behind it [40]. The kinetic energy expended in the process is recovered at the end of it as nothing is lost to viscous drag. If the object now accelerates, not all of the energy expended in parting the flow ahead is recovered when the flow closes up behind it. This additional force required to accelerate the neighbouring fluid around the object is called an apparent mass force. In insect flapping, the fluid around the wing is continually being accelerated or decelerated and these apparent mass forces can be significant. Ellington [41] speculated that for symmetrical half-strokes, the net apparent mass forces were close to zero. This was later shown by Sunada and Ellington [42]. However, work must be done to accelerate and decelerate the apparent mass during each halfstroke, resulting in power expenditure that must be accounted for Kramer effect At the start of each half-stroke, the preceding stroke reversal is coming to an end and the flapping wing is pitching downwards. This causes a decrease in lift [14] the so-called Kramer effect (after Kramer [43]). By the same token, the increase in angle of attack towards the end of each half-stroke, when stroke reversal begins, results in an increase in lift Leading-edge vortex Although a number of unsteady aerodynamic phenomena pertaining to insect-like flapping flight have been identified above, they are still unable to explain the high lift required to sustain flight. This remained a mystery essentially until 1996 when Ellington and his co-workers discovered the leadingedge vortex [44]. This work used Ellington s flapper a scaled-up model of the hawkmoth Manduca sexta for their experiment and released smoke from its leading edge while it flapped with insect-like kinematics. Ellington s flapper is a 10:1 electromechanical model that uses a gearbox and a system of bevel gears, driven by several servomotors. In an attempt to preserve both the Reynolds number 3 and the Strouhal number, 4 Ellington et al. preserved the ratio fl 2 between the real insect and the model, where f is flapping frequency and l is wing length. In fact, this ratio preserves neither Reynolds nor Strouhal numbers but does preserve their product in air. When scaling up for constant Reynolds number Re in air so that kinematic viscosity n is unchanged, lu is preserved but not necessarily l=u (where l and U are a characteristic length and velocity, respectively), so that shedding of vortices may occur at a frequency not preserving the Strouhal number St. In their experiments on the Robofly (a scaled-up fruit fly), Dickinson and coworkers used oil as the fluid medium and preserved Re (see e.g. [27,45]). In scaling up geometrically and also moving from air to oil (i.e. n also changes), while a multiple of lu was preserved, St was not. If St in a scaled-up model is not preserved, then an important aspect of unsteadiness is not reproduced. Since both leading- and trailing-edge vortices are shed periodically in insect-like flapping, this may affect the validity of the results thus obtained. Despite these concerns, several sets of experiments with the flapper verified the leading-edge vortex (LEV) [46 48]. Although this mechanism had been observed in earlier experiments [49 53] it was not until the work of Ellington et al. [44] that the significance of the LEV for insect flapping flight received proper recognition. Ellington and co-workers reported that the LEV, which persisted through each half-stroke, existed on the wings and proposed that it was responsible for the augmented lift forces. The overall structure of the LEV has been likened to that observed on lowaspect-ratio delta wings [49,47]; it is produced and fed by a leading-edge separation. In view of this, Birch and Dickinson [45] suggested that spanwise transport of the leading-edge vorticity served to remove energy from it, and hence, limited its growth and shedding (see also [54,55]). The augmented lift is similarly analogous to the vortex lift on delta wings. The leading-edge vortex starts close to the wing root and spirals towards the tip where it coalesces with the tip vortex and convects into the trailing wake [44]. It remains attached to the wing 3 Re ¼ Ul=n, where l is wing span, U is a characteristic velocity, and n is kinematic viscosity. 4 St ¼ fl=u, where f is flapping frequency, l is wing span, and U is a characteristic velocity.

8 136 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) for most of the half-stroke and is shed at the end when the wing rapidly rotates in pitch. Willmott et al. [46] speculated that the LEV was formed either due to the rotational motions prior to translation or via a dynamic stall mechanism. Although dynamic stall has been suggested by others (e.g. [56,34]), this is unlikely [57] since a dynamic-stall vortex breaks away almost immediately and rapidly convects as soon as the wing translates [58]. A detailed account of the nature of the LEV and its possible origins can be found in Ansari [15]. LEVs have been seen on the wings of both large insects [44] and small ones [59]. Liu et al. found that because of the presence of the LEV, the wings of a hovering hawkmoth were able to generate vertical forces up to about 40% greater than required to support its weight [60]. The LEV is, therefore, fundamental in explaining the large forces generated by insect-like flapping wings. Whereas the LEV may be common to most insects [51,38,46,45], its spanwise flow characteristic seems to vary with insect size (and hence Reynolds number). Recent experiments by Dickinson and coworkers [45,36,61] on a dynamically scaled mechanical model of the tiny fruit fly Drosophila melanogaster (dubbed Robofly and using Ellington s approach to aerodynamic scaling) revealed the presence of a strong bound LEV but they reported only weak spanwise spiralling (unlike for the hawkmoth above), prompting them to conclude that the precise flow structure of the LEV depends critically on Reynolds number [61]. This is supported by [62] who suggested that spanwise flow exists at all relevant speeds but its spiralling nature becomes less discernable as the Reynolds number decreases. Insect flight aerodynamics is a combination of the phenomena identified above, the exact proportions of the various effects being determined by the wing kinematics and flight condition. The manner in which the wake is shed and how the wing interacts with it is key to the forces and moments generated, the latter being particularly so during hover when the wing remains predominantly in the vicinity of its shed wake (see Fig. 4). The scenario shown in Fig. 4 is for a 2D section through a typical flapping wing in the hover [15]. As the wing approaches the end of a half-stroke, it slows down and pitches up, producing stopping and starting vortices, respectively (in accordance with Kelvin s law), that may result in no net vorticity (a) (b) (c) (d) End of half-stroke Start of stroke reversal End of stroke reversal Start of new half-stroke Fig. 4. Typical wake generated by an insect-like flapping wing in the hover (after [15]). Vortices originating at the leading edge are shown as dotted lines while those from the trailing edge are solid lines (online version shows these in blue and red, respectively). (a) End of half-stroke; (b) Start of stroke reversal; (c) End of stroke reversal; (d) Start of new half-stroke. (Fig. 4(a)). However, as stroke reversal proceeds, the predominant motion is the pitching rotation of the wing and a rotation vortex forms at the trailing edge (Fig. 4(b); see also Fig. 3) while, at the same time, the previous LEVs pass over the leading edge. A starting vortex then joins the rotation vortex at the trailing edge as motion begins for the new halfstroke in the opposite direction (Fig. 4(c)) while a

9 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) new vortex also forms at the leading edge. As the translational motion gets underway, one of the previous LEVs joins the starting vortex at the trailing edge (Fig. 4(d)). Concurrently, contrarotating vortices shed previously from the leading and trailing edges convect under their mutual influence and interact with the wing at later times (not shown). The wing then moves towards the other end of the half-stroke to meet previous shed vortices there and the process repeats. The shed wake and its repeated complex interaction with the wing is the distinguishing feature of insect-like flapping. Any successful aerodynamic model must capture these complex interactions to be applicable in the hover. As will be seen in Section 3, the lack of this feature appears to be responsible for the inadequacy of some of aerodynamic modelling approaches found in the literature CFD studies Reynolds-averaged Navier Stokes (RANS) CFD codes have been used to analyse insect-like flappingwing aerodynamics, notably by Liu and Kawachi [60,63] and by Sun et al. (e.g. [64,65]). Liu and Kawachi used a 3D, incompressible, laminar RANS code (with a remeshing technique for the moving grid) to study a hawkmoth wing in hover. Their wing section was of constant thickness but with smoothed elliptical curves at the leading and trailing edges. They were only able to validate their results against 3D flow visualisation and 2D force data. They used their CFD results to look in detail at the flowfield associated with the LEV. They noted, however, the lack of 3D experimental unsteady force data and aeroelastic deflection data. Sun and co-workers have developed a 3D, unsteady RANS code which represents insect-like wing motions by oscillating the background flow [64 68]. They generally consider 12%-thick elliptical wing sections and have validated their code against some steady-flow wing data and Dickinson s Robofly data [27,36]. Agreement with the latter was only moderate. In a series of papers, Sun et al. have used this code to investigate the aerodynamics of various insects, including the fruit fly in hover [64] and in forward flight [66], reporting calculations of power requirements and effects of wing kinematics. They also investigated the dragonfly in hover [65] and in forward flight [69]. Using the same CFD code for a bumblebee in hover, [67] calculated aerodynamic derivatives and thence analysed longitudinal dynamic stability. This appears to be the only CFD study that attempts to assess flight dynamics but it is limited by the linearised equations of motion used (see [70]). There appear to have been no studies reported on the use of CFD for manoeuvring flight and there is clearly a lack of experimental data for validation of detailed CFD results. Ramamurti and Sandberg [71] used a finiteelement flow solver and an Arbitrary Lagrangian Eulerian formulation to model a 3D Drosophila wing undergoing a Dickinson-derived flapping motion. They showed good agreement on lift with Dickinson s experimental data but poorer agreement on thrust. Only one wing was modelled, with a symmetry plane to represent the other wing in Dickinson s experiments. The wing cross-section was a flat plate with radiussed leading and trailing edges. Ramamurti and Sandberg investigated the effect of advanced wing rotation (i.e. increasing incidence before the end of a half-stroke) and showed that this gave increased overall lift. They visualised the LEV and were able to speculate on the various lift mechanisms which gave rise to the observed time-variation of forces. In common with other CFD studies, however, there was no definitive calculation of the various fluid dynamic mechanisms which contribute to force generation. This is a key advantage of aerodynamic modelling techniques (discussed below). Finally, the authors used their computational results to calculate aerodynamic forces, moments and power requirements on an MAV wing having the same shape and kinematics as the modelled Drosophila. A more extensive design study, investigating wing geometric and kinematic parameters using such CFD techniques would require considerable time and resources. This, again, is where aerodynamic modelling techniques currently offer advantages. Ho et al. [21] recently reviewed flapping flight and the application of flow control technologies. Their own work links CFD with FEM (finite-element modelling) to analyse wing aeroelastics and attempts to optimise the stiffness distribution for maximum lift and thrust (using a Gur Game algorithm). They validated their CFD results against their own experimental flow-visualisation data. In a separate development, Isogai and co-workers have modelled the hovering flight of the dragonfly Anax parthenope julius using 3D Navier Stokes calculations [72]. They employed a multiblock

10 138 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) method so that each of the fore- and hind-wings could be moved independently. By means of algebraic functions, they mapped the physical domain into computational space (where the wings were rectangular) and the domain was re-gridded at each time step. Comparison with an experimental setup that crudely mimicked dragonfly kinematics showed moderate agreement [72]. However, using a more advanced experimental apparatus that reproduced the dragonfly kinematics more accurately, Yamamoto and Isogai [73] found better comparison with their CFD results, and concluded that their Navier Stokes code might be a useful design tool for insect-like micro air vehicles. They also found that varying the phase angle between the flapping of the fore- and hind-wings had little effect on the time-averaged forces. In another study, Kurtulus et al. employed a direct numerical simulation (DNS) code to find optimum values for some parameters (e.g. aerofoil incidence during the linear motions, location of start of incidence change, location of start of velocity change and location of pitch axis) to maximise lift [74,75]. They ran 2D CFD calculations on a NACA0012 aerofoil, flapping in incompressible, laminar flow at Re Inspired by the work of Pedersen [37,76], they also made some analytical calculations, based on the Wagner [33] and Ku ssner [77] functions for unsteady potential flow, and the Rankine Froude momentum theory. Results from this were compared with those from the CFD model and Kurtulus et al. reported good agreement. They found that positive lift was always obtained for wing angles of attack in excess of 30 during the translational phases of the flapping cycle. Whereas it revealed the effects of various flapping-wing parameters and the influence of Reynolds number, the 2D nature of their study failed to elucidate any 3D behaviour. Eldridge [78] presented a viscous vortex particle method and used this to simulate flow about a rigid, 2D, elliptical-section aerofoil in pitching and plunging motion (no sweeping). In principle, this technique offers advantages of speed and efficiency over grid-based methods. The method is considered in some detail later (see Section 3.5). Although Eldridge calculated body forces and presented flow visualisation, no experimental validation was offered. Extending this technique to 3D and deforming surfaces is an ongoing and non-trivial exercise, although Eldridge claims that many of the techniques required have been developed in previous work Aerodynamic modelling The essence of aerodynamic modelling is to provide a mathematical framework, consistent with flow physics, for describing the main flow phenomena while avoiding the extremes of mathematical oversimplification and intractable complexity [57]. Aerodynamic modelling progresses from first principles, but introduces several simplifications to the basic equations of fluid mechanics, justified by the known flow phenomenology and/or geometric and kinematic symmetries. The resulting mathematical model must have usable predictive capabilities and verifying those on experimental data is crucial for assessing whether the simplifications made are acceptable. In aerodynamic modelling, rather than using directly the Navier Stokes equations, the flow to be modelled is first studied experimentally and the observed phenomena and symmetries are used to simplify modelling. In practice, even simplified Navier Stokes equations are difficult to handle, so the usual starting point is an inviscid setting, the Euler equation, in which viscosity is accounted for indirectly by imposing a boundary condition. A typical example is the celebrated Kutta Joukowski condition which is essentially an empirically justified choice out of infinitely many theoretical possibilities. Finally, it often happens that a 2D approximation of the flow is justified, or is used within a wing-element framework as an approximation. Mathematically, this means dealing with the Laplace equation in the plane with appropriate boundary conditions and constraints added. The 2D Laplace equation is a convenient and wellunderstood mathematical tool as its solutions are harmonic functions and powerful methods from Complex Analysis [79,80] can be employed. Since the Laplace equation is linear, the superposition principle applies and different contributions to the flow can be considered separately and then added together. The possibility of linearly combining the contributions is not only mathematically convenient, but also gives important insight into the flow structure which is particularly useful in aerodynamic design. Finally, the presence of a wing section (aerofoil) is accounted for by the non-penetration constraint, while viscosity and flow separation can be modelled with the Kutta Joukowski condition.

11 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Justifiable simplifications, consistent with flow physics, are present in special flight regimes, especially hover. Experimental observations of insect hover allow defining the following phenomenology. It is assumed that the flow is incompressible, has low Reynolds number and is laminar, and that the following phenomena dominate: (i) bound LEV, which models leading-edge flow separation; (ii) vortex sheet shed from the trailing edge and (iii) the attached part of the flow following from the wing motion. The kinematics is assumed to be periodic and reciprocal, exhibiting spatial and temporal symmetries. Even with these simplifications the resulting problem is non-trivial, because the flow is unsteady and vortical. Flapping-wing flows necessitate capturing the separated flow due to the LEV and the trailing-edge wake, and their complex interaction with the wing. An aerodynamic model of insect-like flapping in hover will be acceptable if it manages to capture these critical elements. The main advantage of having an acceptable aerodynamic model for insect-like flapping is availability of a quick turnaround wingdesign tool which, for a given wing geometry and kinematics, will predict the resulting aerodynamic force and moment whilst providing useful insight into, for example, the components of the forces and moments. The technique of aerodynamic modelling saw its biggest development in the 1930s when analytical methods for unsteady aerodynamics were developed, especially with reference to wing flutter problems [81 83,77,84 89]. Methods were required that were analytically tractable and yet could predict, with reasonable accuracy, the complex unsteady aerodynamic behaviour. Despite recent advances in CFD methods, this technique still finds favour in the helicopter community [90] Important features to model In developing an engineering model, the ultimate goal is to predict quantitatively the behaviour of some physical phenomenon. The process begins with constructing a mathematical model that captures the essential features of the problem by simplifying the physics to a tractable, yet physically realistic, level. This is followed by a process of analysis where every possible tool is tried (and even some new tools developed) in order to understand the behaviour of the model as thoroughly as possible. Finally, the results are interpreted and validated against real-world facts to understand better the workings of the observed physical phenomenon. As noted by Thwaites, any new theory must take into account as far as possible what appear to be the most important physical characteristics ([91], p. 512). Thwaites further noted that for wing aerodynamics the positions of the separation lines, in particular, must be taken into account. Therefore, those properties and flow features that need to be represented in a model for insect-like flapping wings must first be identified. From the review above, the following conclusions can be drawn. The flow associated with insect-like flapping comprises two components attached and separated flow [57]. The attached flow on the aerofoil refers to all flow characteristics associated with the free-stream flow on the aerofoil as well as the effects of unsteady motions (sweeping, heaving and pitching). In insect-like flapping wings, flow separation is usually observed at both leading and trailing edges due to the high angles of attack and the severity 5 of the kinematics. These wakes affect the forces (and moments) generated by the wing due to their interaction with the wing in the form of inhibitory or favourable effects (e.g. the Wagner and Kramer effects). The back and forth motion of the wings also gives rise to effects associated with wake re-entry. The LEV is bound to the wing for most of the duration of each halfstroke and flow remains more or less attached in all other regions of the wing. The trailing-edge wake leaves smoothly from the trailing edge (except perhaps during stroke reversals when the smooth-flow condition may not hold). All of the above features must be included in any representative model. 3. Aerodynamic modelling methodologies found in the literature Hoff [92] was probably the first to attempt to analyse insect flight. He based his explanation on previous data and tried to bring insect flight into the realm of conventional aerodynamics. His analysis was based on the forward flight of insects and did not account for the flapping motion of their wings. Demoll [93] was quick to refute Hoff s work and proved that in certain cases, the required force coefficients were excessively high. Hoff had, however, pointed out the similarity between insect 5 Severity implies rapidity and magnitude of change.

12 140 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) actuator disk partial actuator disk wake wake (a) (b) Fig. 5. Actuator disks used in simple momentum theory. flapping flight and a spinning propeller, arguing that the analysis had to hold from momentum considerations. Experimental work, as may be expected, had been lacking in this flight regime mainly due to the size of insects and the instruments available for analysis. Nevertheless, as early as 1934, Magnan was able to make a detailed investigation into insect flappingwing flight. However, he still remained unable to explain his results theoretically [31]. In this section, we review various aerodynamic modelling techniques found in the literature. These can be classified into those employing theoretical aerodynamics only steady-state (Section 3.1), quasi-steady (Section 3.2) and unsteady (Section 3.4) and those with some experimental input, usually in the form of empirical coefficients (Section 3.3). Finally, a short section discussing some of the other techniques available is presented (Section 3.5) Steady-state methods Although the methods described here may be considered quasi-steady, they have been categorised as steady-state because the analysis applies once a steady-state has been reached. As noted by Hoff [92], an actuator-disk-type analysis must hold from momentum considerations and several authors have used this basis for proposing their theories (e.g. [94 97,41]). An actuator disk is an idealised surface that continuously imparts momentum to a fluid by maintaining a pressure difference across itself. Momentum theory accounts for both axial and rotational changes in the velocities at the disk but does not attempt to explain how these changes occur. Assuming that, while remaining confined to their stroke planes, insect wings beat at high enough frequencies (so that the forces appear more or less constant), their stroke planes (see Fig. 2) approximate to actuator disks (Fig. 5(a)) and Rankine Froude theory 6 may be applied. Weis-Fogh [96] derived the induced downwash velocity w i for a hovering insect at the actuator disk or stroke plane as sffiffiffiffiffiffiffiffiffiffiffiffiffi W w i ¼ 2prR 2, (1) 6 This momentum-based theory uses Bernoulli s equation for steady flow to compute induced velocity at the actuator disk and the jet velocity in the far wake downstream.

13 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) where W is insect weight, r is air density and R is wing length. The jet velocity w in the far wake downstream is twice that at the disk, i.e. w ¼ 2w i. Although he acknowledged that w i varied through a half-stroke and that stroke amplitude F was rarely 180, Weis-Fogh claimed that experimental measurements on the beetle Melolontha vulgaris were in agreement with Eq. (1). Instead of a circular disk, Ellington [41] proposed a partial actuator disk (see Fig. 5(b)) of area A ¼ FR 2 arguing that it reflected the problem better. Using this modified expression for area, induced power P i is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 3 1 W P i ¼ 2rFR 2 ¼ W, (2) cos b 2r cos b A where cos b takes the component in the direction of gravity (b is stroke-plane angle; see Fig. 2). The quantity W =A is disk loading and controls the minimum power requirement. Bound circulation around a flapping wing joins up with tip vortices at each end which then trail into the wake [98]. In accordance with Helmholtz s second theorem [99], these, in turn, connect with the starting vortex from the start of the half-stroke, thus forming a closed loop or a vortex ring. In accordance with Helmholtz s first theorem [99], vorticity must be constant along the vortex ring this becomes possible once the vortex is free after being shed. Thus, shed vortex rings carry information about the circulation and, hence, forces around the flapping wing. Ellington postulated, therefore, that vortex sheets (which rolled-up at the edges to form vortex rings) were shed during each lifting half-stroke, giving rise to a vortex wake, an array of regularly shed vortex sheets. He also noted that due to the timeperiodic nature of flapping, a pulsed actuator disk may be more representative of the hovering problem. Using the force impulse of a vortex ring, he showed that the circulation G of the vortex rings in the far wake downstream was related to the jet velocity thereby G ¼ w2 2f, where f is shedding frequency. Remembering that w ¼ 2w i, G may be related to the vertical lifting force using Eq. (1). Having drawn inspiration from Magnan s and Ellington s [100] earlier work, Rayner [97] also proposed an analysis of insect flight based on its vortex wake. His analysis was much more detailed than Ellington s [41] but did not include the effects of stroke amplitude F or stroke-plane angle b. He represented the wake of a hovering insect by a chain of small-cored coaxial vortex rings (one produced for each halfstroke; see also [38]) that stacked one upon the other 7 (see also [101], p. 139). The approach bypassed the need to determine lift and drag coefficients and ignored the mechanism by which the vortex wake was generated. Rayner concluded that circular vortex rings could efficiently represent wakes shed by hovering insects. Sunada and Ellington [42,102] devised a more advanced method that modelled the shed vortex sheets in the vortex wake as a grid of small vortex rings. The shape of the grid was determined by wing kinematics and all forward speeds (including hover) could be handled. Collocation points were defined at the centres of these grid cells and the Laplace equation was satisfied there by requiring that the induced velocity due to the circulation of all vortex rings equalled that due to wake convection. In this way, they estimated induced power from the added mass of the vortex rings and reported good agreement with the Rankine Froude momentum theory [42,102]. As noted by Ansari [15], the simple momentum theory is rather limited as a tool for aerodynamic modelling as it singles out stroke-plane angle and disk loading as the only important parameters (Eq. (2)). It does not allow, for example, estimation of lift forces for given wing kinematics or wing geometry. Rather, mean induced velocity and mean power must be reverse-engineered from weight (or lift) using some basic kinematics. Nevertheless, it has its uses, especially when coupled with a bladeelement-type approach (see below). Although the vortex-wake method is more advanced, it has similar limitations as it essentially ignores how the vortex sheets are generated, which is fundamental to any wing-design study Quasi-steady methods The first major contribution in the area of quasisteady methods is usually attributed to Osborne [94]. He used quasi-steady aerodynamics to model insect flight the forces on the insect wing at any 7 A similar pattern of vortex rings is observed in the wake of fish [34].

14 142 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) point in time were assumed to be the steady-state values that would be achieved by the wing at the same velocity and angle of attack. Although such methods have been found to be somewhat limited in accuracy, they may offer some insight into the forces (or some of their components) generated and allow comparison between different types of geometries or kinematics [15]. From momentum theory for an ideal fluid, power required for wings to flap must equal rate of change of kinetic energy passing into the slipstream. Osborne used this to calculate average values for mean lift ( C L ) and mean drag ( C D ) coefficients. He used blade-element analysis to discretise the wing into chordwise sections and used quasi-steady aerodynamics to compute forces and power. He analysed two extreme cases one where both upstroke and downstroke contributed equally to the forces and the other where lift was produced by the downstroke alone and found that the former required much larger values for C L and C D. On this basis, Osborne concluded that in reality, most insects derived more of their forces for flight from the downstroke. He also argued that, in order to minimise power required for flight, wing tips must follow a figure-of-eight (Fig. 2). Osborne s [94] analysis yielded rather large values for C L (Oð5Þ) and C D (Oð2Þ), lift and drag being defined with respect to the mean stroke plane. He attributed these to the enormous accelerations involved because of the distinct correlation he found between p ðc 2 L þ C2 D Þ and advance ratio8 of insect flight. In 1956, Weis-Fogh and Jensen laid out the basics of momentum and blade-element theories as applicable to insect flight. However, they used pre-existing models (e.g. [94]) to analyse quantitatively data on wing motion and energetics available at the time [95]. In most cases, fast forward flight was considered and the quasi-steady assumption appeared to hold for the reason that, as flight velocity increased, unsteady effects reduced in comparison to quasi-steady ones. Weis-Fogh went on to pursue this avenue of research further [96,103]. Beginning with the simple momentum theory (see Section 3.1), he coupled it with a blade-element method to compute forces on chordwise wing segments along the wing. He used these to compute the torque Q exerted at the wing 8 Advance ratio is the ratio of flapping velocity to forward flight velocity. root and, hence, power expended [96], thus Z P ¼ f Q df, F where f is flapping frequency, F is wing stroke amplitude and Q is the sum of the aerodynamic and inertial torques. In this way, he showed that most insects must have a system for elastically storing flapping energy. Otherwise, it was metabolically impossible to sustain flight [103]. He also found that values for mean lift coefficient C L for real insects were generally much more conservative (Oð1Þ) than those predicted by Osborne [94]. The aim of the studies was to use quasi-steady methods to generate quick estimates of flight fitness for various insects and birds. Apart from Pringle s [104] suggestions on quasisteady methods for analysing insect flight (somewhat similar to [94]), this area of scientific study remained relatively dormant. It was not until the mid-1980s that Ellington s seminal work rejuvenated research into insect flapping flight [105,8,25,30,41,106]. He presented a number of theories for insect flight using actuator disks, vortex wake ([41], see discussion above), quasi-steady methods [30] and even some insight into unsteady aerodynamics [30]. Building on a blade-element method, Ellington combined expressions for lift due to translational and rotational phases. Using thin aerofoil theory and the Kutta Joukowski theorem, he put forward that bound circulation due to translation was given by G t ¼ pcu sin a, (3) where c is chord length, U is incident velocity and a is angle of attack corrected for profile shape. Using the method of Fung [107], he then derived a similar expression for circulation due to rotational motion by computing incident velocity at the 3 4-chord point and satisfying the Kutta Joukowski condition, giving G r ¼ p_ac a, (4) where _a is rotational (pitching) angular velocity and a is the point about which rotation is being made (pitch axis), normalised with respect to c. Combining the above two expressions, Ellington obtained the quasi-steady lift coefficient, thus C L ¼ 2p sin a þ _ac U 3 4 a.

15 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) The effect of any heaving motion was not included in this expression explicitly. In a later study in the context of helicopter aerodynamics, van der Wall and Leishman [108] presented a more generalised form (see also [90]), thus " h C L ¼ 2p sin a þ _ U þ _ac # 3 U 4 a, (5) where h _ is heaving velocity and U may be timevarying. Ellington also presented some methods for estimating wing circulation for rotation-based lift mechanisms, such as clap, peel and fling [30]. In order to determine lift and power requirements for hovering flight, Ellington [106] sought estimates for mean lift coefficient C L through the flapping cycle. He found an expression for mean lift coefficient based on nondimensional parameters which reduces to 8L C L ¼ 4rs 2 ðdf=dtþ 2 upon restoring dimensions, where L is mean lift through a half-stroke, ðdf=dtþ 2 is mean-square flapping angular velocity and s 2 is second moment of wing area. Some useful inferences can be drawn from this equation. For low C L, both flapping frequency and aspect ratio need to be maximised. This has been corroborated by the more complex quasi-steady model of Ansari [15, see below] and the unsteady model used in Ansari et al. [109]. In his book, Azuma [29] presented a number of techniques for modelling the flight of insects. On the basis of equilibrium considerations, he derived a number of relations from momentum and bladeelement methods. He also used some relevant unsteady aerodynamic methods, notably that of Theodorsen [82]. Although Azuma presented numerous modelling techniques, his emphasis remained on computing power requirement and its extraction, and so did not exploit fully the various models he proposed. In their more recent contribution, Azuma et al. [110] reiterated some of Azuma s [29] earlier work with a number of new additions, especially in terms of unsteady aerodynamic modelling (see Section 3.4 below). Again, they did not pursue the models presented to extract results but instead left it to others (e.g. [111]). Ansari [15] presented another variation on quasisteady models for insect-like flapping flight. Based essentially on a blade-element method, he coupled a quasi-steady aerodynamic model with a Glauerttype analysis to model the tip vortex. The latter has hitherto not been attempted in the context of insectlike flapping wings. Each wing element was subjected to the usual quasi-steady treatment but now with the addition of downwash due to the tip vortex. This has the effect of reducing local angle of attack (hence, lift) and also provides an estimate for induced drag. As in Glauert [112], circulation was represented by means of a Fourier series but modified to accommodate the radially varying nature of flapping-wing flow. The analysis required judicious use of the Glauert integrals and the expression obtained for downwash was " w i ðyþ ¼ fr _ X1 cos y sin ny na n sin y n¼1 # þ X1 ða n cos nyþ, n¼1 where y is a parametric variable of a general point along the wing. The total lift L and (induced) drag D i can then be found from L ¼ Z R R rug dr and D i ¼ Z R R rw i G dr, respectively, where circulation G has been modified to include the effect of downwash w i. On the basis of this model, the effects of various wing kinematics and wing geometry were investigated. Some of these results are shown in Fig. 6. Despite being a quasi-steady model, the approach shows particular promise especially as a means for comparing and contrasting various geometric and kinematic parameters associated with insect-like flapping wings. In a later study (in preparation), a revised form of the model using the more generalised version for quasi-steady lift in Eq. (5) is compared with the more accurate unsteady aerodynamic model developed later ([13,14], see Section 4 below). A number of other quasi-steady approaches can be found in the literature (e.g. [ ]) but, as these include semi-empirical corrections, they are discussed separately in Section 3.3 below. Support for quasi-steady methods has fallen over time, especially for describing the hovering flight of insects. This is the result of over-simplification and exclusion of essential unsteady aerodynamic effects. For example, Dudley and Ellington [116] found that for bumblebees, the quasi-steady estimates of lift

16 144 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) lift [N] 2 lift [N] (a) flapping frequency [Hz] Mean lift vs. frequency (b) wing length [m] Mean lift vs. wing length Fig. 6. Variation of mean lift for various wing shapes with (a) flapping frequency and (b) wing length using the quasi-steady model of [15]. and power requirements fail at all flight speeds. In a study by Wakeling and Ellington [117], the free flight of the dragonfly Sympetrum sanguineum and the damselfly Calopteryx splendens was analysed, and the required mean lift coefficient C L was reverse-engineered using a quasi-steady approach. In each case, C L was found to exceed the maximum lift coefficient C L max possible in steady flow. One feature that is a common drawback to a large number of quasi-steady methods is the extensive use of small-angle approximations (e.g. C L ¼ 2pa). As explained in the earlier discussion on insect wing kinematics (Section 1.1.1), angle of attack is seldom below about 35 so that making use of these approximations is unjustifiable. However, Pedersen dispensed with the approximation in the quasisteady element of his aerodynamic model [37], see also [118]. Nevertheless, quasi-steady methods cannot be discounted altogether, especially at high forward speeds. At these speeds, wings move longer distances at lower angles of attack and quasi-steady aerodynamics are able to account for the forces quite reasonably [51,119,113]. In fact, Lighthill [120] commented that the upper limit for quasi-steady methods was a reduced frequency 9 k ¼ 0:5. 9 Reduced frequency is based on Strouhal number St and is defined as k ¼ fc=2u, where f is flapping frequency, c is mean wing chord, and U is flight velocity Semi-empirical methods Simplified aerodynamic models (such as quasisteady methods) are unable to predict accurately the nonlinear forces generated during an insect s flapping cycle. A possible remedy is to introduce some form of empirical correction to allow more accurate predictions. These are described as semiempirical methods and are discussed in some detail here. Without such corrections being applied, the results do not agree well with experiments. Using simple formulae and compensating for their simplicity by empirical data has a great appeal for the user as it apparently dispenses with the complexities of unsteady, vortical flows. However, a price to pay is that the empirical coefficients lump together contributions of different types of flow (and at different times). More importantly, the predictive power of a semi-empirical model is often questionable since it does not reflect properly the relevant flow physics and relies instead on data points. Interpolating between the data points, and indeed extrapolating beyond them, may not always be valid, a critical issue for design. While the previous discussions on steady-state and quasi-steady methods followed a logical progression in Sections , the semi-empirical studies do not, in general, build on earlier works and are, therefore, addressed individually in turn. Walker and Westneat [121] presented a semiempirical model for insect-like flapping flight which they described as unsteady due to the inclusion of

17 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Wagner s [33] function and apparent mass effects. They used a blade-element method to discretise the flapping wing and computed forces on the wing elements. The forces comprised a circulation-based component and a non-circulatory apparent mass contribution. They first calculated the normal and tangential flow velocities due to wing translation and rotation at the 3 4 -chord point (v n and v t, respectively) and used these to compute the resultant incident velocity v i ¼ p ðv 2 n þ v2 t Þ. With this velocity, the semi-empirical lift dl 0 was calculated, thus dl 0 ¼ 1 2 rv2 i cfc L dr, where c is chord length, dr is section width and lift coefficient C L was empirically determined using data from Dickinson et al. [27]. A similar expression was used for semi-empirical thrust (or drag) dt. Walker and Westneat also included the Wagner function f to account for the delay in growth of lift due to the presence of starting vortices, and used the approximation due to Garrick [86] for its computation. Lift and drag were then resolved normal and parallel to the chordline as df n and df t, respectively. These were then resolved in the vertical and horizontal directions (with respect to gravity) as dl c and dt c, respectively, where the subscript c denotes circulation-based forces. The non-circulatory apparent mass force was computed from the normal velocity v n at mid-chord [107], thus df ¼ 1 4 pc2 _v i dr. This force, which acts normal to the chordline, was then resolved in the vertical and horizontal directions (as done previously for the circulation-based forces) as dl a and dt a, respectively, where the subscript a denotes forces due to apparent mass. As a simple approximation, Walker and Westneat also included the effect of skin friction by computing its additional drag 1 2 rv2 t cc sf where C sf is coefficient of skin friction computed as C sf ¼ 1:33= p Re. The total vertical force was then found by taking the sum of the circulatory and non-circulatory components, thus L ¼ L c þ L a and similarly for the net horizontal force T. Using a similar analysis, Walker and Westneat computed power requirements, dividing power into circulation-based and non-circulatory components. From these, they computed mechanical efficiency Z, thus Z ¼ TU P, where T and P are thrust and power, respectively, U is forward flight velocity and where the bar symbol denotes mean values. The aim of their work was to evaluate the relative efficiency of rowing and flapping. They validated their model against experiments with motor-driven rowing (as in fish) and flapping (as in birds or insects) wings and reported good agreement. They also found that flapping was mechanically the more efficient. Using this model, Walker [114] investigated the existence of a Magnus-effect-like force [122] due to the rapid pitching motions experienced in insect-like flapping motion. He divided total circulation G into components, thus G ¼ G t þ G r þ G M where G t was circulation due to translational motion (including heaving/plunging), G r was circulation due to rotational motion and G M was Magnus circulation. He noted that such a Magnus force was not covered by the quasi-steady circulation G r because it was independent of angle of incident flow and centre of rotation a (see Eq. (4)). Instead, it depended on rotational speed and was given by the product of added mass, translational velocity and angular rotation, thus df M ¼ 1 4 rpc2 _ f_ar dr. The expression quoted by Walker had a discrepancy in the calculation of the translational velocity term; this has been corrected in the expression above. Walker claimed that the effect of a Magnus force was minimal. Given the error in its computation identified above, these results may be inconclusive and warrant further investigation. Sane and Dickinson [113] presented a quasisteady model to describe the forces measured in their earlier experiments on the Robofly, a mechanical, scaled-up model of the fruitfly Drosophila melanogaster [36]. The experiments were carried out in a tank of mineral oil where the Robofly wing executed various insect-like kinematics. In their aerodynamic model, Sane and Dickinson decompose force F into four components, thus F ¼ F t þ F r þ F a þ F w, (6) where the subscripts are t and r for translational and rotational quasi-steady components, respectively, a

18 146 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) for added mass and w for wake capture. A bladeelement approach was used whereby the Robofly wing was divided spanwise into chordwise strips. The forces on each strip were computed individually and then integrated along the span. The translational quasi-steady forces F t were computed from empirically fitted equations from a previous study [27]. In that study, Dickinson et al. moved the wing through 180 arcs at various fixed angles of attack. After an initial startup spike, the forces more or less levelled out; these were then averaged over a fixed time period and used to fit the equations for the translational forces. In these equations, forces varied only with incident velocity U and angle of attack a. In terms of coefficients of lift C L and drag C D, these empirical fits were given by [113] C L ¼ 0:225 þ 1:58 sinð2:13a 7:2 Þ, C D ¼ 1:92 þ 1:55 cosð2:04a 9:82 Þ, where angle of attack a is expressed in degrees. In order to determine the rotational quasi-steady force F r, Sane and Dickinson set the forces due to added mass and wake capture to zero, i.e. F a ¼ F w ¼ 0, by removing any accelerations and avoiding wake re-entry, respectively. The wing was moved at constant translational and rotational speeds and for one forward stroke only. Eq. (6) reduced to F ¼ F t þ F r and F r was then evaluated from the total measured force F and the empirically predicted translational force F t. An analytical expression for added mass F a was derived in an earlier study [36] using the method of [123]. 10 Knowing these three components, Sane and Dickinson evaluated the forces due to wake capture F w from experiments with wake reentry and subtracting the components F t, F r and F a from the total measured force F [113]. In their approach, Sane and Dickinson assumed that, after accounting for the quasi-steady (translational and rotational) and added mass forces, all remaining forces were due to wake capture. As shown by others (e.g. [87,124,15]), the effect of the far wake also influences forces experienced by the 10 The expression used by Sane and Dickinson for the computation of added mass is strictly valid only for the case of pitching about the quarter-chord point, but other practical pitching positions are unlikely to change F a dramatically [15]. wings. Therefore, wake capture only accounts for some of the remaining forces. Wake capture is a function of wing kinematics and wake dissipation. Therefore, reverse-engineering it, as did Sane and Dickinson [113], only indicated its effect for the dataset in question: it could not be used to predict wake capture effects for given kinematics. The model is, therefore, somewhat incomplete from a design perspective. Nevertheless, it has provided some insight into some of the component forces associated with insect-like flapping flight. The work has also proved useful in further studies [61,125,126]. Traub [115] presented a quasi-steady method for generating estimates of the lift of a flapping-wing insect in hover. The focus of the work was on estimating time- or stroke-averaged, rather than instantaneous, forces so that the model did not account for any peaks or troughs through a flapping cycle. The method was essentially quasi-steady but with some empirical corrections to give good timeaveraging. Unlike most other approaches, the blade-element method was not employed. Rather, the effect on the entire wing is computed. Pure sinusoidal flapping motion and a mean angle of attack of 40 was assumed throughout. By simplifying the flappingwing flow to consist of two principal components attached and vortex flow Traub used quasi-steady aerodynamics to derive an analytic expression for lift. Using simple actuator disk theory, an expression for lift and average downwash at the wing surface was derived. To this downwash, an additional component due to the vortex flow was added. To compute the effect of the vortex flow, Traub used Polhamus leading-edge suction analogy [127]. First, an expression for the induced drag on the wing was derived and, from this, the leading-edge suction force was computed. The latter was then rotated through 90 in the manner proposed by Polhamus (see Fig. 7) to account for the vortex lift. Having used up the leading-edge suction force, the lift due to the attached flow was reduced by a corresponding amount. To allow for empirical input to the model, Traub defined a pseudo-advance ratio J. Because the flapping-wing system is in the hover and not advancing, this was defined using the zero-penetration condition at the wing surface so that J / c=rf where c is chord length, R is wing length and F is stroke amplitude. Using this, the total lift on the

19 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) F s wing F s attachment point spiral vortex (a) Leading-edge suction force (b) Suction force rotated through 90 Fig. 7. Polhamus leading-edge suction analogy (F s is leading-edge suction force). flapping wing in hover was then found to be L ¼ qsk þ ðqskþ3=2 cos 2 a 1 p U ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, (7) 2rR 2 F cos b where q is dynamic pressure, S is wing area, a is effective angle of attack after correcting for vortex flow, U is stroke-averaged incident velocity, b is stroke-plane angle and r is air density. K is a constant of proportionality that is a function of the pseudo-advance ratio, i.e. K ¼ f ðc=rfþ, and whose value is determined empirically. The first expression of the right-hand side is lift due to attached flow (less leading-edge suction) and the second is vortex lift. Using data presented by Weis-Fogh [103] and substituting insect weight for L in Eq. (7), Traub found that K was more or less invariant with a mean value of K ¼ 2:99 (see Fig. 8). Traub also used the model to infer the relative magnitudes of forces due to attached and vortex flow. While lift due to attached flow did not appear to vary much with incidence, the effect of vortex lift was found to be more pronounced. Traub concluded that since lift through a wing beat was not uniform, additional mechanisms, such as rotational circulation and wake capture, must play an important roˆle and have to be accounted for. The main drawback of Traub s [115] method was its dependence on small-angle approximations. It was also unclear how well the nonlinearity of the flapping-wing problem had been handled and comparison with more experimental datasets is needed. Nevertheless, the method provided a useful analytical expression for the stroke-averaged lift of a hovering insect. In a recent development, Tarascio et al. [128] presented a rather simple method for modelling the free-wake shed off an insect-like flapping wing to K compare with their experiments on such a device. As the wing flapped through air, it shed vorticity into the wake from the trailing edge, equal to the change in bound circulation around the wing. The latter was determined empirically from lift vs. angle-ofattack characteristics of a flat plate. The released vortices were modelled as vortex blobs [129] and convected at the local induced velocities. These velocities were computed for all vortices according to the Biot Savart law and convection was effected using an Adams Bashforth integration scheme. Tarascio et al. noted that as layers of opposite vorticity are laid one upon the other, a recirculating pattern developed Unsteady methods insect weight [N] Fig. 8. Values of constant of proportionality K for various insects from Traub s [115] method. In this section, analytical methods that rely purely on unsteady aerodynamics are reviewed. Key to this

20 148 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) class of methods is modelling the wake and, in particular, two separations one from the leading edge for the LEV and one from the trailing edge to model the conventional wake. A common feature of these approaches is that the twin separations are paired with the requirement to enforce the Kutta Joukowski condition at both wake-inception points. Although there are some similarities between some of the approaches, these methods are relatively new and, therefore, a progression of method development cannot be identified. For this reason, each of these independent studies is reviewed separately (like the semi-empirical methods above). Azuma [29] proposed some methods for handling the calculation of the unsteady forces and moments associated with a flapping wing based on coefficients for lift, thrust and aerodynamic moment, using results from the earlier works of Theodorsen [82] and Garrick [85]. The objective of the exercise was to calculate the thrust due to beating wings (for forward flight) and the associated power expenditure. Since no account was taken of the LEV, the method falls short of addressing insect flapping flight. In a more recent contribution, Azuma et al. [110] extended these ideas to include effects due to starting flow (Wagner s [33] function), gusts due to the returning wake (Ku ssner s [77] and Sear s [130] functions) and the effect of previous shed wakes (Loewy s [131] function). However, they failed to include any effects due to the LEV and, as noted for their work on quasi-steady methods in Section 3.2, they did not present any calculations or offer any results. In his work, Wu [132] mentioned some of the above works in the context of modelling aquatic as well as aerial animal locomotion. His discussion relied mainly on the unsteady forces associated with oscillatory motion [82] and the delay in the growth of lift [33,87], and he even made reference to some of the nonlinear extensions proposed by McCune et al. [124] and Tavares and McCune [133]. However, Wu s [132] emphasis was on aquatic locomotion and, hence, forward flight. Because of the low angles of attack, therefore, he also precluded LEV effects. Having noticed the absence of LEV effects in theoretical models for insect flapping flight, Żbikowski [57], proposed a framework that took account of this using Polhamus s [127] leading-edge suction analogy, while still keeping the essential elements of the other unsteady aerodynamics models. Pedersen and Żbikowski ([118], for full details see [37]) applied this framework to the development of a model for insect-like flapping flight in the hover. The model linearly superposed the unsteady forces due to the attached flow and the LEV while also accounting for effects due to the shed wake. The flapping wing was divided into rectangular strips that extended from the root to the tip. For each strip, the flow was solved as a 2D problem. The workings of the model are shown schematically in Fig. 9. The model was not iterative in its solution but the analytical expressions derived by Pedersen and Żbikowski [118] were made mathematically tractable using a number of simplifying assumptions. For the wing section, they used a thin, flat plate. In addition, the LEV was shed at the end of each half-stroke and was assumed to dissipate immediately. A new one was formed on the other side of the aerofoil at the start of the next halfstroke. The flow was treated as inviscid and the usual Kutta Joukowski condition was satisfied at the trailing edge. The total lift on the model was computed from three separate contributions non-circulatory lift, quasi-steady circulatory lift and unsteady (wakeinduced) circulatory lift. Quasi-steady circulatory lift was calculated from the quasi-steady lift of each 2D aerofoil section and that induced by the presence of the LEV. By satisfying the Dirichlet boundary condition, the attached flow around the aerofoil was computed using the classical velocity-potential approach, 11 maintaining the zero-through-flow condition at the aerofoil surface. Theodorsen s [82] method and the later generalisation of van der Wall and Leishman [108] (removing small-angle approximations) were used for this purpose. The effect of the LEV was computed by making use of Polhamus s [127] leading-edge suction analogy: the force vector due to the leading-edge suction was rotated through 90 on the leeward side of the aerofoil (see Fig. 7), thus adding to the lift contribution. Owing to a non-zero angle of attack, there was also a significant drag component of the calculated normal force. This method has been used successfully to model the LEV observed on delta wings [127, ]. Non-circulatory lift accounted for the added mass also set in motion due to the repeated accelerations 11 This approach is the basis of thin-aerofoil theory and is usually credited to Munk [134].

21 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Wing Geometry Wing Kinematics NON-CIRCULATORY Added Mass CIRCULATORY Attached Flow Quasi-Steady Polhamus LEV Primary Wake Wagner Unsteady Secondary Wake Loewy.. Kussner Total Lift Fig. 9. Schematic of the model of Pedersen and Żbikowski [118]. and decelerations of the flapping wing, and was based on Sedov s [123] expression for a flat plate. The wake-induced circulatory component was calculated using modified forms of existing methods. The model handled the primary wake (the wake from the current half-stroke) and secondary wakes (all other previous shed wakes) separately. In computing the effect of the primary wake, they considered the inhibitory effect of starting vortices due to startup and changes in angle of attack using Wagner s [33] function. For simplification, they used existing analytical approximations for these functions (e.g. [88]) and used the Duhamel integral ([139], see also, [140]) for their computation. To compute the effect of the secondary wake, Pedersen and Żbikowski used a modified form of the method of Loewy [131]. Loewy modelled the effect of previous wakes on a helicopter blade which, for steady flight, were taken to be vortex sheets at regular distances beneath one another. Pedersen and Żbikowski modified this to take account of the fact that consecutive vortex sheets were of opposite vorticity because they were generated from opposite motions (up- and downstrokes, respectively). The flow induced by the secondary wake resulted in a velocity vector induced at the wing section. This was decomposed into horizontal and vertical components, and handled using Wagner s [33] and

22 150 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Pedersen Pedersen Dickinson 4 Dickinson time [s] time [s] (a) Lift (b) Drag lift [N] Fig. 10. Comparison of forces from Pedersen and Żbikowski s [118] theoretical model and Dickinson s [193] experimental data. thrust [N] Ku ssner s [77] functions, respectively. All of the above effects were combined to give the total lift (see Fig. 9). The force predictions from this model were compared with experiment (see Fig. 10). Although the comparison for lift was good (9% error in mean lift; see Fig. 10(a)), capturing the essential features of the force history, Pedersen and Żbikowski [118] remarked that drag predictions were rather poor (153% error; see Fig. 10(b)). The explicit nature of the analytical expressions, however, makes for quick computation and the model has aided other design studies [141,142]. Minotti [143] presented a 2D unsteady aerodynamic model for flapping wings using an analytical approach. Using a flat plate to represent the flapping wing, the formulation was such that the wing remained at rest in a noninertial co-ordinate system, with the origin at the wing pitch axis, so that all flow was relative and moved around the aerofoil. Minotti defined the problem as a flat-plate aerofoil with a concentrated LEV above the wing. In fact, this vortex was positioned rather arbitrarily to give best correspondence with experiment and was, in some phases of the flapping cycle, closer to the trailing edge. In this way, the resultant force was always normal to the flat plate. Forces F were evaluated using the Blasius Chaplygin formula F ¼ F x {F y ¼ {r I do 2 dz þ {r q I O dz, 2 dz qt where O is complex potential, r is density and the overbar symbol implies complex conjugation. Conformal transformation was used so that the flow was evaluated in the circle (Z) plane (where it is simpler to compute) and then transformed into the physical (z) plane, as necessary, using the Joukowski transformation z ¼ a þ Z þ R2 Z, where R is radius of circle and a denotes the offset of the pitch axis from the aerofoil midpoint. Minotti formed the complex potential of the flat-plate problem using Milne-Thomson s circle theorem (see [99]), where the Neumann boundary condition (tangential flow velocity at aerofoil surface) is automatically satisfied. He then introduced a LEV at z lev and recomputed the expression for complex potential. The unbounded velocities at the two ends of the flat plate were regularised by the circulation of the LEV G lev and by the introduction of bound circulation G around the flat-plate aerofoil. The coordinates of the LEV z lev (which consist of an abscissa and an ordinate) were used to compute G and G lev simultaneously for all time, based on the instantaneous values of translational (U) and rotational (_a) velocity. To model the 3D problem of real insect-like flapping, Minotti used an approach commonly found in rotorcraft design to compute a representative translational velocity, thus U ¼ fr _ p ffiffiffiffiffi ^s 2, where f _ is sweeping angular velocity, R is wing length and ^s 2 is nondimensionalised second moment of wing area. The evaluation of chord length

23 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) was made less obvious but it is likely to have been mean chord also weighted by the second moment of wing area. With the model complete, Minotti validated it against experiment to find the best value for z lev. In that sense, the model of Minotti [143] is semi-empirical. He used results from Dickinson et al. [27] and found that z lev ¼ cð 0:3125 þ {0:12Þ gave the best fit, where the origin is located at the midchord position. Although this was the default location of the LEV, the specific location was dependent on the incidence angle of the relative airflow and, hence, could be at either of z lev or z lev, where the overbar signifies complex conjugation. Minotti postulated that the LEV was always located on the leeward side of the aerofoil and closer to the edge which made an acute angle with the incident flow. In doing so, for some part of the flapping cycle (especially during stroke reversal), this vortex appeared near the trailing edge. As a consequence of using the Blasius Chaplygin formula to compute forces, time-derivatives of bound circulation dg=dt and LEV circulation dg lev =dt appear in the computation, together with a number of other terms. Minotti speculated that values of dg=dt due to the dynamics of shed vorticity were much smaller than the corresponding changes in the conditions of the flow. Hence, it may be possible to set dg=dt ¼ dg lev =dt ¼ 0, thus making the approach quasi-stationary. Unsteady effects due to apparent mass were still included. Having calibrated the model using previous data [27], Minotti presented comparison with other data [36] and reported good agreement (see Fig. 11). While being able to capture the forces reasonably through a half-stroke, the model was unable to capture the peaks in lift (and drag) following stroke reversal (Fig. 11). Minotti attributed this to the incapability of the model to handle wake capture. He also presented comparisons with the dg=dt and dg lev =dt terms included and found that this produced superfluous peaks (Fig. 12). The plots for drag had a characteristic kink during stroke reversal (Figs. 11(b) and 12(b)) due to the switch in position of the LEV from one end and/or side of the aerofoil to the other, according to the angle of incident flow. Minotti s [143] method is rather undemanding computationally (it requires flow computation for only one wing section and does not involve expensive wake-convection algorithms) and is, therefore, attractive. At the same time, it does have some notable drawbacks. There is no method of representing vorticity shed by the wing except through changing bound and LEV circulations. For this reason, it is not capable of handling wake capture. The rather arbitrary placement of the LEV is also of concern, although Minotti has discussed its stability in a later publication [144]. Finally, it is not possible to extract flow-visualisation data from the computed results. In his paper on the unsteady separated flow of an inviscid fluid around a moving flat plate, Jones [145] used a boundary integral representation for the 1 Minotti Dickinson et al lift [N] drag [N] 0 0 (a) time [s] Lift (b) -0.5 Minotti Dickinson et al time [s] Drag Fig. 11. Comparison of forces between experiment [36] and Minotti s [143] model with dg=dt ¼ dg lev =dt ¼ 0.

24 152 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Minotti Dickinson et al lift [N] drag [N] 0 0 (a) time [s] Lift (b) -0.5 Minotti Dickinson et al time [s] Drag Fig. 12. Comparison of forces between experiment [36] and Minotti s [143] model with dg=dt and dg lev =dt terms included. velocity field. He claimed that the method had advantages over conformal mapping in that it could be extended more naturally to three dimensions and could handle flexible wings. Although the model was not developed for insect-like flapping (and cannot handle it), it is worthy of mention in passing since it considers a very similar flow. In fact, the startup problem is identical to that found in insect flapping wings. Jones posed the problem of a flat plate with two separations using three connected vortex sheets one bound to the flat plate and two free vortex sheets (one each emanating from the leading and trailing edges) and defined their evolution in terms of their velocity field q at any point z using a boundary integral of the form qðz; tþ ¼ 1 I I g tew 2p{ tewz z dz þ g foil foil z z dz I g þ lev z z dz, lev where g and z denote vortex strength and location, respectively, and where the subscripts foil, tew and lev refer to the flat-plate aerofoil, trailing-edge wake and leading-edge vortex, respectively. Note that the dependence of the integrals (which are defined in the sense of the Cauchy principal value) on time t and of the vorticity on time and location have been omitted for conciseness. The problem of finding the function qðz; tþ was then posed with the usual boundary conditions that the flow was discontinuous across the vortex sheet, that the zero-throughflow condition on the plate was satisfied, that the perturbation at large distances from the plate was zero and that q was bounded everywhere together with Kelvin s theorem that total circulation is conserved. The resulting algebra is rather involved and eventually produces an expression for qðz; tþ which is singular at the two edges of the flat plate. Restoring boundedness finally results in the pair of simultaneous equations p_a 2pð l _ sin a þ h _ cos aþ I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! z þ R lev z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dg lev z lev z I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! z R tew z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dg ¼ 0, ð8þ tew z tew z where z refers to the co-ordinates of the leading and trailing edges, respectively, l _ and h _ denote lunge and heave velocity, respectively, where R implies real part of and where circulation has been made the Lagrangian variable, thus dg ¼ g dz. Again, time- and location-dependence of the variables have been suppressed. Jones used a modified form of the unsteady Bernoulli equation (using the time rate-of-change of circulation and derived from the Euler equation) to form an expression for the pressure difference across the plate. From this, expressions for force and moment were derived. He also decomposed force (and moment) into three components a quasi-steady one, an unsteady one and a wakeinduced one. Apparent mass effects were not given

25 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) explicitly but, rather, were absorbed in the unsteady component. The formulation of the problem was such that, initially, only g tew and g lev were known and computed for all later times as part of the solution. All other quantities are unknown for all times and computed as part of the solution. This was done by means of a time-marching algorithm that tracked the evolution of the vortex sheets according to the Rott Birkhoff equation [146,147]. In this way flow visualisation was generated automatically. Due to the ill-posedness of the differential equations developed, Jones used a vortex-blob method to discretise the vortex sheets. He used a weighted trapezoidal quadrature rule to integrate the relevant quantities over the free vortex sheets. For the integrals over the flat plate, a Chebyshev polynomial series was used so that the integrals were evaluated exactly. Jones validated the model against the experimental work of Keulegan and Carpenter [148] on a fixed plate in a fluid oscillating sinusoidally and normal to the flat plate. He reported some agreement for his validation case of Keulegan Carpenter number 3.8 and Re ¼ (see Fig. 13). Although the force predictions were of the correct order, there was a phase difference which was attributed to the different conditions in each case: while the experimental measurements were taken once the flow had become time-periodic, the theoretical predictions included an impulsive startup which gave rise to the delay. Jones s [145] method is not suitable for insect-like flapping flight because his numerical scheme cannot normal force Jones Keulegan & Carpenter time Fig. 13. Comparison of Jones s [145] method with the experiment of Keulegan and Carpenter [148]. handle the wing interacting with its own wake. This is a fundamental drawback in terms of insect-like flapping-wing applications where such interactions are ubiquitous (see Fig. 4). Further, the occurrence of a numerical event prevented the time-integration of the evolution equations beyond a certain finite time [145, Section 7.1], presenting yet another limitation. The method is also not capable of handling thick or cambered aerofoils. On the premise that each half-stroke of insect flapping begins with the wing at rest and then accelerating, Pullin and Wang [149] studied the flow and forces associated with a flat-plate aerofoil during such a startup manoeuvre. They presented a completely theoretical investigation of the problem of a 2D wing at various angles of attack in inviscid starting flow using two approaches an analytical model and a CFD one. Due to our interest in aerodynamic modelling, we shall focus on the former here. Pullin and Wang used the similarity approach whereby vortices are modelled using spiral vortex sheets. They argued that, for small times, this scheme gave more accurate results. The approach was somewhat similar to that used by Graham [150] to model the starting vortex shed by a wing accelerating from rest. It differs from the latter, however, in that two vortex sheets are shed one each emanating form the leading and trailing edges to depict the flow observed on insect wings. By using conformal mapping, Pullin and Wang mapped the flow around the flat plate to that around a circle. Then, by introducing the complex potential, they formed expressions for the force on the wing-wake system. In doing so, they derived separate expressions for the force due to the attached flow and that due to the two vortex sheets. The evolution of these two wakes was governed by the Rott Birkhoff equation where circulation G was used as the Lagrangian marker. The resulting vortex-dynamics equation was highly nonlinear and would normally warrant solution by numerical methods due to the complex nature of the integration kernels involved. Instead, Pullin and Wang used an analytical approximation in the form of a similarity expansion. Such an approximation, however, limits the range of validity of the solution. Pullin and Wang state that it is valid only for sufficiently small times such that the maximum distance of any point on a vortex sheet from its point of inception is much smaller than half the chord length. The similarity expansion then

26 154 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) yielded expressions for the evolution of the vortex sheet and its circulation. Pullin and Wang also investigated the problem using a different method whereby the two wakes were represented by point vortices joined to the leading and trailing edges with branch cuts (as in [150]), instead of using the Rott Birkhoff equation. Such a method was originally used in the context of delta wings [151], see also, [152,153] but has since been used in many other applications [154,150]. Again, by using a similarity expansion, explicit solutions were formed. Comparing their results with Navier Stokes calculations on a 2D elliptical section, Pullin and Wang found some agreement. They also found that maximum vortex-lift occurred at an angle of attack of a 52:2 while that for attached flow occurred at a ¼ 45 so that angle of attack for maximum lift was in the range 45 oao52:2. A number of features of their method are of concern. Pullin and Wang essentially validated their method against their CFD calculations (although they had validated their CFD method previously against experiment [125] but with mixed results). Due to a limitation in the CFD method, the flow around an ellipse was used to compare with the case of the flat plate. Pullin and Wang note that their analysis was strictly only valid for distances such that the length of the shed vortex sheet was much smaller than the aerofoil semi-chord. They nevertheless used the method to compute the lift generated by a wing through a half-stroke where the wing travelled a distance of 5 semi-chords. They plotted some of their predictions together with experimental data from Dickinson and Go tz [52] but the comparison was poor. The method of Pullin and Wang [149] is essentially a sophisticated form of an impulsive start where the wing does not move back into its previously shed wake as in insect flapping flight (see Fig. 4). Because of these concerns, the method is of limited use in application to insect-like flapping flight. It is still useful, however, in providing insight into the acceleration phase of such wing motions. Yu et al. [155] presented a purely analytical, unsteady aerodynamic method with no empirical fixes, similar in some ways to the approach used by Ansari et al. [13,14]. Following loosely the approach outlined in Żbikowski [57], Yu et al. represented the insect flapping-wing problem by a flat plate with separated flow at both edges. They solved the Laplace equation for unsteady inviscid flow and, in doing so, modelled the wing as an array of sources (or sinks, depending on sign) while the separated flow comprised of free point vortices. The 2D wing was given translational (sweeping) and rotational (pitching) degrees of freedom. A timemarching algorithm was employed whereby a pair of vortices one each at the leading and trailing edges was introduced at each time-step and the wake due to the free vortices was convected. The velocity potential j was formed from two perturbation potentials that due to the sources (or sinks) j 1 and that due to the vortices j 2 such that it satisfied simultaneously the Kutta Joukowski condition at the leading and trailing edges. With the aid of conformal mapping, Yu et al. found expressions for j 1 and j 2. The Kutta Joukowski condition was imposed simultaneously at the leading and trailing edges by equating velocity there to zero, thus giving the pair of equations qj 1 qy þ qj 2 qy þ q qy R X 2 n¼1 G n 2p{ ln ðc=4þe {y r n e {yn ðc=4þe {y c 2 =ð16r n e {y nþ ¼ 0; y ¼ 0; p, ð9þ where c is chord length and where the bracketed term concerns the newly shed vortices in which n refers to the new pair of vortices added, y is angular displacement on the circle, ðr n ; y n Þ and G n are the polar co-ordinates and the unknown circulation of the newly added vortex. These were solved simultaneously for the latest trailing-edge wake and LEV circulations. The new vortices were then absorbed into the j 2 term and all free vortices were convected at the local induced velocity given by rðj 1 þ j 2 Þ, where r is the gradient operator. Forces were computed using Kelvin s method of impulses [156, see Section 4.1.2], and Yu et al. divided these into three components depending on their origin: source (or sink) distribution, LEVs and trailing-edge vortices. Although the net effect of the presence of the LEV was to increase lift, decomposing the forces showed that the component of lift due to the LEVs was actually negative for a significant portion of the wing stroke while that due to the trailing-edge vortices remained positive throughout. Ansari [15] also made a similar observation. Yu et al. offered some comparisons with experiment. The case of an impulsively started flat plate studied by Dickinson and Go tz [52] was used first!

27 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Yu et al Dickinson & Gotz 4 3 Yu et al Dickinson et al 2 C L 2 C L (a) time [s] normalised time with Dickinson and Götz (1993) (b) with Dickinson et al (1999) -1 Fig. 14. Comparisons of lift coefficient C L from the model of Yu et al. [155] with experiments of Dickinson et al. and Yu et al. reported good agreement (see Fig. 14(a)). The transient associated with the impulsive start was captured well but as the aerofoil velocity steadied, the general trend was that predictions for lift coefficient C L were over-estimated by their model. Ansari [15] also reported a similar finding. Yu et al. also showed results for insect-like flapping motion by comparing with the experiment of Dickinson et al. [27, see Fig. 14(b)]. Again, they found good agreement but, crucially, failed to comment on how their model was made 3D. Although their results showed both under- and over-estimation, the forces were of the correct order and more or less in phase with experiment. No comparisons for drag were presented in either case. The kinematics used were all estimated and, therefore, may be of dubious accuracy. In a later development, they have used the model to investigate stroke asymmetries and slow forward flight [157]. It would appear that a purely 2D version of the model was used (they abstained again from commenting on this aspect). In Żbikowski [57] a number of methodologies for the aerodynamic modelling of insect flapping flight were proposed in outline. While restricting himself to hover, Żbikowski laid out some fundamentals for attacking the problem using a circulation-based approach. Ansari et al. [13,14] adopted and significantly advanced that approach, culminating in the development of a model capable of predicting, with good accuracy, the forces (see Fig. 22) on insect wings in the hover (for full details, see Ansari [15]). The method was also successful in providing comparison of flow visualisation against experiment (see Fig. 20) and is described in more detail in Section 4 below Other methods In a review of this scope, one should mention some of the less analytical aerodynamic models, such as panel methods, found in the literature. There are also other techniques that model only certain aspects of insect flapping flight. In this section, we review some of these methods. The origins of modern aerodynamic panel methods are usually traced back to [158] and [159], who used a distribution of sources to model the steady flow past a body of revolution. Since then, panel methods have been extended in several ways (by the addition of circulation, for instance) to model a plethora of flows, both steady and unsteady. They are sometimes preferred over the more analytical approaches due to their simplicity of formulation. For example, Zdunich [160] arrived at similar results to the analytical approaches discussed above for the unsteady separated flow around a thin aerofoil but without recourse to complex algebra. However, their computational implementation can be rather cumbersome. Also, it is generally not possible to extract component forces (as in Eq. (4.1.2) below) and, hence, gain extra insight into the flow physics. Details of most current panel methods can be found in Katz and Plotkin [161]. In the context of insect flight, however, the application of panel methods has met with limited

28 156 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) success, owing principally to the difficulties of modelling the leading-edge vortex and the returning wake. Nevertheless, a number of studies can be found in the literature. Vest and Katz [162], for example, used a 3D, incompressible, potential flow panel method to study bird flight. They modelled the wing as a tessellation of vortex panels that shed a wake from the trailing edge while enforcing the Kutta Joukowski condition there. The flapping motion was sinusoidal in pitch and heave, and there was no stroke reversal. They investigated the case for high advance ratios (J ¼ 4:31) and reported good results. For flapping flight at slow forward speeds, Vest and Katz noted that the pressure distributions at the leading edge were such as to create a high suction peak there, thereby inherently causing the flow to separate. This would violate the assumptions of their method (which did not model the LEV) so they introduced dynamic twist along the wing span. They used various values of twist coefficient to study the efficiency of the flapping wing. In a somewhat similar study, Smith et al. [163] used an unsteady panel method to study the aerodynamics of a moth wing in forward flight. The model was validated against some quasi-steady calculations (which have their limitations; see Section 3.2 above) and reported some correlation. In a related study, Smith [164] extended the model for flexible wings with the introduction of a finiteelement formulation into the panel method. However, he validated the model against another panel method (for pitching and vertically plunging aerofoils, from [161]) but conceded that the work only laid the bare foundations for more advanced studies in the area. In a more recent study Fritz and Long [165] presented a vortex lattice method that differed from previous models by taking into account vortex stretching, free-wake relaxation and vorticity dissipation. However, the absence of a LEV renders the model unusable for hovering insect-like flapping flight. Eldridge [78] used a viscous vortex particle method to study flapping-wing flows but these were not quite insect-like: he used an elliptical section for the wing which only oscillated in pitch and heave (no reciprocating sweeping motion). Vortex particles were laid initially on a regular grid but then allowed to move at the local induced velocities according to the Biot Savart law. Both normal (zero-through-flow) and tangential (no slip) wall boundary conditions were enforced. The viscous form of the Navier Stokes equations was used, thus do dt ¼ ~u roþnr2 o, where the first term on the right-hand side is the convective term and the second is the diffusive term. A viscous-splitting algorithm was implemented whereby the convection of each vortex particle was computed first, followed by its diffusion using a 1.5 clap 1 wings g (α cf ) 0.5 2α cf fling α cf [ ] (a) Clap-and-fling (b) g (α cf ) Fig. 15. Lighthill s [167] mechanism of the clap-and-fling characteristics.

29 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) particle-strength exchange mechanism (see e.g. [166]). The reason for including it here is because it differs from conventional mesh-based CFD methods in that it is essentially grid free. Eldridge presented force prediction and flow-visualisation for some synthetic flapping kinematics but did not show any validation against experiment. Also, due to a modelling limitation, an elliptical section was used to represent the flapping (heaving and pitching) aerofoil. There are some aerodynamic modelling techniques that have been confined to certain aspects of insect-like flapping flight. For example, Lighthill [167] analysed the aerodynamics of Weis-Fogh s [103] clap-and-fling mechanism (see Fig. 15(a)) for insects (as in butterflies) using potential methods. He used a conformal mapping technique and found that the circulation developed by the wings G was given by G ¼ O 2 cgða cf Þ, where O is the fling angular velocity, c is wing chord, a cf is the half-angle between the wing pair and gða cf Þ is a symmetrical function that was smallest (but not zero) for a cf ¼ 90 (see Fig. 15(b)). Lighthill s [167] method was essentially quasisteady so that no vortices were shed during the fling, although the function gða cf Þ accounted somewhat for such an effect. Edwards and Cheng [168] applied their vortex-and-branch-cut method for delta wings [151,153] to extend Lighthill s [167] method to model directly the effect of shed vortices. More recently, Iima and Yanagita [169] used a discrete vortex method to tackle this problem. 4. The method of Ansari et al. This section presents the model of Ansari et al., which appears to be the most comprehensive and successful modelling approach for insect flapping flight to date. The essence of the method is given in Section 4.1, its numerical implementation is described in Section 4.2 and validation in Section 4.3. The remainder of this introduction provides the theoretical context of the model by summarising the most relevant approaches to unsteady aerodynamic modelling. The concept of unsteady aerodynamics first received major attention when the problem of wing flutter was encountered early in the 20th century. Notable early works on the topic were by Theodorsen [82] and Garrick [85]. Although Glauert [81] was probably the first to consider the airloads on an oscillating aerofoil, pioneering work in this field is usually attributed to Theodorsen. 12 He used potential-flow methods to derive relations for the unsteady forces and moments experienced by an aerofoil executing simple harmonic motion in pitch and heave (plunge). Earlier on, Wagner [33] considered another important problem in unsteady aerodynamics the impulsive start of a thin aerofoil. He derived the socalled Wagner function which estimates the lift generated by such an aerofoil. The Wagner function, however, does not have a convenient analytic form due to the complex nature of the integral equation involved, and simple analytical relations have been proposed to approximate the function (e.g. [86,89]). Ku ssner [77] considered a similar problem the transient flow associated with a sharp-edged gust. He used a method similar to Wagner to derive the so-called Küssner function (which was later rectified for an error by von Ka rma n and Sears, [87]). But again, due to the complexity of the equations and lack of closure, exponential fits have been suggested (see e.g. [140]). All the above analyses used a velocity-potential approach to tackle the problems associated with unsteady flow, which brought with it a lot of complexity. So much so, that Morris remarked that the general formulae obtained thus far had become so complex that they failed to convey the underlying physical significance [170,171]. In view of this, von Ka rma n and Sears [87] introduced the circulation approach. They represented the surface of a thin, flat-plate aerofoil by a thin vortex sheet a system of vortices with a continuous distribution of vorticity (this property differentiates this method from a panel method; see e.g. Katz and Plotkin [161, Chapter 9]). As alluded to earlier, Żbikowski [57] presented a modelling framework for insect flapping flight and, in particular, highlighted the circulation-based approach. This approach built on the seminal work of von Ka rma n and Sears [87] but with certain generalisations and improvements. Ansari [15] advanced this approach significantly in a number of ways to develop a successful model for hovering flapping flight. This is described below. 12 We also find in the literature that similar work was also carried out independently by Keldysh and Lavrent ev [83] and Sedov [84].

30 158 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Description of the method As noted already, the history of unsteady modelling using the circulation approach is usually traced back to the seminal work of von Ka rma n and Sears [87] who considered the case of an aerofoil in nonuniform motion with a flat wake shed from the trailing edge. McCune et al. [124] extended the model to include a nonlinear wake. They considered the case of aerofoils in severe manoeuvres and removed any small-angle approximations. In a somewhat related study, Tavares and McCune [133] studied the unsteady motion of delta wings. These have two separation points, one from each leading edge. Insect flapping flight involves severe unsteady manoeuvres, large angles of attack and, hence, separation from both leading and trailing edges (see Section 2.1). Żbikowski [57], therefore, suggested a circulation approach that consolidated these previous works. Working along these lines, Ansari et al. [13,14] modified and extended the circulation approach for modelling insect-like flapping wings in the hover (for full details, see [15]). It is based on the original approach of von Ka rma n and Sears [87] together with the nonlinear extensions proposed by McCune et al. [124] and Tavares and McCune [133] but with further significant extensions which will become apparent below. Ansari et al. [13,14] considered the case for hovering flight Methodology The problem of insect-like flapping was realised using an aerodynamic modelling approach. Associated with this were a number of physical and aerodynamic flow characteristics and the resulting assumptions. These are discussed in turn below. The model developed is quasi-three-dimensional by means of a blade-element approach, the problem was reduced essentially to an array of 2D wing (aerofoil) sections extending from the root of the flapping wing to the tip. In each 2D crossplane, the flow was analysed individually and the combined effect was obtained by integrating along the span. Because of the rotational nature of flapping motion, velocities and distances increased radially from root to tip. Although the true nature of insect-like flapping gives rise to a spherical problem, the formulation used here reduced it instead to a cylindrical one. Apart from simplification, this also had the advantage that hover was handled automatically. Two physical properties are + centre of rotation Fig. 16. Radial chord. sweeping motion a direct consequence of this radial chords and cylindrical cross-planes and are described now. For wings of high aspect ratio, such as blades in helicopter rotors, the ratio of blade area to disk area (the so-called solidity) is very small (about 7.5% for a rotor with four blades of aspect ratio 17) and it is reasonable to assume that the blade elements are straight (i.e. normal to the rotor spar). In the case of flapping wings such as those of insects, solidity is much higher (about 39% for the hawkmoth used by Wakeling and Ellington [172] and 55% for the fruit fly scaled-wing used by Birch and Dickinson, [45]) and the straight-chord assumption is less tenable. Instead, as also noted by Ansari et al. [173], the concept of a radial chord is more suitable so that each section of the wing chord still sees a normal incident velocity (Fig. 16). A consequence of combining a blade-element method with the concept of radial chords is that consecutive wing sections lie in cylindrical crossplanes (Fig. 17a). In the approach used by Ansari et al., each cylindrical cross-plane was unwrapped into an equivalent flat cross-plane (Fig. 17b) where the flow was solved as a 2D problem. The flow around a 3D flapping wing was treated in a quasi-3d manner by adopting a blade-elementtype approach. In each 2D section, the aerofoil was represented by a continuous distribution of bound vorticity. 13 Two wakes were shed in the form of free vortex sheets one each emanating from the leading and trailing edges which were also continuous distributions of vorticity. The flow was assumed to be entirely inviscid but viscosity was introduced indirectly through flow separation and the Kutta Joukowski condition at the points of inception of the wakes. The flow was assumed irrotational 13 This property differentiates the method from panel methods where the aerofoil is divided into panels each with two distinct points one for vortex position and one for collocation point (see e.g. [161], Chapter 9).

31 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) centre of rotation φ Φ wing section ξ η η ~ ~ ξ cylindrical wing- wing section η ξ ~ η ~ ξ (a) Actual 'wrapped' cylinderical cross-plane (b) cylinderical and cross -planes Fig. 17. Cylindrical and flat cross-planes. everywhere except at solid boundaries and in the free vortex sheets. The flow was solved for using potential methods by satisfying the kinematic boundary conditions (zero-through-flow on the aerofoil surface), the Kutta Joukowski conditions at the wake-inception points and by requiring that the total circulation in a control volume enclosing the system must remain constant (Kelvin s law; see e.g. [32]). The exact 3D nature of the LEV found on insect flapping wings has remained elusive. Whereas Ellington et al. [44] reported spanwise velocity in the LEV, Birch and Dickinson [45] observed only weak entrainment. The model of Ansari et al. [13] was quasi-three-dimensional in that it could not treat spanwise flow. Hence, there was no communication between adjacent wing sections, a view supported by Glauert [112] in the context of propellers. An extension of this assumption was that tip-vortex effects were also ignored. The validity of this assumption was established a posteriori it was found that the loss of lift (and thrust) due to 2D vortex breakdown in the outboard sections of the wing was equivalent to the drop in lift (and thrust) that would have been experienced in those regions due to tip-vortex effects Analysis Having formulated the methodology for tackling the insect-like flapping problem for hover, Ansari et al. [13] set out to solve for the flow using potentialflow methods. A conformal mapping technique with the following transformation: z ¼ Z þ ð1 ÞR2 Z þ R3 2Z 2 (10) was used to map the flow in the physical (z) plane to that in the circle (Z) plane, where R is radius of circle in the Z-plane. is the aerofoil-shape parameter and given by ¼ðt {sþ=r where t and s are thickness and camber parameters, respectively. For a flat-plate aerofoil, t ¼ s ¼ 0. The advantage of using this transformation was that arbitrary Joukowski-type aerofoil shapes (with thickness and camber) could be handled (see also [174]) and conjugate function theory (see e.g. [175]) could be used to evaluate the unsteady Neumann boundary condition at the aerofoil surface. The main driver for using a conformal transformation was the simplicity of the problem in the circle plane. As necessary, relevant elements (e.g. wake convection and flow visualisation) were converted into the physical plane where they had more meaning. By assuming that the flow is inviscid and incompressible, the Navier Stokes equations reduce to the Euler equation and potential-flow methods can be used. Further, if the flow is assumed to be irrotational everywhere (except at solid boundaries and discontinuities in the wake) then it can be solved for using Laplace s equation, which in complex-number notation requires that r 2 ðj þ {cþ ¼0,

32 160 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Flapping Wing Aerodynamic Model Quasi-Steady Component Unsteady Component Freestream Unsteady Motion Leading-Edge Vortex Trailing-Edge Wake Fig. 18. Components of the flapping-wing model of Ansari et al. [13]. where j is velocity potential and c is stream function. The nature of Laplace s equation means that the principle of superposition can be applied. In other words, the effects of various components contributing to the fluid motion may be computed separately and the overall effect obtained by taking their sum. This was the cornerstone of the approach of Ansari et al. In their approach, Ansari et al. exploited the linearity of Laplace s equation to describe the flow associated with an insect-like flapping wing. The problem was subdivided into wake-free and wakeinduced components (see Fig. 18). The former was subdivided further into effects due to free-stream velocity and those due to the unsteady motion of the flapping wing itself. Since these excluded any wake contribution, they were collectively termed the quasi-steady components. The wake-induced components were also further subdivided into the contributions from the leading-edge vortex and the trailing-edge wake (Fig. 18). These components induced the remaining forces and moments on the wing and were henceforth referred to as the unsteady components. The wing was given three degrees of freedom lunge (horizontal), heave/ plunge (vertical) and pitch (angular) motions and this unsteady nature of the problem made each of the component contributions functions of time t. Using Milne-Thomson s circle theorem (see [99]), quasi-steady circulation due to free-stream flow was calculated by satisfying the Kutta Joukowski condition at the trailing edge. The Neumann boundary condition was used to compute quasi-steady circulation due to the unsteady motion again by simultaneously satisfying the Kutta Joukowski condition at the trailing edge. As a result, the total quasi-steady circulation G 0 was found to be G 0 ðtþ ¼2p 2R ðð lðtþ U _ 1 Þ sin aðtþþ hðtþ _ cos aðtþþ þ_aðtþ 1 2 t2 þ 1 2 s2 2R ðr þ aþ, ð11þ where U 1 is free-stream velocity (U 1 ¼ 0 for hover), l _ and h _ are lunge and heave velocity, respectively, a and _a are angle of attack and pitch rate, respectively, a is position of pitch axis aft of the leading edge, and t and s are aerofoil-shape parameters. A rather complex formula for quasisteady surface vorticity was also derived Constraint equations. The shedding of two wakes one each from the leading and trailing edges disturbs the zero-through-flow boundary condition and the Kutta Joukowski condition. In addition, since flow is also separating from the leading edge, flow must stagnate there. This follows from the fact that there is no load across the vortex sheet emanating there (free vortex sheets are unable to sustain Kutta Joukowski forces [176]) and, hence, local velocity must be zero. In order to restore the Neumann boundary condition, image vortices were added inside the aerofoil (or circle in the Z-plane). More vorticity was then added on the aerofoil surface to satisfy the Kutta Joukowski condition at the trailing edge, while satisfying Kelvin s law that total circulation must remain unchanged. The result of this analysis yielded the first constraint equation [13], thus I G 0 ðtþ ¼ R Z tew þ R g tew Z tew R tew dz tew I þ R Z lev þ R g lev Z lev R lev dz lev, ð12þ

33 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) where g is vorticity and the subscripts lev and tew refer to the LEV and trailing-edge wake, respectively, and where the integrals are defined in the sense of a Cauchy principal value. This is the general wake-integral equation for the flappingwing problem and is also the circle-plane rendering of a generalised, nonlinear form of Wagner s [33] equation. In his work, Wagner presented a simple, linear form of this equation. Later, von Ka rma n and Sears obtained the same linear form but using the circulation approach [87, Eq. 38]. More recently, McCune et al. [124] generalised the form of the equation derived by von Ka rma n and Sears to include the effect of a nonlinear deforming wake [124,133]. Eq. (12) is a further generalisation that is still nonlinear but now includes the effect of a second wake the LEV. It is also a statement of Kelvin s law that the vorticity developed around the aerofoil must be reflected in the same amount of vorticity shed into the wake. As noted earlier, separation at the leading edge implies stagnation there. By enforcing this condition while upholding Kelvin s law, Ansari et al. [13] derived a second constraint equation, thus 1 A 1 A 2 þ 1 R 2 A 3 þ A 4 2U 1 sin aðtþ ¼ 1 I R Z tew R g 2pR Z tew þ R tew dz tew where I þ tew g lev dz lev, ð13þ R Z lev R lev Z lev þ R A 1 ¼ sð _ l cos a _ h sin aþþð2r tþð _ l sin a þ _ h cos aþ þ aðt 2R Þ_a, A 2 ¼ sð l _ cos a h _ sin aþþtð l _ sin a þ h _ cos aþ at_a, A 3 ¼ 4R ðt R Þ_a, A 4 ¼ 1 2 ðt2 þ s 2 4R tþ_a, where t and s are thickness and camber parameters, respectively (see Section 4.1.2). Eqs. (12) and (13) are two simultaneous equations that were used to solve for g tew and g lev at any point in time t and, together with Eq. (11), they fully describe the vortical-wake system. They are analogous to Eqs. (8) and (9) met earlier. They are exact (within the limits of the assumptions made) but highly nonlinear. Solutions were, therefore, found by numerical methods. In integrating them numerically, the values of previously-shed vorticity (g tew and g lev ) and their respective locations must be known. Ansari et al. [14] used a discrete vortex method to implement a solution to these equations (see Section 4.2 below) and the locations of the vortices served as the Lagrangian markers Z tew and Z lev in the above integral equations. It was, therefore, also necessary to compute the paths described by these discrete vortices. This was done by implementing vortex convection according to the Rott Birkhoff equation Forces and moments. Computing forces and moments for unsteady flow can be approached in a number of ways. For incompressible and irrotational flow, common methods include the unsteady form of the Bernoulli equation and the Blasius Chaplygin formula. The method adopted by Ansari et al. [13], however, was the momentumbased method of vortex pairs used by von Ka rma n and Sears [87]. This method often referred to as the method of impulses was first suggested by Kelvin [156] and has since been employed by many workers [32,87,175,177]. The idea is that for every shed vortex, there exists an equal and opposite vortex on the wing and the two constitute a vortex pair (see Fig. 19). The momentum per unit span of the system can be expressed by the sum of the momentum of the vortex pairs that constitute the system from which force and moment data can then be extracted. bound vortex impulse wake vortex freestream -Γ x +Γ Fig. 19. Vortex pairs.

34 162 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Although the formulation may appear different from the other two approaches mentioned earlier, they are all derived from the same Euler equations. Ansari et al. derived the relevant equations in the vector formulation, converted the results to complex-number notation and finally transformed them into a co-ordinate system whose origin moved with the pitch axis of the flapping wing but whose orientation remained fixed with respect to gravity. They found, for force F FðtÞ ¼{r d I ^zgð^zþ d^z (14) dt and for aerodynamic moment M MðtÞ ¼ r I I d j^zj 2 gð^zþ d^z I {ru ^zgð^zþ d^z, 2dt (15) where r was air density, U was incident velocity of the moving co-ordinate system (denoted by the caret ^ ), g and z denoted strength and location of vorticity, respectively. By dissociating vorticity into its constituent elements, Ansari et al. decomposed force into four components, thus FðtÞ ¼F 0 ðtþþf 1 ðtþþf 2 ðtþþf 3 ðtþ, where F 0 ðtþ ¼ {ru 0 G 0, F 1 ðtþ ¼{r d Z ^zg dt 0 ð^zþ d^z, foil F 2 ðtþ ¼{r d Z ^zg dt 1 ð^zþ d^z, foil F 3 ðtþ ¼{rU 0 G 0 þ {r d Z ^zg dt tew ð^zþ d^z tew þ {r d Z ^zg dt lev ð^zþ d^z. lev If the effect of the LEV is removed, then the expression above reduces to the form derived by McCune et al. [124] for a nonlinear deforming wake. Further, if small-angle approximations are introduced and the shed wake is assumed to be planar, then the form presented in the classical work of von Ka rma n and Sears [87] is obtained albeit without the F 3 -term (in their approach, the relevant components in F 2 and F 3 above were combined into the F 2 -term). By defining lift L and drag D parallel and perpendicular to gravity, respectively, Ansari et al. wrote F ¼ D þ {L from which lift was dissociated as LðtÞ ¼L 0 ðtþþl 1 ðtþþl 2 ðtþþl 3 ðtþ. Here, L 0 referred to the quasi-steady element of lift, generated in the absence of any wake. L 1 denoted contribution due to apparent mass. The change in bound circulation (and hence, lift) due to the presence of the LEV and the trailing-edge wake was responsible for the L 2 -term. Finally, L 3 represented the lift deficit experienced in the presence of the two wakes due to changes in wake vorticity. L 0, L 2 and L 3 were termed circulatory forces because they depended on circulation while L 1 was described as non-circulatory as it is the result of impulsive pressures alone. Drag was also decomposed similarly. In accordance with d Alembert s paradox [178], quasisteady drag D 0 would be zero in the absence of heaving motion. (Since drag was defined as the horizontal force, in the sense of being perpendicular to gravity, heaving motion could generate a force that had a component in this direction, giving rise to a drag force inspite of d Alembert s paradox.) However, the separated flow on the flapping wing gives rise to a normal force which ensures that the sum contribution of D 1, D 2 and D 3 was generally non-zero. Aerodynamic moment M was also separated into equivalent components with similar meaning. Ansari et al. also defined a thrust force T which was parallel to drag but always in the direction of travel, i.e. F ¼Tþ {L. This was used for validation purposes (see Fig. 22(b)). The model accounted for the flow on the whole wing by integrating along the span: a so-called quasi-three-dimensional model with no communication between adjacent wing sections and no tip vortex, but in keeping with the radial chord methodology (Fig. 16). Sane and Dickinson [113] found remarkable similarity between 2D and 3D flapping-wing flow in the Reynolds number range , and speculated on the validity of using simple strip theory to extend 2D flows to 3D Implementation Ansari et al. [14] used a discrete vortex method to implement numerical solutions to the equations developed in Ansari et al. [13]. They formulated the solution as an initial-value problem where initial conditions were known and the flow was solved for all subsequent times. A time-marching algorithm

35 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) was employed using a simple forward-euler scheme and a fixed time-step. During stroke reversals, when kinematic changes were very rapid, sub-time-steps were used to resolve better the flow trajectories and improve accuracy. The relevant equations were first discretised (as necessary). In addition, circulation G was made the Lagrangian variable using the conversion dg g dz so that Eqs. (12) and (13) were modified accordingly. The solution procedure was as follows. All kinematics and quasi-steady vortex strengths g 0 and G 0 were pre-computed since they did not depend on the wake. At each time-step, two new vortices were introduced into the wake, one each from the leading and trailing edges. These were positioned such that they followed the trace of the shedding edge and the previously shed vortex. Having known these locations, the discrete versions of Eqs. (12) and (13) were solved simultaneously for the latest values of dg lev and dg tew. The presence of the wake modified the vorticity distribution on the aerofoil, so that wake-induced vorticity g 1 was recomputed. The impulses necessary for the calculation of forces and moments (Eqs. (14) and (15)) were stored at this stage. Finally, all free vortices in the wake were convected at the local Kirchhoff velocities (after [176]) according to the Rott Birkhoff equation (which essentially is an extension of the Biot Savart law), and the entire procedure was repeated for the next time-step. Ansari et al. used a vortex-blob method [129] and employed the kernel proposed by Vatistas et al. [179] to regularise the singularity at the centre of point vortices due to its similarity to the Lamb Oseen vortex [32,180]. Since the problem size increased by 2 at each time-step n,4n 2 computations were required to evaluate local induced velocities. In the interest of reducing computation time (CPU flops), vortex amalgamation was implemented whereby distant vortices that were of like-sign and in close proximity to each other were merged according to the usual rules [ ]. A number of numerical experiments were conducted to ascertain the best values for the various parameters, e.g. vortex core size. Ansari et al. also found that in some cases, especially during wake re-entry, spurious values for shed circulation occurred which then led to erroneous solutions. These were the result of rigidly enforcing stagnation at the wake-inception points. Situations arose when vortices in the vicinity of these points induced a velocity field that required an extraordinarily high vortex strength to enforce stagnation. In such cases, therefore, it was necessary to relax the Kutta Joukowski condition. These events also occurred during stroke reversal, leading Ansari et al. [14] to speculate that the commonly accepted smooth-flow condition at the trailing edge may be of dubious validity in such situations. 14 It should be noted that Tavares and McCune ([133], see also [190,191]) also used a vortex-blob method for their implementation. McCune et al. [124] used the discrete point vortex but distributed the shed vorticity over circular arcs using a Fourier series (see [192]). Such a method, however, gets into problems for insect-like kinematics when the aerofoil re-enters its own shed wake and cuts through these arcs. It was for a similar reason that the implementation used by Jones [145] was unable to handle the returning wake. Jones s [145] method for the computation of the wake integrals (Eq. (8)), however, was better than that used by McCune et al. or Ansari et al. [14] he used a Chebyshev series instead of a simple trapezoidal rule Validation Ansari et al. [14] established the validity of their unsteady aerodynamic model by making successful comparisons with experiments. Recalling that their model yielded both flow-visualisation and force and moment data, Ansari et al. compared output from their model with the flow-tank experiments of Dickinson and Go tz [52] and with force data from Dickinson [193]. In their experiment, Dickinson and Go tz [52] moved a rectangular wing (5 cm chord and 15 cm span) through a glass aquarium filled with a 54% sucrose solution. They traversed the wing at constant angle of attack between a pair of baffles to limit any 3D tip flow. The wing was started impulsively from rest, accelerated at 62.5 cm/s 2 to a constant speed of 10 cm/s and then rapidly brought 14 Crighton [184] considered the validity of the Kutta Joukowski condition in unsteady flows. In practice, the condition appears to hold (see e.g. [185]), especially for helicopter aerodynamics where the kinematics are not too severe [90]. The unsteady Kutta Joukowski condition was also investigated theoretically by Giesing [186] and Maskell [187] and corroborated through experiments by Poling and Telionis [188,189]. They studied the shedding of vortices from the trailing edge but none of these works studied cases as extreme as the flow associated with stroke reversals in insect flight.

36 164 ARTICLE IN PRESS S.A. Ansari et al. / Progress in Aerospace Sciences 42 (2006) Fig. 20. Comparison of flow visualisation between the experiment of (Dickinson and Go tz [52] left) and predictions from the model of Ansari et al. (right). Numbers indicate distance travelled in chord lengths. to rest after travelling 37.5 cm (or 7.5 chord lengths). The Reynolds number for the experiment was 192 (based on chord). Although they ran experiments for angles of attack ranging from 9 to þ90, flowvisualisation photographs were only provided for the case of a ¼ 45. This was the case used by Ansari et al. [14] for comparison and is shown in Fig. 20. As can be seen, there was marked agreement between theory and experiment. After one chord length of travel, flow began to separate from both leading and trailing edges (Fig. 20(a),(b)). Flow separating at the leading edge rolled-up tightly under the influence of bound vorticity while that emanating from the trailing edge rolled-up less tightly, being further from the aerofoil. This roll-up continued as the wing moved, with the LEV becoming stronger and lifting off slowly from the wing surface (Fig. 20(c),(d)). This pattern progressed as the wing continued to move, causing the now-strong LEV to induce a second trailingedge vortex that rolled-up forward on to the wing s surface (Fig. 20(e),(f)). This roll-up eventually caused distortion of the LEV and vortex breakdown