[ 221 ]AERODYNAMICS OF FLAPPING FLIGHT WITHAPPLICATION TO INSECTSBy M. F. M. OSBORNENaval Research Laboratory, Washington, 20 D.C.(Received 6 November 1950)(With Fifteen Text-figures)INTRODUCTIONThe problem of natural flight is one of long-standing interest, both intrinsically andin connexion with mechanical flight. Rayleigh (1883) was perhaps the first to givea satisfactory explanation of the soaring flight of birds, and Walker (1927) gave asatisfactory quantitative discussion of the flapping flight of birds in a particular case.Hoff (1919), using the data of Demoll (1918), attempted to bring the flight of insectsinto the domain of conventional aerodynamics by showing that lift coefficients ofreasonable value were required. He used only the linear velocity of flight to evaluatethese coefficients and ignored the motion of the wings. He also pointed out thesimilarity between the flow pattern around an insect and around a propeller oractuator disk, a similarity which must hold from momentum considerations, inde-pendently of the action of the wings (Bairstow, 1939). Demoll (1919) indicatedseveral errors and omissions in Hoff's work, so that in a number of cases the liftcoefficients required were inordinately high. Finally, a most thorough experimentalinvestigation was carried out by Magnan (1934), who was, however, unable to explainhis results theoretically.The difficulty of high lift coefficients has not been removed by other investigationswhich have indicated lift coefficients larger than those normally expected in wind-tunnel measurements. It has, therefore, been believed that insects must utilize somespecial mechanism in flight not present in conventional aerodynamic phenomena.In this paper the mechanism of insect flight will be examined in detail in order tofind some possible explanations for the high lift coefficients, as well as for othercharacteristic features of insect flight. General expressions will be obtained for theforce acting on a wing surface element moving in an arbitrary manner. This force willbe resolved into lift and drag components with respect to the relative wind, andintegrated over the surface of the wing and averaged over the period of a wing beatin order to determine the total average vertical and horizontal forces exerted by theinsect. The former is the required vertical force (weight of insect), the latter therequired thrust (drag of insect's body). Average values of the lift and drag coefficientsof the wings, CL and CD, can then be determined, since there are two known forcesgiven, and two unknown coefficients to be determined.The scalar product of the vector force on the wing element into the vector velocityof the element with respect to the insect, integrated and averaged as before, gives themechanical power exerted by the insect in flight.1EB.28 2 15222 M. F. M. OSBORNEThe absolute magnitude of the total force on the wing element, integrated amiaveraged, will give a lower bound to the force coefficient for the average total force,The power, the lift, drag and total force coefficients computed for the twenty-fiveinsects for which data is available, will, when plotted against the other parametersof insect flight such as mass and ratio of flapping to linear velocity, reveal a numberof interesting and systematic characteristics of insect flight. The flight data, sum-marized in Table i (p. ), were taken from the work of Magnan, and were supple-mented by measurements on specimens of the U.S. National Museum.For those who wish to draw conclusions from this work without verifying themathematical details, it may be said that the principal mathematical problem is tojustify replacing the instantaneous velocities by suitably chosen averages overposition and time. Once this process is admitted practically all of the qualitativeconclusions may be reached by drawing vector diagrams similar to those of Fig. 4,with these average velocities for any particular case. In fact, if the averaging preceptsare admitted, quantitative conclusions can be reached by drawing to scale, and withall small components such as v and w included, diagrams like Fig. 4.DERIVATION OF THE FUNDAMENTAL FORMULAEIt is assumed that the lift force dFs on an element of wing surface c{r)dr is given bydF.^pC^O^IlxWpfllxWyilxWl}, (1)and the drag force is given bydFp=-(|)PC^(r)rfr|lxW|2{(W-W.ll)/|W-W.ll|}. (2)The quantities which appear in these two formulae are defined as follows (see Fig. 1).Vectors are denoted by bold face, their scalar magnitude by two vertical lines.1, m, n are mutually perpendicular unit vectors parallel to the axis of the wing,perpendicular to the axis of the wing in the instantaneous plane of beating, andperpendicular to this plane, respectively. The plane of beating is not the plane of thewing, but the plane in which its axis is moving, p is the air density. W is the relativewind or velocity of the surface element of the (right) wing with respect to the air, andisgivenby W = (nQxlr)-U. (3)Q, is the instantaneous angular velocity of the wing element, and U = (o, — (V+ v), —w)is the velocity of the air with respect to the insect's body. V is the velocity of flightand v, w are the induced velocities. c(r) is the chord, r and t are the independentvariables — distance along the wing and time. In the general case all of the quantitiesappearing in eqs. (1) and (2) (except p and c(r)) are functions of r and t. If the wingis assumed not to twist or bend, the unit vectors and Q will be independent of r.CD evidently represents the profile drag coefficient, since the relative wind includesthe induced velocities.Average values for the induced velocities v, w, or velocity increments of the slip-stream, are given from momentum theory (Durand, 1935). They are independentof the mechanism of the wing action, and are obtained in first approximation byAerodynamics of flapping flight with application to insects 223requiring that v and w at the insect have one-half the final value necessary to providethe necessary vertical force and horizontal thrust:Lift (vertical force): L = Mg = nR2(V2 + w2)*210, (4)Thrust: T=(\)CDbSbPV* = TrR?{V2 + w*?2v. (5)R is the length of the wing, M is the mass of the insect, g is gravity, and Sb the bodycross-section area and CDb the body drag coefficient. Since, by division of the above,Fig. 1. Quantities defining motion of insect in flight.v\w = T/L, and for the insects considered the thrust T is small compared to thevertical force L. v is small compared to w,