quick estimates of flight fitness in hovering flight fitness in hovering animals 173 table 1. shape
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7. Exp. Biol. (1973). 59. 169230 l 6 g With 23 textfigures
Printed in Great Britain
QUICK ESTIMATES OF FLIGHT FITNESS IN HOVERING ANIMALS, INCLUDING NOVEL
MECHANISMS FOR LIFT PRODUCTION
BY TORKEL WEISFOGH
Department of Zoology, Cambridge CBz ^EJ, England
(Received 11 January 1973)
INTRODUCTION
In a recent paper I have analysed the aerodynamics and energetics of hovering hummingbirds and DrosophUa and have found that, in spite of nonsteady periods, the main flight performance of these types is consistent with steadystate aerodynamics (WeisFogh, 1972). The same may or may not apply to other flapping animals which practise hovering or slow forward flight at similar Reynolds numbers (Re), ioa to io*. As discussed in that paper, there are of course nonsteady flow situations at the start and stop of each halfstroke of the wings. Moreover, it does not follow that all hovering animals make use mainly of steadystate principles. It is therefore desirable to obtain as simple and as easily analytical expressions as possible which should make it feasible to estimate the forces on the wings and the work and power produced. In this way one may make use of the large number of observations on freely flying animals to be found in the scattered literature. It may then be possible to identify the deviating groups and to approach the problems in a new way. This is the main purpose of the present studies, which both include new material and provide novel solutions.
Major emphasis must be placed on simplicity. This involves approximations since the true flight system is so complicated as to be unmanageable. However, when we confine ourselves to free flight and make use of the most reliable flight data available, the task is neither as difficult nor the conclusions as unrealistic as one would expect, since it is possible to introduce simple corrections.
Although entitled ' Quick estimates', this does not mean that the approach is super ficial, but rather that a procedure has been devised whereby the flight performance of a given animal can be evaluated quantitatively as well as qualitatively on the basis of only a few accurate observational data and a minimum of computation. In other words, given reliable information about bodily dimensions and wingstroke parameters, the method enables one quickly to arrive at a firstorder approximation so as to assess whether the animal makes use of wellestablished mechanisms or employs unusual or novel principles.
The following procedure is, in essence, the strategy of the present investigation. From measurements of the size and shape of the wings, the geometry of the wing stroke, the frequency of the wing beat and the weight of the animal which is sustained in hovering flight one can calculate, on the basis of steadystate aerodynamics, the minimum coefficient of lift which must be ascribed to the wings. From the same mea surements one can also calculate the Reynolds number under which the wings operate,
170 TORKEL WEISFOGH
Vertical
w Horizontal
Chord, c(r)
Fig. 1. Simplified diagram of normal hovering flight. (A) The instantaneous forces. (B) The animal seen horizontally from the side, and (C) vertically from above.
and from this figure one can obtain (using published data on wings) the maximum coefficient of lift which may be expected under these conditions. If the minimum coefficient of lift for steadystate aerodynamics does not exceed the maximum coeffi cient of lift obtainable at the Reynolds number in question one assumes that steady state aerodynamics are adequate to explain hovering flight, and from then one can go on to calculate the power requirement and other parameters of the flight mechanism. If, on the other hand, the minimum coefficient of lift for steadystate aerodynamics is greater than the maximum obtainable, it is clear that hovering flight cannot be explained by steadystate aerodynamics and that a new approach must be made.
AERODYNAMIC RELATIONSHIPS
The main simplifications are that the animal is assumed to make use of steadystate flow patterns only, that lift is produced at right angles to the direction of the relative wind, that the stroke plane is horizontal with no tilt, that the induced wind can be disregarded because it is relatively small and, finally, that the wing movements are sinusoidal and of similar shape and the same duration for both the morphological
Flight fitness in hovering animals dr
171
Fig. 2. Diagram of the wing parameters (A) and the movements during hovering (B).
upstroke and downstroke. As we shall see (p. 192), these assumptions lead to values for the average lift coefficients CL needed to remain airborne which are slightly higher than in the more complete treatment (WeisFogh, 1972). As to the specific aerody namic power PJ, the estimates are too small by a factor of about 2 but, as we shall also see (p. 195), it is possible in a simple manner to make appropriate corrections. The instantaneous aerodynamic force on a wing element F(t, r) of chord length dr) and its vertical (lift) and horizontal (drag) components are shown in Fig. 1 A. In Fig. 1 and in the text, any quantity X which is a function of time t and distance r from the wing hinge, or fulcrum, is written as X(t, r). Appendix 1 gives a list of symbols and constants used.
The justification for the simplifications is that the majority of insects, birds and bats have been found to tilt the long axis of the body towards the vertical (Fig. 1B) when hovering, so that the wings beat in an almost horizontal plane, usually symmetrically about the average positional angle y of 900 (Fig. 1C).
(a) Coefficient of lift
Fig. 2 A shows how the outline of a wing of total length R, and in particular the wing chord dj), varies with the distance r. In practice I have found that most wing
172 TORKEL WEISFOGH
contours can be described quite faithfully by means of a simple mathematical functionl whether the moving wing is morphologically twowinged as in Lepidoptera and Hymenoptera or consists of only one part as in Diptera. It should be stressed here that there are hovering insects which make use of two pairs of wings which beat out of phase, as is the case in Odonata and Neuroptera, and these insects are not included in the present analysis.
The angular movement of the long axis of the wing in the horizontal stroke plane is
7(0 = y + W>sin(2nnt), (1)
where y is the average angle, the stroke angle, n the wingstroke frequency (number of complete stroke cycles per unit time) and t is time. The angular velocity and acceleration are then
dyjdt = 77710 cos {zrnit), (2)
d*yldt2 = 2nW
Flight fitness in hovering animals 173
Table 1. Shape factor a for the second moment of the area of two whole functional wings (or two pairs of wings) (eq. 7), shape factor rfor the third moment of two whole wings (eq. 50), together with the radius of gyration rg, the chord length ca at distance r^ the factor in eq. (11 b)for the calculation of Reynolds number for the wings shapes considered, and the factor in eq. (35)/or calculating A, the ratio strokearc I wingchord.
(a) hape... Half ellipse

r, \R c, 087c Factor in 87 eq. (116)
Factor in °"575 eq (35)
Parabola '.
Tot
•R/VA 0690 66
0693
(c) tectangle
R
1 I
IOOC
n5
OS77
wTriangle, base Triangle, apex 
c
i
o4ic 33
O995
^ ^
f
oyic ioo
0995
means of which the outline of the wing is described. The chord of the wing c is then defined in each single case, as is explained in Appendix 2 and Table 1, where some shape factors are listed. It should be noted that the third term is a function only of r and not of t.
The fourth and final term in equation (6) is a function of time / and not of distance r, and we therefore have, for a complete quarterstroke, that the average lift L is
L = X I*' c(r)rldrx [ ~ cosi(27mt)dt. J r0 J t0
(8)
A quarterstroke is the time which it takes for the wing to move from an extreme position and until it reaches the middle of its stroke path. We have already dealt with the first integral, which depends only on the wing shape and is equal to S. The second definite integral equals $. Since the movements and forces are assumed to be sym metrical about the middle position (harmonic motion) the average lift over each quarterstroke must equal the weight G of the animal so that the average lift coeffi cient is
(9)
(b) Reynolds number (Re) For a given aerofoil the coefficients of lift and drag depend not only on the angle of
attack but also on Reynolds number (Re). This is particularly so in tiny insects, where it may be as low as 10 or even smaller (cf. Horridge, 1956). The task is to find a simple but representative expression. The general expression is
fl V ' V '
where /* is the viscosity of the air and the ratio (i\p is the socalled kinematic viscosity v. In all systems of measurement, v has the dimension of (length)2/time. At 20 °C and
174 TORKEL W E I S  F O G H
normal atmospheric pressure v is oi4cma/sec in the units used here. In ordinarf fixedwing airplanes the representative velocity is the airspeed and the length is some representative wing chord. In our case we much make a specific choice. During hover ing 80% of the lift is produced during the middle part of a halfstroke, where the average value of cos(27rcif) in equation (2) is 089. As to distance from the fulcrum