Aerodynamics of Hovering Flight in the Long-Eared Bat Plecotus Auritus

Hovering flight is a common habit of the long-eared bat Plecotus auritus, which has crowns of broad-leafed trees as its main feeding habitat. It has a very agile and manoeuvrable flight and is even able to fly vertically up and down for short periods. It often flies very slowly and can therefore make sharp turns in narrow spaces. Hovering is a very power-demanding type of flight. It is consistent with steady-state aerodynamics in hummingbirds and a variety of insects (Weis-Fogh, 1972, 1973), but this is not so in some insects (Weis-Fogh, 1973 ; R. Å. Norberg, 1975) and the pied flycatcher (U. M. Norberg, 1975). Its aerodynamics has not been previously worked out in bats.

The purpose of this investigation is to find out if hovering flight in the long-eared bat, Plecotus auritus, is consistent with steady-state aerodynamics, or if it has to be explained by non-steady-state phenomena. The kinematics, aerodynamics, and energetics of slow horizontal flight in P. auritus can be explained by steady-state aerodynamics (Norberg, 1976). Previously obtained data for the kinematics of hovering flight in P. auritus (Norberg, 1970) have been used in the present analysis and evaluation of the aerodynamics of hovering flight in this species.

Body measurements and flight parameters for hovering flight are given in Table 1. Calculations are based on previously described data from a wing-stroke in hovering flight (Norberg, 1970) (Figs. 1 and 2).

The average force coefficients are estimated with the same method as for horizontal flight (Norberg, 1976). It involves the use of steady-state aerodynamic theory for calculation of average lift and drag coefficients. The magnitudes of the force coefficients thus derived are then used to judge the plausibility of the concept of steady-state aerodynamics in this state of flight. The average force coefficients obtained with this method of calculation must not exceed the maximum coefficients of lift and drag obtainable at the Reynolds number under which the wings operate, if steady-state aerodynamics prevail. The method constitutes a further development of Weis-Fogh ‘s (1972) formulae for calculation of C_L in hovering flight, and of Pennycuick ‘s (1968) formulae for calculation of the resultant force coefficient in horizontal flapping flight. The new development results in an equation system expression from which the two average force coefficients, C_L and C_D, can be solved. The force coefficients thus obtained are the averages that actually prevail if steady-state aerodynamics prevails. Weis-Fogh calculated only the average lift coefficient and assumed a certain lift/drag ratio. He did not separate the upstroke and downstroke, but made calculations for half a downstroke and multiplied the value by 4, since hummingbirds and several insects have a rather symmetrical stroke. The hovering flight in Plecotus is unsym-metrical, i.e. the kinematics of the upstroke differs considerably from that of the downstroke. Hence, the upstroke and downstroke are treated separately in this investigation.

C_L and C_D enter the above two equations via L_vert (r, t), D_vert (r, t), L_hor (r, t), and D_hor(r, t), respectively, which are defined below. These two unknowns (C_L and C_D) thus can be solved from equations (1) and (2).

The wing was divided into three chordwise sections (I–III, Fig. 4) and the area of each was measured. The arm wing (the part of the wing proximal to the fifth digit) is flexed during the upstroke, thereby reducing the wing area. The area of the arm wing was estimated from my films and photographs to be reduced during the upstroke to ca. 65 % of the area of the extended arm wing, and at the reversal points to ca. 80 %. The hand wing (the part of the wing distal to the fifth digit) was not reduced at all during the upstroke, nor at the reversal points. The hand wing thus is maintained fully extended during the entire wing-stroke during hovering, as well as forward flight.

The indices of the horizontal and vertical components of the lift and drag forces were calculated from equations (5–8) for each of the three sections for 13 time-equidistant positions during the downstroke, and 13 during the upstroke. The time of the downstroke is longer than that for the upstroke. This has been taken into account when calculating the stroke frequency and flapping speed used here. The force coefficients were then obtained from equations (1) and (2).

The virtual force index curves for the vertical and horizontal components of the lift and drag forces are shown in Figs. 5 and 6, respectively. The sum of the areas between the L_vert index curve and the abscissa, and between the D_vert index curve and the abscissa, in Fig. 5, times C_L and C_D, respectively, must equal the area under the horizontal hatched line for the weight (cf. equation (1)). The areas between the L_hor index curve and the abscissa and between the D_hor index curve and the abscissa, in Fig. 6, times C_L and C_D, respectively, must equal zero (cf. equation (2)).

Fig. 7 shows the inclinations of the wing chords (the lines joining the leading and trailing edges of given sections of the wing), relative to the horizontal, at 0·20R, 0·50 R, and 0·85R. It was difficult to obtain reliable values from the tracings from the film for these inclinations of the different strips. For strips 3, 7 and 11, however, there are some reference points (elbow, thumb, finger joint and tips) (cf. Fig. 4). These strips were considered to be representative for the proximal (section I), middle (section II), and distal (section III) parts, respectively, of the wing. Only a rough estimate of the inclinations could be made from the film. However, it is clear that the angle between wing chord and the horizontal is positive for every section of the wing for every position of the entire wing-stroke.

During the downstroke, except at the reversal points, the relative wind is meeting the wing from a direction below the horizontal (Fig. 8). Therefore, the effective angles of attack (angles between the relative wind and zero-lift lines (Fig. 8) of the wing sections) are positive for every section of the wing during the entire downstroke, except at the reversal points, where they seem to be negative. During the upstroke they seem to be negative in most phases, but positive for the distal section of the wing in at least one phase (cf. Fig. 8). The sign of the effective angles of attack during the upstroke depends on the size of the zero-lift angle (angle between wing chord and zero-lift line). Since the zero-lift angle is not known we must consider all possible alternative cases regarding the effective angles of attack during the upstroke. Hence, calculations are made under the following alternative assumptions.

1. If one assumes that all effective angles of attack are negative for all sections of the wing during the entire upstroke, which they are during most of the upstroke, then the force coefficients would need to be C_L= 3·2 and C_D= 10·5 (Figs. 5 and 6). These are unreasonable values if steady-state aerodynamics prevails.

2. If the effective angles of attack are positive at section III of the wing in two phases of the upstroke, which is uncertain but estimated from the film (cf. Fig. 8), then C_L becomes 5·0 and C_D 5·8.

3. The estimated angles between wing chord and relative wind are high during most of the wing-stroke (Fig. 8), and so are therefore the effective angles of attack. If one assumes that the high angles of attack give only drag and thus zero lift, the lift force, as averaged over the entire wing-stroke, would become very small, which means that enough lift would not be produced for the bat to be able to fly. If the lift forces are assumed to become zero during the upstroke only and the drag force therefore being assumed to be reduced by 40%, the force coefficients become C_L = 3·1 and C_D = 1·9; and thus L/D = 1·6. These values might be the most probable ones. The 40% reduction of the drag is adopted from Pennycuick ‘s (1971) wind-tunnel studies on Rousettus in gliding flight.

4. If all effective angles of attack are positive during the upstroke, C_L would need to take the value 6·4 and C_D – 5·0, which are impossible values if steady-state aerodynamics prevails.

Taking into account the above alternatives, which cover all possibilities, the calculations based on quasi-steady-state aerodynamics thus result in values for the force coefficients which are not consistent with quasi-steady-state aerodynamics. Thus, non-steady-state aerodynamics must prevail, and another approach must be taken to explain the hovering flight.

Weis-Fogh (1973) devised a generalized, quick method to calculate the lift coefficient in hovering animals. To get the corresponding drag coefficient, and hence L/D ratio, he used experimental lift/drag diagrams of aerofoils of suitable form and aspect ratio, so his drag coefficient refers to stiff aerofoils and not to the actual wing. The lift/drag ratio he used for Plecotus was 6·5. He further assumed that the wingstroke was symmetrical as in hummingbirds. With data from Norberg (1970), i.e. the same set of kinematic data as used here, he then calculated the average coefficient of lift to be 1·3. From this result he drew the conclusion that hovering flight in Plecotus can be explained mainly by steady-state aerodynamics. However, the wingstroke in Plecotus hovering flight is unsymmetrical, so the upstroke has to be treated separately.

Slow horizontal flight (2·35 m s⁻¹) can be explained by quasi-steady-state aerodynamics (Norberg, 1976). It is presumable that the lower the flight speed is, the larger the relative importance becomes of non-steady-state aerodynamics in force generation. In hovering, most or all of the vertical forces have to be explained by non-steady-state mechanisms.

I am indebted to Professor T. Eeg-Olofsson, Chalmers University of Technology, Gôteborg, for making available a high-speed film camera, and to Civ.-ing. K. Pehrsson and Ing. R. Nilsson, Chalmers University of Technology, for filming aid. I also thank Dr C. J. Pennycuick, Department of Zoology, University of Bristol, for valuable comments upon the paper.