36

P. Mukkavilli et al.

U∞

is free stream velocity.

Po

is free stream pressure at stations 1 and 4.

U2

is the velocity at the upstream of the actuator disc.

U3

is the velocity at the downstream of actuator disc.

Uw

is the velocity in the wake region.

Actuator Disc Model for Flapping Flight

The calculation of induced power for flapping flight comes basically from the heli-

copter theory rather than a fixed-wing theory. A helicopter rotor sweeps out a circular

disc, which can be considered as an ‘Actuator Disc Model’, which states that air pres-

sureincreasesinastepwisemannerwhentheairpassesthroughthedisc.Thisincrease

in air pressure imparts a downward velocity to a tube of air that flows through the

disc. The cross section of the tube is the diameter of the rotor in the case of heli-

copter and wing span in the case of an insect wing. The wings of insects do not cover

the full disc area due to the limitation of stroke angle; however, the induced power

is calculated based on this actuator disc model. The actuator disc concept helps in

calculating the induced power during hovering. The methodology has been outlined

by Chari [26].

The following advantages are observed when the actuator disc concept is applied

for flapping wing flight which is termed as ‘Bow-Tie’:

1.

The approach does not require information about the wing structure or wing

mass except the wing span.

2.

Local airspeed is assumed to be uniform all along the wing span, which permits

a clear simplification considering rotary or flapping wings.

3.

The calculation does not require the angle of flapping except the wing area

through which air passes as it is accelerated downward.

4.

Some authors have used wing swept area instead of full circles for calculation of

induced power, as the bird or insect flaps its wings in less than 70°, a relatively

narrow swept area is formed.

This bow-tie concept may not hold good when lifting line theory is considered

for lift generation and induced velocity calculations.

Prandtl’s Vortex System

It has been explained earlier that only by circulatory flow, the lift is generated

by proper vortex strength. The German scientist Prandtl [27] postulated the three-

dimensional wing theory and proposed that the actual wing could be replaced math-

ematically by a vortex filament. A similar concept was proposed earlier by English

engineer FW Lanchester simulating the wing with the help of a vortex line. Prandtl