30
P. Mukkavilli et al.
where
u(x, t)
The Flow Field.
p
Pressure.
ρ
Density of Fluid.
∇
Del Operator.
μ
Dynamic Viscosity.
ϑ
Kinematic Viscosity.
ubd
Velocity at Boundary condition.
us
Velocity of the solid.
The Navier-Strokes equation in a non-dimensional form describes how the
velocity, pressure and density of a moving fluid are related. Hence, it comes out
to be a useful phenomenon to understand the insect flight under various dimensions.
The equation is non-dimensional and the airflow is incompressible.
A non-dimensional form of equation with Reynolds number (Re), wing length (l)
and velocity (V) can be expressed as
Re = Intertial Forces
Viscouse Forces = ρV L
μ
= V L
ϑ
Total Momentum Transfer
Molecular Momemtum Transfer
(3.10)
Compared to an airfoil, the insect wing is very small and it also flaps with high
frequency during flight. The range of Reynolds number in most of the insect flights
varies from 10 to 104. For small insects, the Reynolds number will be very low and
increases proportionally with the increase in the insect dimensions (Table 3.1).
A dragonfly with a mean wing chord of 1 cm, wing length 4 cm and wingbeat
frequency of about 40 Hz with tip speed (u) as 1 m/s operates at a Reynolds number,
Re = uc
ϑ = 103. However, Chalcid wasp with a wing length of 0.5–0.7 mm with a
wingbeat frequency of about 400 Hz operates at a Reynolds number of 25 [24]. On
the other hand, a bigger insect like soapnut bug (Tessaratoma javanica) with wing
length of 2.2 cm and a chord length of 1.15 cm has a wingbeat frequency of 50 Hz
and is having a Reynolds number of above 4000. Similarly, for a Pentatomid bug
(Chrysocoris purpureus), the wing length is 1.2 cm and chord length is 0.65 cm, has
Table 3.1 Flight parameters and Re
S. No
Insect
Wing Chord (Beff) in (cm)
Wing length
(cm)
ϑh (Hz)
Re
1
Dragonfly
1
4
40
40
2
Chalcidwasp
0.5–0.7
400
25
a3
T. javanica
1.15
2.2
50
> 4000
a4
Pentatomoid bug (CP)
0.65
1.2
90
1000
a The compilation of data from various sources