Phase II Final Report - NASA's Institute for Advanced Concepts

Chapter 3.0 Vehicle Design
3.3 Wing Aerodynamics
Consider the forces acting on a thin airfoil, of infinite aspect ratio, moving with constant velocity
in a perfect fluid and with constant angle of attack. From elementary aerodynamics it is
deduced that the net forces acting on such an airfoil are those associated with the circulation of
the fluid around the airfoil. The circulation for the steady state condition is proportional to the
velocity, angle of attack, and chord length.
However, for the case of unsteady motion the lift on the airfoil section is no longer a simple
function of the circulation. For one thing, it can be easily shown that any plate accelerated in a
fluid exerts forces and moments even in the absence of circulation. Thus, even at zero forward
velocity for an airfoil performing simple harmonic motion, the fluid will exert forces and
moments on the airfoil section due to the acceleration and deceleration of the fluid moving with
the airfoil. These can be considered aerodynamic inertia forces. A second important contribution
to the lift for the case of unsteady motion is known as the quasi steady lift. This is the lift which
would be produced by the motion of the airfoil if the circulation pattern behind it (i.e., the wake)
had no effect. It represents the force that would be produced if the instantaneous velocity and
angle of attack of the airfoil were permanently maintained. The lift can then be considered to
vary and to be a function of the instantaneous configuration of the system.
In general, every change of the state of motion of the airfoil is accompanied by a change in circulation
around it. Furthermore, every change in circulation about an airfoil section is accompanied
by a vortex shed from the trailing edge of the airfoil. For the change of a continuously
changing circulation, such as for a simple harmonic motion, a continuous band of shed vortices
develop behind the airfoil section. These shed vortices (or the vortex sheet) produce vertical
velocities in the neighborhood of the airfoil. The periodic force is a function of this vortex sheet,
the distributed vortex strength of which is another periodic function in terms of reduced frequency
parameter (k). This will be explained in more detail below.
Thus from the qualitative discussion above, it can be deduced that lift on a flapping airfoil is a
function of free stream velocity and periodic functions of plunge and pitching motion. Now after
having discussed the problem qualitatively, the different assumptions made to simplify the
model and equations formulated as a result of this analytical model, will be explained. This will
be followed by the results thus obtained.
3.3.3.5 Assumptions
1. The flow is assumed to be by a perfect gas, which behaves in accordance with a perfect
gas law, with constant specific heat values over a reasonable range of temperature and
pressure.
2. No friction is considered internally or externally; so for the required power analysis, a
factor must be added to account for frictional losses.
3. Irrotational flow such that all fluid particles have zero angular momentum about the center
of gravity axis.
4. The basic model only encompasses the two dimensional airfoil effects with infinite span.
This will be modified to include the three dimensional finite span effects and planform
shape by inclusion of Reissner's correction factors.
5. The model is based on non stationery potential flow theory, and no deviations from
potential flow are being considered.
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