Lesson 4: Fluid Flow
Section 4 – Aerodynamic Coefficents
There are several factors from everyday experience that determine the aerodynamic resistance on a body. If you place a hand broadside to a flow outside a car window at 20 km/hr, little resistance is felt, but if one speeds along at 100 km/hr, the force felt is considerable.
VELOCITY
Velocity is one factor that determines the resistance. In fact, considering the flow problems of subsonic flight (high Reynolds number under relatively small viscosities), the resistance depends directly on (velocity) times (velocity) or (velocity)2.
Consider the example of a car traveling at a velocity of 100 km/hr is five times that of 20 km/hr, the aerodynamic resistance will be about 25 times as great at the higher velocity.
If you walk along a beach, there is little aerodynamic resistance to doing so. But try to wade in the water at the same speed. It is considerably more difficult, if not impossible.
The density of water is much greater than the density of air. Density of the fluid represents another determining factor in the resistance felt by a body.
Try another experiment: hold a small piece of cardboard up against a stiff wind. Little resistance is experienced. Now hold a much larger, similarly shaped sheet of cardboard up against the same stiff wind. A considerable resistance is felt. Area (or length times length) exposed to the airflow is another determining factor of resistance.
We can therefore conclude that in the flow of the real fluid, air, about a body, the aerodynamic resistance is dependent on the size, shape, and attitude of the body, the properties of the fluid, and the relative velocity between the body and the fluid (air).
For a real fluid, we should note that viscous, elastic, and turbulent properties are also important. In addition to the shape and attitude of the body the surface roughness has an effect on the force. It may be demonstrated that:
Lift = ρ∞ x V∞2 x S x Factor (α, ρ∞V∞l/μ, V∞/a∞, surface roughness, air turbulence)
Where:
ρ∞ = free-stream fluid density
V∞ = free-stream velocity
S = characteristic body frontal area
l = characteristic body length
a (Greek letter alpha) = altitude of body
μ = coefficient of viscosity
a∞ = free-stream speed of sound of fluid
Note that S is a characteristic body frontal area that is usually chosen to be consistent with a series of comparison experiments. For a cylinder it would be the diameter of the cylinder times its length. For a wing, however, it is usually taken to be the platform area (chord length times wing span for a rectangular wing). Therefore, it is necessary to check the particular definition of S used for a body.
We know that ρ∞V∞l/ μ is the Reynolds number of R. We also know that a∞/V∞ is defined to be the Mach number of M, which you may recall is 740 mph in value.
The Reynolds number is the dimensionless quantity associated with the fluid viscosity whereas the Mach number is associated with the fluid compressibility. Surface roughness was shown to have influenced the transition from a laminar to a turbulent flow.
EQUATION OF LIFT
Air turbulence represents the degree of the wake formed past the separation points. Furthermore, the effects of attitude and shape of a body are lumped together into the factor. Letting the factor be called K, then,
Lift = ρ∞ x V∞2 x S x K
The dynamic pressure of a fluid flow was previously defined as 1/2 pV2 so if a value of 1/2 is included in the equation above and the value of K is doubled to keep the equation the same, 2K may be replaced by CL. Finally,
Lift = CL x ½ ρ∞V∞2 x S
The equation above is the fundamental lift formula for usual aircraft flight. CL is known as the coefficient of lift. The equation states simply that the aerodynamic lift is determined by a coefficient of lift times the free-stream dynamic pressure times the characteristic body area.
It is very important to realize that the lift coefficient CL is a number dependent upon the Reynolds number, Mach number, surface roughness, air turbulence, attitude, and body shape. It is not by any means a constant. CL is generally found by wind-tunnel or flight experiments by measuring lift and the free-stream conditions and having a knowledge of the body dimensions. We can say then,
CL = Lift/ ½ ρ∞V∞2 x S
The aerodynamic drag is the aerodynamic resistance parallel to the free-stream direction. From the two equations, we then get,
Drag = CD x ½ ρ∞V∞2 x S
or-
CD = Drag/½ ρ∞V∞2x S
where CD is the drag coefficient, dependent on the previously enumerated parameters.
MOMENT
We can now consider the moment acting on a body is a measure of the body's tendency to turn about its center of gravity. This moment represents the resultant aerodynamic force times a moment distance. We can derive these equations as follow in a similar manner to those previously,
Moment = CM x ½ ρ∞V∞2 x S x l
or
CM = Moment/ ½ ρ∞V∞2 x S x l
Cm is the coefficient of moment and an additional characteristic length l is necessary for it to be dimensionally correct. Recall that CL, CD, and Cm are dependent on the Reynolds number, Mach number surface roughness, air turbulence, attitude, and body shape.
Bibiliography
Allen, John E. Aerodynamics: A Space Age Survey. New York: Harper & Row, 1963.
Taley, Theordore A. Introduction to the Aerodynamics of Flight. Science and Technical Information Office. Wash, DC: NASA. 1975.
Wegener, Peter P. What Makes Airplanes Fly? New York: Springer-Verlag, 1991.
