Airfoil and wing theory
Airfoil Nomenclature
Introduction: Downwash and Induced Drag
• Infinite wings versus finite wings
Airfoil (Stream line body)
Upper surface Distance > Lower surface distance
Fact : Velocity α distance
Velocity of upper surface > Velocity of lower surface
Fact: Velocity α 1/Pressure
Pressure of upper surface < Pressure of lower surface
Lift
Introduction: Downwash and Induced Drag
• Infinite wings versus finite wings
Introduction: Downwash and Induced Drag
Trailing vortex and downwash
Downwash
Wake turbulence and tip vortices
Airfoil - Downwash
Airfoil - Downwash
The presence of downwash has two important effects on
the local airfoil section, as follows:
• effective angle of attack.
• induced drag.
Differences in nomenclature
For the two-dimensional bodies:
Lift, drag, and moments per unit span have been denoted with primes. For example, L’, D’,
and M’.
The corresponding lift, drag, and moment coefficients have been denoted by lower case
letters, for example, cl, cd, and cm.
For the three-dimensional bodies:
Lift, drag, and moments on a complete three-dimensional body are given without primes.
For example, L, D, and M.
The corresponding lift, drag, and moment coefficients have been denoted by capital letters,
for example, CL, CD, and CM.
Incompressible Flow Over Finite Wings
The Biot-Savart law
The velocity induced at P by the entire vortex filament is
The direction of the velocity is downward
The magnitude of the velocity is given by
The Biot-Savart law
Helmholtz’s vortex theorems
Helmholtz’s vortex theorems:
1. The strength of a vortex filament is constant along its length.
2. A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a
closed path.
Lift distribution
Prandtlˈs Classical Lifting-Line Theory
Prandtlˈs Classical Lifting-Line Theory
Let us replace a finite wing of span b
with a bound vortex, extending from
y=-b/2 to y=b/2. Since a vortex
filament cannot end in the fluid, we
assume the vortex filament continues
as two free vortices trailing
downstream from the wing tips to
infinity.
This vortex is in the shape of
horseshoe, and therefore is called a
horseshoe vortex.
Prandtlˈs Classical Lifting-Line Theory
A single horseshoe vortex.
w approaches -∞as y approaches –b/2 or b/2.
Prandtlˈs Classical Lifting-Line Theory
A single horseshoe vortex.
w approaches -∞as y approaches –b/2 or b/2.
Prandtlˈs Classical Lifting-Line Theory
The velocity dw at y0
Prandtlˈs Classical Lifting-Line Theory
induced angle of attack αi is
w is much smaller
This expression for the induced angle of attack in terms of
the circulation distribution Г(y) along the wing.
Prandtlˈs Classical Lifting-Line Theory
Consider again the effective angle of attack αeff.
The lift coefficient for the airfoil section located at y=y 0 is:
Prandtlˈs Classical Lifting-Line Theory
The fundamental equation of Prandtl’s lifting-line theory
It simply states that the geometric angle of attack is equal to the sum of the effective
angle plus the induced angle of attack.
Prandtlˈs Classical Lifting-Line Theory
Prandtl’s lifting-line theory
The solution Г=Г(y0) obtained from the above equation gives us the three main aerodynamic
characteristics of a finite wing, as follows:
1. The lift distribution is obtained from the Kutta-Joukowski theorem
2. The total lift is obtained by integrating the above
equation over the span
Prandtlˈs Classical Lifting-Line Theory
Prandtl’s lifting-line theory
The solution Г=Г(y0) obtained from the above equation gives us the three main aerodynamic
characteristics of a finite wing, as follows:
3. The induced drag per unit span is:
Prandtlˈs Classical Lifting-Line Theory
Effect of Aspect Ratio:
Induced drag factor δas a function of taper ratio.
Prandtlˈs Classical Lifting-Line Theory
Effect of Aspect Ratio:
There are two primary differences between airfoil and
finite-wing properties.
1. A finite wing generates induced drag.
2. The lift slope is not the same.
For a finite wing of general planform,
equation is slightly modified
Prandtlˈs Classical Lifting-Line Theory – Problem 1.
Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8.
The airfoil section is thin and symmetric. Calculate the lift and induced
drag coefficients for the wing when its angle of attack is 5°. Assume
that δ=τ. Refer the graph for δ
Prandtlˈs Classical Lifting-Line Theory – Problem 2.
Consider a rectangular wing with an aspect ratio of 6, and induced drag factor
δ=0.055, and a zero lift angle of attack of -2°. At an angle of attack of 3.4°, the
induced drag coefficient for this wing is 0.01. Calculate the induced drag for a
similar wing at the same angle of attack, but with an aspect ratio of 10.
Assume that the induced factors for drag and the lift slope, δ and τ,
respectively, are equal to each other. Also, for AR=10, δ=0.105.
