Linear MotionLift Theories

Requirements for a Valid Theory

1) A valid theory is a rational explanation of observed phenomenon

2) A valid theory can be used to predict future observations

3) A valid theory produces numerical results

For a lifting airfoil, the average pressure on the upper surface is lower than the average pressure on lower surface.

The difference in pressure produces the lift.

Generation of Lift

For a lifting airfoil, the surface static pressure varies from top to bottom and from front to back.

The ideal surface velocity varies from top to bottom and front to back. The surface of the foil is a streamline, so Bernoulli’s equation relates the surface velocity to surface pressure.

When we include viscosity, the surface velocity is zero, but the local velocity varies at the boundary layer edge.

Generation of Lift

Why does the surface static pressure vary top to bottom and front to back?

The flow must follow the surface contour (or the edge of the boundary layer). Flow can not pass through the airfoil.

What determines the values for the local surface velocity, and therefore the local surface pressure, and ultimately the amount of the lift?

Generation of Lift

Why does the surface velocity vary top to bottom and front to back?

What determines the values for the local surface velocity, and therefore the local surface pressure, and ultimately the amount of the lift?

Generation of Lift

The flow around the airfoil must satisfy the conservation laws:

Conservation of mass (continuity)Conservation of momentum (2 or 3 components) Conservation of energy

Lift is Generated

Uniform Flow + Doublet + Vortex

Euler Equation Solution

Lift

The Kutta Condition

No Lift

Mapped Uniform Flow + Doublet Mapped Uniform Flow + Doublet+ Bound Vortex

Bound Vortex Theory

Lift = L = V

Uniform + Doublet + Bound VortexBound Vortex Theory

Kutta-Joukowsky Theory

Free stream velocity = V

Vortex strength =

Air density =

Lift Coefficient

Angle of Attack

Theory correctly predictsslope of curve = 2 for thin airfoils

V

Starting (Shed) Vortex

Prandtl’s Experimental Photo of Starting Vortex

Prandtl Lifting Line Theory

Bound Vortex

Tip Vortex

Three dimensional version of Bound Vortex Theory

A continuous line of bound vortices terminating at the wing tips with “tip vortices” that continue downstream to the “starting vortex”.

Photo of Tip Vortices