How is the lift produced - Fluid Dynamics.

DOMINANT PHYSICS:

How is the lift produced - Fluid Dynamics.

For the purpose of this project two explanations will be presented in a general and basic way. These theories are the application of Bernoulli’s Equation and Euler’s Equation for Streamline Curvature Effect.

Bernoulli’s Equation: Po = P1 + ½rv1² + rgy1 = P2 + ½rv2² + rgy2

Variables

Units

Po  Stagnation Pressure

[Pa] or [lbf/ft2]

P    Pressure

[Pa] or [lbf/ft2]

r    Density

[kg/m3] or [lbf/ft3]

V   Velocity

[m/s] or [ft/s]

g    Gravitational Constant

[m/s2] or [ft/s2]

y    Height

[m] or [ft]

Detail of Hydrofoil: a) Pressure Profile    b) Momentum Transfer    c) Circulation     d) Streamlines

  This equation applies to flows along a stream line which can be modeled as: inviscid, incompressible, steady, irrotational and for which the body forces are conservative. Also the difference on the height of the foil (the distance from the bottom section to the upper one) is small enough so that the difference rgy2 - rgy1 is negligible compared to the difference of the rest of the terms. What is left is that the pressure plus one half the density times the velocity squared equals a constant (the stagnation pressure).

As the speed along these streamlines increases,the pressure drops (this will become important shortly).   The fluid that moves over the upper surface of the foil moves faster than the fluid on the bottom. This is due in part to visous effects which lead to formation of vertices at the end of the foil. In order to conserve angular momentum caused by the counter-clockwise rotation of the vortices, there has to be an equal but opposite momentum exchange to the vortex at the trailing edge of the foil. This leads to circulation of the fluid around the foil. The vector summation of the velocities results on a higher speed on the top surface and a lower speed on the bottom surface. Applying this to Bernoulli’s it is observed that, as the foil cuts through fluid, the change in velocity produces the pressure drop needed for the lift. As it is presented in the diagram, the resulting or net force (force= (pressure)(area)) is upward.

This explanation can be enriched with the Principle of Conservation of Momentum. (Momentum = (mass)(velocity)) If the velocity of a particle with an initial momentum is increased, then there is a reactant momentum equal in magnitude and opposite in direction to the difference of the momentums. (See diagram).(Mi = Mf + DM)

Euler’s Equation: d(p+rgy)/dn = rv²/R

Variables

Units

P    Pressure

[Pa] or [lbf/ft2]

r    Density

[kg/m3] or [lbf/ft3]

V   Velocity

[m/s] or [ft/s]

g    Gravitational Constant

[m/s2] or [ft/s2]

y    Height

[m]

...