Lift and drag theory

[DSmith]

The International Moth website has some theory on the lift and drag of foils

http://www.intmoth.com/design/lift.php
http://www.intmoth.com/design/drag.php

In summary it gives the following equations for a symmetrical foil:

L = {1/2 . p . V^2 . S . LCS . alpha} / { 1 + [LCS / (pi . AR)] }
D = 1⁄2 pV^2 S Cd + Cl^2 /π AR (1.4)

Where:
p = density of water
V = boat speed
S = foil area
LCS = slope of CL v alpha plot
Alpha = Angle of attack in radians
AR = Aspect ratio
Cd = Coefficient of drag
Cl = Coefficient of lift

A couple of questions:

1. Are they missing some brackets in equation 1.4 to give D = 1⁄2 pV^2 S (Cd + Cl^2 /π AR)?
2. Is the CD in equation 1.4 the 2D CD?
3. Is this theory adequate for preliminary foil design?

Cheers

Dave

[Tim B]

1) yes, they are missing brackets, and you are correct. ie Drag=1/2 * rho * V^2 * area * Cdtotal

2) yes, as you would get from X-foil. Note. Navier Stokes CFD codes will give Cdtotal.

3) as long as you are careful with the values of Cl and Cd, yes.

Also see Larsson's "principles of yacht design" for more info, or Hoener's book of lift and drag, for a detailed explanation of the effect of tip-shapes on lift and drag.

Hope this helps,

Tim B.

[DSmith]

Thanks again Tim. I will have to start paying consultancy fees to you shortly.

[Tim B]

well, if you'd like me to do an optimization job....

That said, I'd have to finish writing the code first!

Tim B.

[bilgeboy]

The mathematics might work out nicely, but I didn't get past this...


"Due to the curvature on the foil the fluid has a longer distance to flow around the top side. This forces the liquid to stretch and so the density of the fluid drops causing a pressure difference between the two sides. "


I am no physicist, so I ask if this sounds right.

I've had water in a syringe, half full. I can easily pull a vacuum in the syringe, and this is evident by empty space, probably with a small amount of vapor in it. The water definately does not "stretch", though, to fill the void.

Since liquids are incompressible, I would also think they are "unstretchable." As far as I know, the one and only factor affecting density of a liquid is temperature.

I don't ask to be nitpicking, just that I am curious how foils really do work. I am familiar with increase in velocity, Venturi effect, etc, but can't say my understanding is concrete.

Mike

[Tim B]

Basically, in any (ideal) fluid Bernoulli's law will apply to link pressure and velocity. Basically, as pressure drops, velocity increases, and vice versa. So, if you accelerate air on the upper surface of a body, relative to the lower surface, the body will suck itself upwards.

However, in order to get a decent amount of lift, you need a decent change in speed, and that doesn't come as a ratio of distances. The culprit is circulation. Mathematically, it is super-imposing a vortex (top flowing aft) on the quater chord of the wing, with a uniform flow ahead of the wing. Consequently, there is very high-speed flow on the upper surface, and very low-speed flow on the lower surface.

The same is true, of a spinning cylinder in a smooth flow. Due to friction, it will drag air around with it on the surface, and in doing so generate the same sort of mathematical pattern.

If you're interested, I'd suggest you found a few fluid-mechanics books and had a browse through them. The subject matter can go through some rather nasty mathematics at times, though.

Hope this is of some help, at least,

Tim B.

[Skippy]

"This forces the liquid to stretch and so the density of the fluid drops causing a pressure difference between the two sides. "

bilgeboy: I am no physicist, so I ask if this sounds right.

No, that's definately a mistake. Even a gas (e.g. air) won't compress significantly until it approaches the speed of sound. On the upper foil surface, any given cell of fluid will stretch along its flow line and compress in the direction(s) perpendicular to that, always maintaining constant volume at low speeds.

[tspeer]

Originally Posted by bilgeboy

...
"Due to the curvature on the foil the fluid has a longer distance to flow around the top side. This forces the liquid to stretch and so the density of the fluid drops causing a pressure difference between the two sides. "


I am no physicist, so I ask if this sounds right.
...

NNNOOOOOO!!!!! This has got to be the WORST of the "Just-so Stories" ever passed around regarding fluid dynamics. It's the most common explanation you'll find in lay literature, and it's just plain wrong.

Just think of your common experience, and you'll see all kinds of counter examples that disprove this theory:
- The dimestore balsa gliders you flew as a kid had totally flat wings, and yet they fly.
- There's no signficant difference in length between the windward and leeward sides of a staysail, and yet they produce a powerful lift.
- The explanation has no way to account for the effect of angle of attack, yet everyone experiences the fact that lift is proportional to angle of attack - it's the whole basis for trimming sails.
- It doesn't account for stall.
- It doesn't account for the fact that lift is proportional to the square of the freestream velocity.
- If the explanation was right, symmetrical sections wouldn't work - but symmetrical keels and rudders work just fine.

I'm constantly amazed that people will parrot something that is so contradictory to their every-day experience! It can only be because what's written is totally irrelevant to actual practice.

If you want to understand the origin of lift, it's far better to consider the three conservation laws: conservation of mass, conservation of momentum, and conservation of energy.

Conservation of momentum says that the net force on the surface has to be equal to the net change in momentum of the fluid. So when hydrofoil or sail deflects the fluid's direction of flow, the result is a lift force at right angles to the flow. Conservation of momentum also means that if you consider a blob of fluid following a curved path, the pressure on the outside of the curve has to be greater than the pressure on the inside of the curve.

Consider a vertical hydrofoil like a rudder or keel that's producing lift in the horizontal plane. The foil is deflecting the flow toward the pressure side and away from the suction side. If you start recording the pressure far from the pressure side of the foil, you'll find it's essentially unchanged. But as you move closer to the foil, you're moving toward the outside of the curve so the pressure is continually growing until you arrive at the foil's surface. Now start far away from the suction side. The pressure there will also be near ambient pressure, but as you move toward the foil you are moving to the inside of the curve, and the pressure has to be dropping as you approach the surface. As a result, you get low pressure on one side and high pressure on the other side. Integrate that difference in pressure over the surface of the foil, and you have the normal force, which for small angles is close to the lift. The magnitude of the force will be just enough to account for the amount of deflection of the fluid affected by the foil.

The connection between fluid velocity and pressure - Bernoulli's "Law" - is really just a statement of the conservation of energy. In fluid flow, potential energy is represented by the pressure. Kinetic energy is mass times velocity squared - the 1/2 rho V^2 term known as dynamic pressure. If there's no other change in the energy of the fluid, say by changing the temperature or by imparting rotation to the fluid, the sum of local pressure and dynamic pressure (kown as total pressure) has to be constant, because that's the energy and we've assumed no energy is being added or subtracted. So if the local pressure is increased, the local velocity is decreased. And if the local pressure is decreased the local velocity is increased. This is why the velocity is higher on the suction side than it is on the pressure side. It's conservation of energy.

And because Bernoulli's Law depends on no energy being added or subtracted, it doesn't hold where the energy in the flow is changing. A good example is in the wake. The pressure across the wake very quickly equalizes to the ambient pressure as the wake is convected downstream of the trailing edge. But if you measure the velocity across the wake, you'll find it's lower in the center of the wake than it is at the edges. So clearly Bernoulli's law doesn't hold there because you have two diffrent velocities with the same pressure. What's happened is kinetic energy has been drawn out of the flow. Some of it has gone into rotational energy in the vortices left swirling in the wake. Some of it's gone into heating the flow. Another example of where Bernoulli's law doesn't hold is in the boundary layer next to the surface. The pressure is basically constant across the boundary layer, but the speed next to the surface is zero while the speed outside the boundary layer is high. Again, the "missing" energy is in the rotation of the flow because the flow in the boundary layer is highly sheared.

In steady flow, conservation of mass says that the density times the velocity of the flow going into a given volume has to be equal to the density times the velocity of the fluid exiting the volume. Otherwise, you'd be accumulating (or draining) fluid from the volume over time. This basically means that the whole flow "picture" has to be consistent. You can't be pulling fluid out of nowhere, or making it disappear.

The conservation laws do account for the situations the "difference in lengths" explanation does not. Angle of attack is accounted for because the deflection of the flow is determined by the direction of the trailing edge. The effect of thin, cambered sections like a headsail are accounted for the same way, as are symmetrical sections. The effiicency of large spans over short ones is accounted for because while force is linearly proportional to the change in velocity, energy is proportional to the velocity change squared. So for the same force, it's better to make a small change in velocity to a large amount of fluid than it is to impart a large change in velocity to a small volume of fluid. And Bernoulli's law still holds for those parts of the flow that have constant energy.

[Toby P]

Mr Speer is bang on, but if you want a simpler to understand explanation (but still correct) visit arvelgentry.com and look at the magazine articles. Pictures tell a thousand words. There are also some good explanations of the slot effect there.

[LSU SolarSplash]

Originally Posted by bilgeboy

Since liquids are incompressible, I would also think they are "unstretchable." As far as I know, the one and only factor affecting density of a liquid is temperature.

That is not entirely correct. Fluids are indeed compressible. However, from an engineering standpoint, the compressibility is negligible, thus it is considered incompressible.