Lift onset, planing, and lift coefficient

There are no integrals in either Landau and Lifshitz nor Newman which directly relate wing geometry to induced drag. Both have integrals for induced drag in terms of the span-wise circulation distribution, not geometry. The evaluation of these integrals does not require any approximation if the circulation distribution is known or assumed. The fundamental difficulty is determining the circulation distribution from the geometry. Neither Landau and Lifshitz nor Newman provide equations to directly do so without approximations such as those used by the lifting line method.

sandhammaren05 should provide the equation numbers from one or both references of the integrals which he claims to have evaluated.

 

That is correct that they do not mention geometry, the integrals in the theory are correct for all reasonable wing geometries.

The integrals are over the span. It's left to you to determine the circulation density for a given wing (camber and shape). I wrote down the circulation density for a flat uncambered wing of elliptic shape, and for an elliptic wing with hyperbolic camber and zero attack angle. If you combine the two densities then the result is not elliptic: a cambered elliptic wing does not produce maximum L/D.

What the lifting line approximation obtains that I do not is the dependence on aspect ratio of the lift coefficient. In my calculation it is left undetermined. We know only that as the aspect ratio becomes large we should retrieve the 2D lift coefficient, which is the largest lift coefficient possible. My calculation shows that elliptic wing shape alone does not imply elliptic circulation density (and therefore also not optimal L/D).

 

We add camber in the thin wing, small attack angle limit so we do not face limits of large camber. I'm sure that adding large camber will kill the performance, as
it does with a marine propeller.

Exact calculation means no approximations in doing the integrals for lift and induced drag. Measurements are a different matter than calculations. One compares calculations with measurements.

 

It appears that we agree about the integrals.

How did you obtain the circulation distributions which you used?

If the camber of the sections along the wing is the same shape and the magnitude of the camber at a section is proportional to the span at the section then the addition of the camber to a uncambered and untwisted wing is equivalent to a change in the zero lift angle of attack. The circulation distribution remains the same for a given change in angle of attack from the zero lift angle of attack. If the camber at every section is not the same shape with magnitude proportional to the chord at the section then the addition of the camber is equivalent to adding twist to the wing. It is well known that a elliptical wing with twist or variations in camber does not have minimum induced drag for the lift.

The calculation of the integral using induced drag requires the circulation distribution. How do you obtain the circulation distribution for the a half ellipse?

 

The 2D theory is a guide, in part, but the 3D theory is very different. Take axis z is along the wing span. c(z) denotes the chord at span location z. The circulation density is proportional to c(z) for an umcambered wing, but goes like bc^2(z) for a cambered wing at a=0. Here, a is the attack angle and b
is the parabola constant. So adding the two densities is not simply equivalent to a change the zero lift angle of attack change in 3D, although that angle is changed, as in 2D (your expectation is based on the 2D theory). Instead, we get the sum of an elliptic circulation density with a parabolic one. So with circulation density linear in c(z) an elliptic wing shape gives an elliptic circulation density, whereas an elliptic wing shape gives a parabolic circulation density for the case of parabolic camber at a=0 . There is no twist in the wing because the camber parameter b is a constant independent of z. One can then use these densities to calculate the induced drag in all three cases.

In propeller blade design, and a propeller is indeed a twisted hydrofoil (to maintain constant pitch as the radius varies), I use a special condition that sets the parabola parameter b(r) as one moves radially outward from hub to tip. But my wings have no twist. The degree of camber is everywhere the same along the wing span.

 

Sandhammaren05 has made a fundamental error. Circulation distribution along the span of a wing is not in general proportional to the chord distribution of a wing, even if the wing is uncambered, as sandhammaren05 claims. The chord at each section affects the entired. For a flat elliptical shape the circulation density happens to be elliptical, but for other wing shapes the circulation distribution does is not proportional to the chord distribution. For example the circulation distribution of an uncambered rectangular wing (constant chord) is not constant along the span but decreases smoothly to zero at the wing tip.

Similarly the circulation distribution due to camber is not simply proportional to a camber constant multiplied by the chord squared. The camber at each section affects the entire wing.

 

It's simple scaling. No mistake on my part. Produce a calculation to back up your words or hold kjef. Like the silly announcement of Dewey's win, your proclamation is premature and false.

 

Interesting response, but also informative.

 

sandhammaren05: Have you tested the calculated vs measured values?

 

You have made a claim

. The claim is wrong

As I wrote, back up your claim or find another hobby. You're back off the track again with empty claims, a waste of time.

 

No, I'm not into airplane wings. I simply showed that (i) an elliptic circulation density follows from an uncambered wing of elliptic shape, and (ii) that a cambered wing of elliptic shape does not have an elliptic circulation density. It's a standard calculation to show that optimal L/D of a wing is produced by an elliptic circulation density.

 

Ultimately, the whole premise is flawed. There are many double enders that plane. Therefore, the theory is wrong. Further, how does a parabolic shape relate to a planing Vee hull with a wide transom?

 

Your confusion is enormous. You mix worse than apples with oranges.

 

The topic somehow switched from planing hulls to airplane wings (fully submerged hydrofoils).

Let me summarize more precisely than before. First, I have not and will not consider the mathematical case where the product of the 2D lift coefficient
and chord vary spanwise to produce an elliptic circulation density, because that looks to me like a design nightmare (variable attack angle) . Being a simple person I restrict to the spanwise variation of chord distribution. First, consider the lowest order approximation where the aspect ratio is large so that the induced correction to the attack angle can be neglected. This is really the case that I've talked about.
1. For a flat (meaning uncambered wing) the 2D circulation goes like the cord c(z). By either dimensional analysis or by generalizing the 2D
math the circulation density G(z) is proportional to c(z). Therefore an elliptic wing shape gives an elliptic circulation density. Including the induction correction to the
angle of attack in lifting line approximation doesn't change this.
2. For parabolic camber at zero attack angle G(z) again must be proportional to c(z) and also to the dimensionless the factor bc(z) where b is the
camber parameter. I consider only constant camber. So G(z) is proportional to bc^2(z) and an elliptic wing shape gives a parabolic G(z). Adding camber to a flat elliptic wing reduces the lift/drag ratio (I would not take this as advice in surface piercing propeller design, which is my main interest). Adding on the induction correction of a negative attack angle gives a term in G(z) proportional to c(z) but the integral multiplying it also varies with z, the induced angle is not constant along the camber. The resulting G(z) is far from elliptic with an elliptic wing shape.

I admit that an extreme example from the Wright Bros. era is interesting: that of a flat rectangular mean camber surface. Without camber G(z)≈Ca(z)c(z) where C is constant and a is the angle of attack at span location z. We can take c(z)=c=constant (rectangular flat wing) and give the attack angle a(z) the elliptic distribution. The the wing is continuously twisted and the attack angle vanishes at both tips. This wing produces the optimal lift/drag.

 

I am confused. I have operated many double ended boats that planed.