# Lift onset, planing, and lift coefficient

Discussion in 'Hydrodynamics and Aerodynamics' started by sandhammaren05, Aug 13, 2018.

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### sandhammaren05Senior Member

Here's part B (part A is above), your homework assignment: Calculate lift over drag for (i) a half ellipse with straight traing edge and (ii) the same for a straight leading edge. The lift in both cases is half that for an elliptic wing. The integrals for the induced drag are formulated in both Landau-Lifshitz and in
Newman. If you follow Newman then you expand the circulation density in Fourier series and he even provides a table of integrals that you need for calculating the drag. It's an interesting and instructive exercise (good for my hydrodynamics class next fall). Case (i) when applied to a propeller designs an uncambered cleaver, where a radial line and and a half ellipse in the plane are projected onto a helicoid. Your recommended case (ii) applied to a propeller would design an inverted (uncambered) cleaver blade. Design (ii) is not produced as a propeller.

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### DCockeySenior Member

By Newman I assume you mean Marine Hydronamics, J. N. Newman, 1977. Induced drag and Lifting-Line Theory are covered in sections 5.10 and 5.11. (I do not see any tables of intergrals in it. Did I overlook the tables or is there a later edition?)

The planforms of the three wings you describe are illustrated below. "0" is an ellipse, "i" is a half ellipse with a straight trailing edge, and "ii" is a half ellipse with a straight leading edge. The spans, mid-wing chord, planform area and aspect ratio are identical. The spanwise distribution of chord is also identical. To put it another way the chord at any span location is identical for the three wings and is described by equation (145) in Newman. The result is the lift and induced drag from lifting-line theory are identical for the three wings. Perhaps your erroneous conclusion that the lift of of i and ii is half that of o, the elliptical shaped wing, is the result of a math error.

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### sandhammaren05Senior Member

There is only one edition, induced drag is calculated for the case of optimal L/D pg. 200. The circulation distribution is elliptic. That does not tell us what wing shape gives the optimal L/D. So I calculated it and have stated the results above.

The integrals needed are the Glauert integrals on pg. 186.

I have made no error. The lift for an uncambered half-ellipse is one half that for the uncambered full elliptic wing (same span, half the chord in the former case). Easy to compare those results, the question is what is the induced drag, what is L/D. You may not assume that the induced drag is the same for all three or for any two of the cases, you have to calculate the drag for each case separately. I am not the one who is confused, you claim that the induced drag is the same-so just do the calculations. Furthermore, one does not use a lifting line approximation, one calculates both the lift and drag exactly, as did Newman for the elliptic case. No approximations, just exact evaluation of integrals. If you make the crude approximation that a wing is a single vortex line then many mistakes are made.

So let's see the calculations instead of just words.

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### DCockeySenior Member

Incorrect. Induced drag on p 200 in section 5.10 is calculated for an elliptical distribution of trailing vorticity in the Treffitz plane, not an elliptical wing. The entire discussion in 5.10 is about calculating induced drag using trailing vorticity in the Treffitz plane and does not have anything about determining the trailing vorticity as a function of the wing shape.

Okay, those are the definitions of the Glauert intergrals, not "tables". Not relevant to this discussion. I just was wondering if you were looking at a different version of Newman's book.

Yes there is an error. The lift of the wings illustrated with half-ellipse planform shape is not half of the lift of an wing with ellipse planform shape and the same span and aspect ratio.

Where did Newman calculate lift and induced drag exactly for wing with elliptical planform shape? I cannot find it in my copy of his book but perhaps I'm overlooking it. Or perhaps you are confused.

How did you determine that the lift and drag exactly of a wing with half ellipse planform shape? What integrals did you evaluate?

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### sandhammaren05Senior Member

You are totally confused. There is no 'elliptical distribution of vorticity' in the Treffetz plane. What nonsense. The lift and drag is calculated exactly from the circulation distribution and its derivative integrated over the wing.You read up to where Newman mentions 'Treffetz', then you quit reading. Also, reading is inadequate, you have to work through the math.

Newman does not calulcate for an elliptic wing. New does not determine the wing shape at all, neither in form nor in camber. Newman never mentions an elliptic wing. He expands in FS and shows that the highest L/D is when all Fourier coefficients but the first vanish. He then claims without proof that the circulation distribution is elliptic. I went on and showed that. He never mentions elliptic wings. I went further and showed that for an uncambered elliptic wing the vorticity distribution is elliptic, and for a cambered wing it is not. I have not calculated for a half-elliptic wing, and it's clear to me from your remarks that you don't know how
to calculate anything. Because: while Newman mentions calculating far downstream (Treffetz plane) he goes on and derives integrals for the induced drag that are over the wing, not far downstream. I suggest that before you confuse yourself further you learn how to calculate induced drag and the corresponding lift.

Stop writing nonsense; study and learn something.

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### DCockeySenior Member

Further dialogue with sandhammaren05 would not be productive. But I will continue to be available for discussions with others.

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### gonzoSenior Member

If you derive an integral, it gives the original formula. This statement make no sense.

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### Doug HalseySenior Member

Excuse the naïve question, but why are you discussing lifting-line theory & results at all?

Lifting-line theory is only useful for relatively high aspect ratios.

Don't the planing shapes you are considering mostly have aspect ratio smaller than 1?

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### DCockeySenior Member

The recent discussion of wings and related arose after claims about wing shapes for minimum drag.

Lifting line theory is not applicable to most planning craft. The applicability of wing theory in general to planning craft is generally limited to high Froude numbers such as a racing hydroplane at speed. This was discussed in a previous thread about sandhammaren05's theory of planning. Also, as Doug points out most planing shapes have low aspect ratio, though the actual planing areas of stepped hulls can be relatively high at higher speeds.

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### sandhammaren05Senior Member

You then evaluate the integral for various cases: e.g., (i) an uncambered elliptic wing and (ii) a parabolic mean camber surface

at zero attack angle. Those calculations are exact and show that adding camber reduces the lift/drag ratio.

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### sandhammaren05Senior Member

DCockey got off on the tangent of lifting line theory. Elliptic wing shapes have nothing to do with planing on the water. I
in any case never used lifting line theory. I evaluated the integrals for lift and induced drag exactly.

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### sandhammaren05Senior Member

1. Lifting line theory is inapplicable to planing period.
2. The theory of lift is applicable to all planing hulls as soon as the trailing vortex is shed. It is, however, not possible to write down a lift coefficient independent of the depth Froude nr. Hydroplanes at high speeds are merely one case. There is both experimental and theoretical evidence that the onset of lift occurs at about F≈2.3 where F is the depth Froude nr. The experimental evidence is for a v-bottom boat. The theoretical model was calculated for a flat bottom.
3. We generally avoid stepped hulls in racing. E.g. in tunnel boat and v-bottom racing at speeds to of 60 to 150 mph. The pad-V kilo record is about 130 mph.

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### sandhammaren05Senior Member

Summarizing the facts, as known from calculations (my own):
1. An elliptic distribution of chord is an ellipse.
2. The circulation density is reduced by 50%, for a given span and semi-major axis (chord) if you take half the ellipse. You can easily compute L/D exactly from the integrals for the induced drag, and it's an interesting case for me (with elliptic leading edge, straight trailing edge). A flat fully-submerged hydrofoil with a rounded rather than squared transom would correspond to the opposite case. Induced drag applies to propellers but not to boat bottoms, where lateral waves and spray creation dominate the drag.
3. Earlier calculations of L/D relied on lifting line approximations. But I notused that aproximation, instead I have evaluated the induced drag integrals exactly: Only an uncambered wing of elliptic shape produces an elliptic circulation density (aside from a case that fails the Kutta condition).
4. There is no 'ventilation', the transom is either submerged (before lift onset) or completely dry (after lift onset). Planing occurs at a greater speed (or depth Froude nr.) than lift onset.
5. Tunnel boats have square (but sharp on both bottom and top) trailing deck edges and thereby generate more drag and less lift than were the 'wing' trailing edge sharp. Cleavers have a blunt trailing edge. A good prop man (there are few) knows that the trailing edge of a cleaver should be very sharp on the high pressure side, never smooth or rounded. Rounding the edge reduces the lift.
6. A sharp trailing edge is demanded in all cases for optimal lift, whether wing, boat bottom, or propeller blade.

Lifting line approximations are unnecessary for computing L/D for either a flat wing or an uncambered parabolic mean camber surface. The integrals for the induced drag (Landau-Lifshitz, Newman) can be and have been evaluated exactly for two interesting cases, there are no approximations. The only thing left out is that without solving the entire boundary-value problem with the span-wise and chord-wise circulation density we do not get the aspect ratio dependence of the lift coefficient.

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### gonzoSenior Member

Are you claiming there is no limit to the reduction of lift/drag ratio by adding camber? Also, exact calculations would mean that there are no variations between the theoretical and measured values. Is that also your claim?

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### sandhammaren05Senior Member

Where did your first question come from? Not from me!

Do you understand the difference between exact calculation and approximation?

Do you understand that calculation has nothing to do with measurement?