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I'm working on automating some aerodynamic calculations for a particular multidisciplinary design tool and in the course of setting it up I ran across this formula from the input theory manual $$ {CL_\alpha} = \frac{cl_\alpha}{1+cl_\alpha (1+\tau)/(\pi AR)} $$

a mostly typical formula for estimating the 3D lift curve slope from the 2D lift curve slope with one addition - $$ \tau $$ casually mentioned "where tau accounts for non-elliptical loading" and it defaults to 0.25. I've played around with the formulation and what it's doing makes sense but I'm wondering if anyone has any references for this? I've looked through some of the Trefftz plane wing loading calculations, but I haven't gotten my hands dirty yet - I'm quite sure someone has already done this but I have yet to come across it. I need to include some sort of justification or method of determining the deviation from an elliptical lift distribution for the future user.

Thanks!

Edit to Add:

I found this formulation in a coefficient of drag equation from Estimating the Oswald Factor from Basic Aircraft Geometrical Parameters

$$ {C_{D,0}} + \frac{C_L^2}{\pi AR e} = {C_{D,0}} + \frac{C_L^2}{\pi AR }(1+ \delta) $$

I'm wondering if this is what I'm searching for aka $${\tau} = {\delta}$$ Any other sources, derivations pointers in the right direction would be very welcomed!!

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  • $\begingroup$ Is this related to the Oswald efficiency factor, en.wikipedia.org/wiki/Oswald_efficiency_number? But at wikipedia at least, that specific number is only related to Drag. $\endgroup$
    – U_flow
    Jun 28, 2022 at 8:25
  • $\begingroup$ @U_flow I would think that they must have some relationship since they both are a measure of the wings lift distribution compared to ideal aka elliptical lift distribution. Oswald or span efficiency is well documented I would think that there must be something out there for tau $\endgroup$
    – spacegirl1923
    Jun 28, 2022 at 13:10

2 Answers 2

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There are multiple formulas based on whether the aspect ratio is high or low, whether there is wing sweep, whether the flow is incompressible or if you need to add a compressibility factor, etc. The formula you listed is correct but not one size fits all.

The formulas and their explanations can be found in the following books:

Aircraft performance and design by John Anderson

An introduction to aircraft performance by Mario Asselin

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  • $\begingroup$ Could you perhaps indicate for which condition the Formular holds? $\endgroup$
    – U_flow
    Jul 7, 2022 at 19:07
  • $\begingroup$ The formula in the question holds for incompressible flow (M < ~0.3), high aspect ratio (AR >= 4), straight wings. Some of the equations for the other cases are long or require reading values from graphs, so it would be hard to post here, which is why I listed the books. It is correct when he said tau = delta. The oswald efficiency factor, e, is given by e = 1/(1 + delta), where e is roughly 0.95. The values are both found with prandtl's lifting line theory and depend on the wing geometry (taper ratio and aspect ratio). $\endgroup$
    – user1801359
    Jul 8, 2022 at 1:40
  • $\begingroup$ what she said. Also I don't think this is quite right. Both quantities can be derived from lifting line theory but tau /= delta. I don't have the Mario Asselin text but I have several Anderson texts and he uses a value e1 and makes it clear this is Not the same oswald efficiency factor. e1 = (1+tau) and e = 1/(1+delta). I'll answer this question with charts and resources for future searchers. $\endgroup$
    – spacegirl1923
    Sep 7, 2022 at 18:37
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I wanted to post the formal answer here for any one who stumbles across this search and needs the answer and resources.

First:

$$\tau \neq \delta $$

Both values can be derived by examining circulation $K$ and for a rectangular wing

$$\ 1+\delta = \frac{\sum_n A_n^2}{A_1^2}$$

and

$$\ 1+\tau = \frac{1}{\mu}\left(\frac{\mu\alpha}{A_1} - \frac{\pi}{4}\right)$$

where the A coefficients are from the series for circulation. For those not wanting to calculate these the following charts will be most useful:

Glauert AerofoilAirscrewTheory Figure 85

and

Glauert AerofoilAirscrewTheory Figure 86

and

Glauert AerofoilAirscrewTheory Figure 88

For references and to look at the effects of taper, twist and such please see Glauert, "Aerofoil and Airscrew Theory" pages ~ 146-155

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  • $\begingroup$ This reference will be helpful - Airplane Aerodynamics by Dommasch, Sherby, and Connolly, 1967. Pitman Publishing Corp., New York. 4th ed. Pay special attention to Chpt 5, Wing Theory. Fully read Sect 5-2 regarding application of tau & del to convert one wing to another of different aspect ratio. Here is a link to exactly the same, 3rd ed text: Go to Viewability, select Full View, then Jump to Section 7 - 149. Use this link... catalog.hathitrust.org/Record/001039657 $\endgroup$
    – Thomas Perry
    Sep 7, 2022 at 21:57

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