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I mean the optimal taper ratio for an unswept trapezoidal wing for minimising induced drag.

I see a lot of places recommending a value of 0.45 to give as close to an elliptical distribution, therefore maximising Oswald efficiency (to 1). However, Hoerner (Fluid-Dynamic Drag, chapter 7-2) has this plot which suggests the optimal taper ratio is about 0.356 (graphical line fit).

Taper ratio induced drag relation

Similar graphs are produced in this paper: Induced Drag of High-Aspect Ratio Wings Tommy M. Chen and Joseph Katz

This exact taper ratio is used by BAe regional jets also.

I haven't read about this thoroughly enough. Maybe this value considers root chord and wing area to be fixed, therefore as taper ratio is decreased the aspect ratio must increase to maintain a constant area. The increased AR (really the increased span) reduces the induced drag.

Does anyone know the answer? I'd really like to be able to justify my taper ratio for an aircraft design assignment.

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The excerpt you posted mentions a further modification of the Oswald efficiency/taper ratio curve which is not applied to the graph in the excerpt. With that modification (which I believe is included in the following pages of the book), the optimum point may move to 0,45. For an assignment, the other places you found 0,45 from and the BAe wings are sufficient motivations to set it at 0,45 IMHO.

I'd like to mention that taper ratio is not the only way to adjust spanwise lift distribution, you can change the airfoils (which you mentioned you have excluded from your design) or you can add twist to the wing so that the wingtips work at a lower AoA. This has the added benefit of making the wing root stall before the wingtips where the ailerons are located.

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    $\begingroup$ Besides loss of authority from the ailerons in a stall, a tip stall is likely to induce a spin, which a root stall will not. ‘Wash-out’ towards the wingtips is desirable as a safety feature as well as reducing parasitic drag. $\endgroup$ – Frog Dec 12 '20 at 2:08

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