Does the quarter-chord-angle rule still hold for wings with very long chords or large differences between leading- and trailing-edge sweep angles?

Question

For most fixed-wing aircraft, the effective wing-sweep angle (i.e., the angle that the aircraft aerodynamically behaves like its wings are swept at) is equal to the angle of the quarter-chord line (the line along the wing formed by the points 25% of the way from the leading edge to the trailing edge).¹ This is because the aerodynamic effects of wing sweep are a product of the spanwise flow along the wing (the airflow from root to tip, or vice versa), which is a product of both the leading-edge sweep angle and the trailing-edge sweep angle; as the contribution from the leading-edge sweep angle is somewhat greater than that from the trailing-edge sweep angle, the effective sweep angle is closer to that of the leading edge than to that of the trailing edge.

One can, however, imagine aircraft where the effects of the leading-edge and trailing-edge sweep angles on the airflow over the wing are less strongly coupled. For an aircraft with a very long wing chord (a very large distance between the leading and trailing edges), one might reasonably expect that air passing over the wing would be affected by the leading edge, then start to return to straight-rearwards flow, and only then be affected by the trailing edge, with the great distance between the leading and trailing edges serving to effectively decouple the effects on the airflow over the wing of the two edges. Likewise, for an aircraft with a very large difference between leading-and-trailing-edge sweep angles (imagine, for instance, an aircraft with its wings' leading edges swept rearwards at 45° and their trailing edges swept forwards at 45°), the aerodynamic effects of these would (one would think, at least) tend to couple together less efficiently, again producing a situation where the airflow is affected by the leading edge and trailing edge in sequential order, rather than all together.³

Does the rule that "effective sweep angle equals quarter-chord sweep angle" still hold for wings with very long chords, large differences between leading-edge and trailing-edge sweep angles, or both? If not, how does one determine the effective sweep angles of these wings?

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¹: For a wing with straight-line,² unkinked leading and trailing edges, calculating this angle is a matter of simple trigonometry. For a wing made up of discrete segments with leading-/trailing-edge sweep angles varying from segment to segment (but remaining constant within each segment), the angle of the quarter-chord line should be calculated for each individual segment. For a wing with curved leading and/or trailing edges, the quarter-chord angle is much harder to calculate by hand, but the overall rule still holds.

²: "Straight" here meaning "uncurved", not "unswept" or "heterosexual".

³: This would be accentuated by the fact that any wing of significant size with greatly-different leading-edge and trailing-edge sweep angles will, by necessity, have a very long chord along at least part of its span, due to the constraints of simple geometry.

Answer 1

Simplifications help to make valid approximations. But they should not be applied when the boundary conditions make those simplifications invalid. Your question is a good example.

This is because the aerodynamic effects of wing sweep are a product of the spanwise flow along the wing

Not really.

Take supersonic flow. Now the leading edge sweep is much more important than quarter chord sweep. A subsonic leading edge will cause leading edge thrust and thus lower drag. At transsonic speed most Mach effects are delayed because the acceleration and deceleration of air flowing over the wing depends on the contour along a line orthogonal to the local sweep angle. Yes, that means that highly tapered, swept wings will have more sweep effect near the leading edge and less towards the trailing edge.

In the end, the acceleration and deceleration of the local airflow depends on the local pressure gradients, that is, it is perpendicular to the local isobars. Those isobars follow the local sweep angle, at least at mid-span. At the center of the wing and at the wingtip the sweep angle of the isobars is lower than local sweep but good designs change the local airfoils such that the isobars are perpendicular to the local sweep angle. Of course, once this is done, the simplification with quarter chord sweep holds true over the full span.

Does the rule that "effective sweep angle equals quarter-chord sweep angle" still hold for wings with very long chords, large differences between leading-edge and trailing-edge sweep angles, or both?

No, because it is a simplification which helps to approximate the flow at mid-span and can be applied mostly to high aspect ratio wings. Once the wing becomes similar to a delta wing, it is no longer valid. On low aspect ratio wings the mid and tip effects will dominate the picture