Why should Mach wave angle be less than wing sweep angle?

This is from my lecture notes:

I see that this is so you can have subsonic flow over the wing planform, but why is this desirable? And are there more reasons?

I can think of heat being an issue if you have a strong shock wave forming on the wing. But I recall that pressure will also increase after a shock, so perhaps there is a reason why you might want the shock to occur inside the wing area? Just guessing at this point.

TLDR; Subsonic perpendicular flow over the wing's front edge lets you optimise the wing for the subsonic take-off and landing.

Don't be mistaken! The flow itself is still supersonic and you will still have to deal with shockwaves on your wing! You will not have (or at least it is very undesirable to have a strong shockwave in front of the aircraft) subsonic conditions over the wing. The perpendicular flow however is not and that what counts for the leading edge.

The leading edge of the wing should be as sharp as possible in a supersonic regime. Curves cause severe drag as multiple infinitesimal shockwaves form and combine into one large one. (In theory it would converge to adiabatic compression which is not at all true for the real case) However, a plane has to take-off in subsonic flight and if you don't have a TWR>1 you'll need your sweet sweet lift, and for this case, a sharp edge is incredibly unfortunate as it will create a very turbulent airflow or even worse separation due to flow from the bottom to the top over the front edge. Therefore, if the perpendicular flow is subsonic you can use a rounded front edge, helping you with take-off and landing.

If you want a very fast plane, (e. g. Ma > 3) you will have troubles as super swept wings won't help you with TO and LDG. Some fighters therefore have adjustable wings but that's a nightmare for safety reasons. In civilian aviation that's not gonna happen any time soon. -> You'll have to consider all your regimes during a mission. A plane that's super efficient in Ma>5 with it's SCRAMJET engines that's not able to TO on it's own is pretty much useless. Same with the wings.

• That's at least the first thing that comes to my mind reading the question. In case I missed something I'll extend the answer. Jul 17 at 18:15
• Thanks for the answer. I think perhaps my lecturer has simplified things. Since we're using Mach number flow tables we always look at examples of strong shock, and therefore flow is actually subsonic in most of the examples given. Jul 21 at 21:00
• That's quite the simplification. The only place a strong shock is desired is in RAMJETs but ongoing research tries to eliminate that, since it's very inefficient, but supersonic burn isn't easy either. I remember we got a delta arrangement with 3D profile and had to calculate all the way through and then once again inside the RAMJET with reflections and all. It isn't complicated but just a lot of work... Jul 22 at 7:55

Slightly longer answer: Subsonic flow has the capacity to negotiate the leading edge, resulting in a suction peak which is missing in supersonic flow. Sweep allows to keep that flow pattern for as long as the flow speed component perpendicular to the leading edge is subsonic.

This suction peak, acting on the forward-facing surface of the leading edge, pulls the wing forward, hence the expression "leading edge thrust". This is missing once the perpendicular speed component exceeds Mach 1, exposing the forward-facing area of the wing to the increased pressure aft of the shock.

This characteristic can be maximized by conical camber, a cambering of the most forward part of the airfoil which increases with span, such that the outer forward wing has a conical surface. This has been introduced with the F-106 and is used on many supersonic designs.

F-106 at an angle to the sun which nicely shows the cambered leading edge (picture source)

• Two questions please: 1. The linked source says the "[LE thrust occurs] as air flows from a stagnation point on the undersurface of the wing around the leading edge to the upper surface." Why isn't that force integrated with the rest that result in the net aerodynamic force? / 2. Why does the cambered LE increase the more outboard (IOW: what's special about the "outer forward wing [having] a conical surface"?) / If those go beyond the scope of this Q&A I'm happy to ask a separate question(s). Thanks!
– ymb1
Jul 18 at 19:18
• @ymb1 1) It is integrated and partially explains the d'Alembert paradox. 2) The outer LE has the most distance to the shock so here the subsonic pattern is "purest" and allows to use LE thrust the most. Jul 18 at 20:13
• Thanks! Why is this effect harnessed more in supersonic aircraft when surely it could be applied to a transonic aircraft with an equivalent leading-edge-perpendicular Mach number? I feel like there is more to the rule of "flow perpendicular to the leading edge is what's important". Jul 21 at 21:07
• @RoryMcDonald Yes, there is. Please read this for an explanation what is so special about Mach 1. Sweep will lessen and smooth out the drag peak at the sound barrier. That is the most important reason for sweeping wings. And of course all sub- and transsonic airplanes use leading edge thrust. That is one reason why subsonic flight is so much more efficient; the other is of course the compressibility expansion effect at supersonic speed which is explained in the linked answer. Jul 21 at 22:06

As explained in another answer, keeping the wing inside the shock cone maintains the subsonic pattern of sideways flow which causes air to flow up around the leading edge. This allows you to use a rounded leading edge, which also works well at low speeds.

But allowing the shock to meet the leading edge is not all bad. This is the condition in which the waverider operates, the Mach 3 capable North American XB-70 Valkyrie being perhaps the most notable example. The trick is to use the shock pressure on the underside to create lift, but to avoid it on the upper surface.

• Thanks, just asked Peter Kampf if leading edge thrust could be used at low speeds. By "meet the leading edge" you mean to just touch the leading edge, correct? Not go past it? Can the same effect be created with a bow shock, or is there too much drag involved? Jul 21 at 21:09
• @RoryMcDonald attaching along the length of the leading edge maximises wave lift. The whole leading edge creates its own shockwave along itself; as I recall any outward-propagating shock merges with it and doesn't "get past" it (not entirely sure). A bow shock cannot attach to the inner leading edge unless the plane is a flying wing or lifting body with no fuselage forward of the lifting surface: most if not all modern experimental waveriders are of this type. Jul 21 at 21:35