Lift direction at non-zero angle of attack for a swept wing, or for a straight wing in sideslip

Question

When the direction of forces is popularly explained, they say "the lifting force is directed perpendicularly, and the drag force is parallel to the oncoming flow".

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But what if it's a swept wing? In that case, which variant is more correct and why? Will the chosen variant apply to all objects affected by aerodynamic forces and, if not, what are the exceptions?

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Also, if in reality the forces are directed as in the second variant, are most of the further formulas created in such a way as to obtain a mathematical model of the first variant? For example, the lift curveslope of a swept wing will be less than that of a straight wing - is this the result of a conversion between models? I would appreciate as much information as possible, because I want to have a solid idea.

Answer 1

Lift is defined to be perpendicular to the flow. Drag is defined to be in the direction of the flow.

What your intuition is missing is that these are not the only two aerodynamic forces. There is also a third component of force -- typically called sideforce.

In many situations, the side force is zero or minimal, so we often work in the '2D' frame of just lift and drag. Most aircraft are symmetrical -- when they fly in a symmetrical condition, the side force generated by the left half of the aircraft cancels with the side force generated by the right half of the aircraft.

Side force is only present when the something is asymmetrical.

We can resolve any force into three perpendicular components -- Lift, Drag, Sideforce or X, Y, Z, or East, North, Up. At different times we make different choices of how to resolve a force because it makes analysis easier down the road.

Because of this, Lift, Drag, Sideforce is just one possible choice. It is the choice defined with lift perpendicular to the flow and drag parallel to the flow. One benefit of this choice is that sideforce is usually zero.

Answer 2

the direction of forces is popularly explained

Luckily enough science is not democratic: definition are not chosen by show of hands every 5 years but so that they make sense and cannot be mistaken.

When an object moves in a fluid, a fluid dynamic force arises. If we consider a 3D object moving in a 3D space, then this force can be decomposed in... well, 3 components.

One component is quite easy to choose: since a part of the fluid dynamic force is always pointing in the main direction of the flow (so called freestream or $V_{\infty}$), then one of those three components is simply aligned with the flow and termed drag.

Therefore the other two components are perpendicular in respect to the drag (i.e. in respect to the flow). How are they fixed? If the object is an aerodynamic object, then it makes sense to define one of those two perpendicular components as going from the belly to the upper surface of the body; this component is termed lift.

Finally, the third and last component points laterally from the right side to left side of the object.

So, variant 1 is the correct one.

The picture in your variant 2 is normally used to schematise how locally the airflow bends as it approaches the leading edge of a swept wing and is used to explain the reduction in transonic drag given by swept wings in respect to a straight one. Here there's a good explanation of this phenomenon.

Answer 3

It is not a “popular explanation”, it is the actual definition. To be more precise, there is some aerodynamic force and we call the component along the direction of airflow (relative to the wing) drag, and the component perpendicular to the flow lift. This applies in three dimensions, not just two.

Now two parallels are parallel in all views, so for drag, variant 1 is the only one possible.

For lift it is a bit more complicated though, because parallel to the flow might mean up, sideways or any oblique angle in between. For a symmetric aircraft the sideways components cancel, so the total lift will be up, but for each wing in isolation it may be slanted sideways.

But sweep does not slant it, dihedral does. Pressure always acts perpendicular to the surface, so lift is approximately perpendicular to the wing. And since horizontal swept wing is still horizontal, the lift will still be vertical. But if the wing has dihedral—is canted from horizontal laterally—then the lift will be “toed in” by about that angle.

Answer 4

With "Variant 2", the statement that "the lifting force is directed perpendicularly (to the oncoming flow), and the drag force is parallel to the oncoming flow" would be accurate from a side view, but not from a top view, and also not from a head-on view or tail-on view.

So "Variant 2" is not an accurate depiction of the definitions of the Lift and Drag vectors for a swept wing.

"Variant 2" tends to beg the question of whether or not sweeping a wing causes it to generate some aerodynamic sideforce, acting toward the wingtip. Sideforce being a third aerodynamic force, defined as acting perpendicularly to both lift and drag. That is really an entirely separate question. This would have interesting consequences during sideslip, where one wing becomes "more swept" than the other wing, in relation to the actual direction of the airflow. Generally speaking, the answer to that question appears to be "no" -- and all-wing aircraft¹, including swept-wing all-wing aircraft, are well known for generating minimal aerodynamic sideforce during a sideslip². But I have seen a reference to the idea that "leading-edge suction" would in fact tend to cause that effect to some degree.³

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Footnotes:

1. By "all-wing" aircraft, we mean flying-wing aircraft lacking tails and fuselages, whose cross-sectional area as seen in a side view is low. Such as the XB-35, XB-49, and B-2.

2. See for example this passage from the 35th Wilbur Wright Memorial Lecture which Jack Northrop read to the Royal Aeronautical Society on May 29, 1947:

Side force effects: All-wing airplanes, particularly those without fins, have a very low crosswind derivative; thus a low side force results from sideslipping motion. Some crosswind force is probably important for precision flight, such as tight formation flying, bombing runs, gun training maneuvers, or pursuit. This importance arises because with low side force it becomes difficult to judge when sideslip is taking place, as the angle of bank necessary to sustain a steady sideslipping motion is small. This lack of side forces has been one of the first objections of pilots and others when viewing the XB-35. After flying in the N-9M or XB-35 the objection is removed, except for some of the specific cases mentioned above. For the correction of the lack of sideslip sense, a sideslip meter may be provided for the pilot or automatic pilot, and for very long-range aircraft there is a valuable compensating advantage in being able to fly under conditions of asymmetrical power without appreciable increase in drag.

3. Email or on-line forum statement by Stephen Morris, co-designer of Bright Star Millennium and Swift rigid-wing ultralight hang gliders.