# Why does induced downwash vary across a wing in a sideslip?

in this excellent answer, it states this :

A well-known effect of wing sweep is the variation of induced downwash along the span from the trailing wake that produces an additional lift distribution characterized by increased loading on the aft wing and reduced additional lift on the forward wing. For a wing with no twist or bend, this results in a significant rolling moment, tending to roll the forward wing downward. There is also a yawing moment from the asymmetrical distribution of induced drag that tends to unsweep the wing.

My question is, what makes the induced downwash along the span vary? Also, when it says "trailing wake", does that mean the wake turbulence from the aircraft, and if so how would that interfere with the induced downwash? (the 2 big vortices formed by the aircraft is what I mean by wake turbulence)

• A bit unrelated, but I was thinking about this also, so I’ll put it here. Say your left wing for whatever reason has no tip vortex. I always thought that would make you roll to the right, as the downwash from the vortices would be “pulling” the right wing down. Is that correct? Commented Mar 23 at 2:55
• Good question. Wouldn't the absence of the tip vortex indicate the left wing was producing no lift? As indicated by the downwash, isn't the net displacement of momentum downward on the right wing? In that case, with lift indicated on the right wing, wouldn't the aircraft roll to the left? Commented Mar 23 at 3:41
• Yes that would definitely indicate that, but pretend that the lift is still being created but somehow the tip vortex isn’t. Isn’t possible in real life, but just a thought experiment. Commented Mar 23 at 4:19
• @ThomasPerry Also, did you mean downwash from the tip vortices (induced downwash) or just the plain downwash from the wing? Commented Mar 23 at 4:56
• Conceptually, downwash and the presence of tip vortices are all part of one continuous process. Commented Mar 24 at 3:13

This effect is really only substantial for swept wings.

Here is the lift distribution on an un-swept rectangular wing at 10deg AoA at 0 and 10 deg Beta.

Notice the general slight reduction in lift - but also the slight asymmetry introduced.

If you were to plot the wakes, they are pretty uninteresting. The 10deg Beta wake is essentially sheared 10deg to mostly align with the freestream.

However, let's look at the same wing with 20deg of sweep. If you draw some velocity diagrams, you'll find that the left and right wing seem to end up seeing substantially different angle of attack.

It is clear that the two half-wings are seeing very different flow situations. This results in a dramatically changed lift distribution for the two.

Notice the large change in lift distribution on the two wing halves. It is as if the right wing sees a higher angle of attack than the left wing.

The wake is the sheet of vorticity that trails behind a lifting surface. It is stronger at the wingtips than across the rest of the wing, but it exists everywhere there is circulation.

Edits:

It looks like I opened a can of worms with the angle of attack analogy.

I based my comment on foggy memories of a classical argument for a $$C_{l,\Beta}$$ derivative (dihedral effect, rolling moment wrt. sideslip) and how it depends on wing sweep angle.

For example, from: Etkin B. and Reid, L.D., "Dynamics of Flight; Stability and Control", Third Edition. John Wiley & Sons 1996.

quiet flyer -- My memory was wrong -- they make an argument based on velocity magnitude perpendicular to the wing -- not based on velocity triangles changing the local angle of attack. I think this tracks better with your understanding.

Interestingly, the dihedral effect due to dihedral is calculated with the velocity triangles I remember.

Amusingly, he goes on to make an argument about the oblique wing in this discussion...

Sophit - you're right, I read the OP's question without clicking on the link. So I was answering for actual swept wings.

For Robert DiGiovanni, Here is a set of load distributions for the straight wing at 0,30 deg Beta and 5,10 Alpha.

And the 30deg Beta, 10deg Alpha plot of delta Cp and the wake -- rotated about 30 degrees to make it look like an oblique wing.

• Not sure if I’m not picking up on something, but is the reason the 2 half’s of the wings end up at different AoA because of the tip vortices? Commented Mar 23 at 5:11
• Please note that in the linked answer, with "swept wing" they actually mean "oblique" wing... The present answer, albeit being technically perfect, is therefore formally wrong. Commented Mar 23 at 6:37
• @RobMcDonald please run the unswept wing at 30 degrees beta and also try 5 degrees AoA (to continue work on oblique wings). The swept case reveals that a yaw unsweeps one and sweeps the other more (good for Dutch Roll studies). I am curious to see if the effect of sweep on lift is linear, especially at higher beta. Commented Mar 23 at 7:20
• Seperate question: in the view from above of the swept wing, if the airflow is crossing from right to left in the aircraft's reference frame, why do the airflow lines right at each wingtip seemed to be kinked the opposite way? I.e. on the right wingtip the line bends slightly outboard but on the left wingtip it bends slightly inboard? Is that because the right wing is generating more lift than the left wing? Commented Mar 23 at 14:52
• I would agree that points 1 and 2 in the other post are not different phenemona. When you take the simple sweep theory approach of 'normal velocity matters', you can book-keep the change as a change in chord, velocity, alpha, or whatever. If the whole expression is multiplied by $\cos(\Lambda)$, then you can choose to interpret that effect on any single term. I would argue that this is a fundamental geometric effect -- it is not an effect of induced downwash. Commented Mar 25 at 16:13