Will two counter rotating vortices cancel each other?

Suppose that an aircraft flies with its wing through a tip vortex of another aircraft which flew in the opposite direction. Suppose that the shed wing tip vortex of both aircraft are exactly the same in strength, only rotating in opposite directions?

What will the net effect be? Will the angular momentum be canceled?

I would like to place this in the light of the existence of a root vortex experienced by wind turbines. From Burton's Wind Energy Handbook (3rd edition, p78) :

For example, on a two blade rotor, unlike an aircraft wing, the bound circulations on the two blades shown in Figure 3.27 are opposite in sign and so combine in the idealised case of the blade root being at the rotational axis to shed a straight line vortex along the axis with strength equal to the blade circulation times the number of blades.

With the aircraft example the circulation is opposite in sign as well. So why is the root vortex a summation of the two blades instead of cancellation?

• Well, suppose we view aircraft 1 from the back and look to the left wing. The resulting rotational motion will be clockwise. Then, when aircraft 2 approaches, we see it from the front. When we look at its left wing (which is on the right form our perspective) it will be a counter clockwise rotation. Commented Feb 1, 2023 at 16:01
• Keep in mind that a vortex is a 3D shape and generally it grows in diameter as the distance between it and the wingtip increases. So in order to fully cancel, you must have two vortices of identical shape, size, and rotational characteristics EXCEPT that the direction of the rotation is opposite. Under these very unusual conditions the two would cancel. Outside of a laboratory environment, I don't believe creating these conditions is possible. Commented Feb 1, 2023 at 16:43
• Even in a lab, it'll only happen on the third Tuesday of the month, and when the cameras aren't working. Commented Feb 1, 2023 at 17:02
• @jwh20 I do not think those conditions can be met unless both wings are in the same place. Usually that makes a big explosion and then there are lots of vortices. Commented Feb 1, 2023 at 18:24
• @CamilleGoudeseune I'd say the cameras will be permanently not working. Commented Feb 1, 2023 at 18:24

Obviously having two perfectly identical and opposite vortices generated in exactly the same space is just an ideal mathematical exercise. I suppose it would something like having two billiard balls moving one against the other along the exactly same trajectory and at exactly the same speed... and that multiplied for all the air molecules!

Anyway something similar actually happens also in reality: with contra-rotating propellers. And luckily enough, this interesting report is just right for our purpose. It shows flow measurements behind naval contra-rotating propellers. Measurements are done in three steps: with propellers stopped, with only one propeller rotating and with both propellers contra-rotating. Unfortunately it doesn't deal with isolated propellers since also the stern of the ship is present in the model. So first of all we have to understand how the water flows due to the hull when the propeller are not rotating:

The (black small) arrows in the plot tell us in which direction the water is locally flowing while the background colours tell us how fast it is flowing (blue is slow, red is fast). The three big white arrows "connecting the dots" are my own work and show more or less the path of the water flow. Following them we see that what the water does in this case of not rotating propellers, is basically closing behind the hull: it goes up and then bend to rejoin in the centre of the picture just behind the propeller axis where the ship ends and then the flow just "dies" there. This picture of the hull should help in understanding this movement (C is the plane where the speeds are measured):

Now, what happens when only one propeller rotates?

We see that the propeller causes the water to rotate anticlockwise around it: the same three white lines of before now clearly bend around the propeller. This whole movement of the wake basically resembles the movement of the tip vortex at the wingtip as seen from behind.

Finally, in the next picture both propellers are in movement:

The three white lines are now again like in the first picture: the contra-rotation of the second propeller has more or less restored the symmetry of the water flow, cancelling the anticlockwise dragging of the first propeller. This image resembles indeed the "smiley face" in the first one: obviously here we have more red than blue (i.e. higher speeds) since the propellers are doing their job accelerating the water, but the anticlockwise rotation given by the first propeller is now more or less balanced by the second one.

Will two counter rotating vortices cancel eachother?

Yes, almost completely, even in reality.

• Your answer is good, but answers not all my questions. So given that counter rotating vortices of equal strength cancel eachother, why then are the root vortices added to a single line vortex. Clearly, the vortex from the root of each blade rotates in the opposite direction. Hence, if they're shed along the same line, which happens when the blade extends to the center of rotation, why don't they cancel? Commented Feb 2, 2023 at 10:51
• Your question is if two contra-rotating tip-vortices cancel each other and the answer is yes (more or less). The vortex structure in a helicopter/wind turbine wake is much more complicated than a wing's tip vortex due to the rotating nature of the entire wake. Note that the two root vortices can never be perfectly aligned due to the hub cutting the blades; plus they have a very weak intensity due to the small lift's change toward the roots. Commented Feb 2, 2023 at 12:17
• The actual conclusion stated by your reference is completely different from your answer. "While the tip vortex generated from forward propeller was almost disappeared, the tip vortex and shed vortex generated from aft propeller were observed clearly. Through the axial vorticity distribution on the longitudinal plane along the propeller shaft, it was noticed that the tip vortex of aft propeller became stronger and its contraction increased due to the interaction between the tip vortices of forward and aft propellers." Commented Feb 3, 2023 at 1:47
• @AnonymousPhysicist: in my answer I'm dealing with the whole vortex shed from the propeller. The whole vortex shed from the propeller resembles the tip vortex shed from an airplane's wingtip. I stated it in my answer: "These movement basically resembles the tip vortex at the wingtips". Anyway I'll modify this sentence to make it clearer 🖖 Commented Feb 3, 2023 at 6:57

Now let's assume the two airplanes do not collide but two of their trailing vortices happen to have their core along the same line. Yes, those two vortices will cancel each other.

But you forget that both planes have two trailing vortices, one on the left and one on the right side. So what is left after half of those vortices cancel each other are two vortices at twice the distance of the vortices of one airplane.

If we neglect dissipation and assume that those two vortices will stay there for some time, they will look just like the vortices left by a single airplane of twice the wingspan and four times the mass of those two which did not collide. However, the downwash between the remaining vortices will look differently and reveal that two separate airplanes were involved in its creation.

For an airplane in level flight, the angular momentum of the aircraft remains zero. As a result, the air the plane flies through is not gaining angular momentum.

Vortices are not vectors and are not subject to any kind of law of superposition. In other words, you cannot add them together. The Navier-Stokes equation is the right way to calculate flow.

• Saying "the wing makes a tip vortex" is already an extreme oversimplification. Commented Feb 1, 2023 at 18:25
• What do you mean with "the air the plane flies through is not gaining angular momentum"? Commented Feb 1, 2023 at 20:52
• No, but the wing tip vortices (or the parcels of air that are typically denoted with tip vortices) DO have angular momentum! Moreover, vortex line codes are derived from the NS stokes equation and the singularities which construct the solution for Laplace equation are these vortex lines (amongst other type of solutions). Therefore, the flow is represented by vortex lines. Basically Kutta, Helmholz, Biot and Savart are the holy fathers here. Commented Feb 2, 2023 at 7:18
• @|Windy As mentioned in the other answers, if you consider the vortices from both sides of the same plane, the angular momentum is zero. The question asked for the "net effect" and I interpret that as including all the air. Commented Feb 3, 2023 at 1:38

Will two counter rotating vortices cancel each other?

• Yes

Suppose that an aircraft flies with its wing through an tip vortex of another aircraft which flew in the opposite direction. Suppose that the shed wing tip vortex of both aircraft are exactly the same in strength, only rotating in opposite direction?

Wheather the planes fly in the same direction or opposite directions will have no effect on the vortex directions - remember that the vortices are symmetrical on the lateral axis. Given that the wing produces a positive lift, the left and right vortices will always rotate clockwise and counterclockwise respectively, regardless of the point of view. Look at the following picture, it shows that the vortices rotate in the same direction.

With the aircraft example the circulation is opposite in sign as well. So why is the root vortex a summation of the two blades instead of cancellation?

As seen in the picture, the signs are same, not opposite. The vortices will therefore add up, not cancel out.

A wing that produces negative lift can produce vortices of the opposite direction (left counterclockwise, right clockwise). If a positive lift producing wing was to intercept its wake, then indeed the vortices from the two wings will be opposite and tend to cancel out.

However, there is one more way of producing opposing vortices - without involving a negative lift producing wing. The clockwise vortex of one wing's left-side can cancel out the counterclockwise vortex of another wing's right-side. Indeed the vortices will tend to cancel out, and the two wings will basically act as one larger wing.

• I think your picture does not match the intent of the question. I think the idea was that the flight paths are offset by one wing span, so that the wing tips are aligned and the fuselages are not. Commented Feb 3, 2023 at 1:35
• @AnonymousPhysicist I'm not sure if question indicates that the flight paths are offset by one wingspan... Commented Feb 3, 2023 at 1:51
• Yes it is indeed as @AnonymousPhysicist says: I also did the same mistake at the beginning and asked for clarification. So, both left (or right) wing tips overlap. Anyway the question isn't that clear... Commented Feb 3, 2023 at 9:18