How can larger wingspan decrease the strength of wingtip vortices?

Question

In the last paragraph from the link below, it states that Vortex Strength is inversely proportional to Wingspan. Why is this?

http://avstop.com/ac/flighttrainghandbook/wingtipvortices.html

EDIT: From https://howthingsfly.si.edu/aerodynamics/vortex-drag: "The farther a vortex is from the main body of the wing, the less influence it has on the wing." Again: why?

Answer 1

The strength of the wing tip vortices depends upon the pressure differential between the top surface and the bottom surface. In general larger heavier planes produce stronger vortices.

Now consider two planes with the same weight but one with shorter wing span and one with longer wing span. The pressure differential between the top and bottom surfaces of the wing for the plane with the longer wingspan will be less ( pressure=force/area). Thus the strength of the wing tip vortices for the plane with the longer wingspan will also be less.

Answer 2

Wingtip vortices, as from the wake rolling up at the wingtips, become larger with increasing lift and decreasing aspect ratio. At equal wing area, the low aspect wing must deflect the airflow to a higher angle, relative to the free air stream.

Answer 3

For a useful comparison, we need to ensure each wing has the same lift coefficient $C_L$. As span increases, assuming chord is constant, then the wing area and aspect ratio ($A$) also increases proportionally. Since wingtip vortex strength is related to the induced drag, you may have seen this relationship:

$$C_{D_i}=\frac{C_L^2}{\pi e A}$$

The larger the span, or aspect ratio, the smaller the induced drag (i.e. wingtip vortex) for the same lift coefficient. But why?

From the circulation theory of lift, any change in the circulation along the span will shed trailing vortices. In turn, these trailing vortices will induce downwash on the wing span. At the wingtip, the downwash rolls over and creates the wingtip vortex. Now if the span is short, the change of circulation will be larger, thereby inducing higher downwash all along the span, and the rollover at the tip will be stronger. If the span is longer, the change is circulation is gentler and less downwash/wingtip vortex is induced.

In the limit where the span becomes infinite, assuming the same $C_L$ for the whole wing, the circulation distribution is stretched to a constant value, such that the change in circulation along the span is zero, then no trailing vortex exists and the induced drag becomes zero as well.

Answer 4

It's because the "leakage zone", where air is flowing around the tip, is smaller relative to the total wing area if you add more span without increasing chord.

Best way to visualize it is to take it to the extreme; look at a sailplane wing, where the area of the wing tip is a pretty small area in relation to the total area, so the leakage and resulting vortice is much smaller relative to the overall package of air displaced by the wing.

In other words, more span takes you a little bit closer to the theoretical infinite span wing that has no vortices at all because there is no tip.

Of course, adding span without reducing chord adds wing area. If you want to compare tip losses of two wings with the same area, they should really be talking about aspect ratio.