A wing is like a vortex-holder, it holds a vortex. The combination of the vortex along the wing and the parallel stream perpendicular to the wing creates lift (the wing flying in the air).
Unfortunately a vortex can not just end at the wingtips, it must drip off (vortices don’t just end, they are or closed in themselves or ending on an infinite wall or ending at infinity). If it would be possible to end the vortex at the wingtip there would be no induced drag.
All vorticity of the wing is also kept behind the wing. Some of it is left behind along the trailing edge of the wing, but most of it is dripping off the tip-region of the wing. This vorticity behind the wing is inducing velocities onto the wing (law of Biot and Savart) which reduce the effective angle of attack of the wing. The delta angle of attack (the induced angle = CL/(Pi A e)) has the effect of tilting the lift backwards.
In other words, Lift being defined as the force perpendicular to the direction of incoming airflow, is now not perpendicular to the direction of flight any more, but slightly tilted backwards by the induced angle.
This creates a drag force, the induced drag ~equal to Lift * induced angle
= Lift * CL/(Pi A e)) = CL* .5 * airdensity*V^2 * S * CL/(Pi A e))
V is the airspeed, S is the area of the wing, Pi = 3,141592…, A is b^2 / A, b is wing span, CL is lift coefficient of the wing.
e is a factor depending on the wing shape. When the wing is elliptical e=1, else e < 1.
For a given wing span b, an elliptical wing has minimal induced drag. It is useless to put a winglet on such a wing if the only intention is to further reduce the induced drag of that wing. Else, if the wing is not elliptical (e<1) a winglet can further reduce drag, but you need to design the winglet together with the wing.
Every wing needs its own special winglet!.
Regards,
Onemorechance