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According to the theory I know, lift is produced by accelerating air downwards at the trailing edge (downwash). Hence, as downwash increases so does lift. Induced drag is due to vortices at the wing tips and they also create downwash. So according to the above theory induced drag should also increase lift.
But, the books say that induced drag increases downwash and reduces the effective AOA, which subsequently reduces the lift. Does the downwash from induced drag increase or decrease lift?
It is not "this causes that" - all is happening together. Let me explain:
For me to understand aerodynamics, it helped to disregard all that talk of vortices and induction, but focus on the pressure field around a wing. When the theory of flight was developed, electricity was new and exciting, and it just happened that electric induction could be transferred to lift. Now every author still copies the explanations from a century ago, but they are totally unintuitive.
Every air molecule is in a dynamic equilibrium between inertial, pressure and viscous effects. Inertial means that the mass of the particle wants to travel on as before and needs force to be convinced otherwise. Pressure means that air particles oscillate all the time and bounce into other air particles. The more bouncing, the more force they experience. Viscosity means that air molecules, because of this oscillation, tend to assume the speed and direction of their neighbors.
Now to the airflow: When a wing approaches at subsonic speed, the low pressure area over its upper surface will suck in air ahead of it. See it this way: Above and downstream of a packet of air we have less bouncing of molecules (= less pressure), and now the undiminished bouncing of the air below and upstream of that packet will push its air molecules upwards and towards that wing. The packet of air will rise and accelerate towards the wing and be sucked into that low pressure area. Once there, it will "see" that the wing below it curves away from its path of travel, and if that path would remain unchanged, a vacuum between the wing and our packet of air would form. Reluctantly, the packet will change course and follow the wing's contour, but not without spreading out (= pressure loss). Spreading happens in flow direction - the packet is distorted and stretched lengthwise, but contracts in the direction orthogonally to the flow. This fast-flowing, low-pressure air will in turn suck in new air ahead and below of it, will go on to decelerate and regain its old pressure over the rear half of the wing, and will flow off with its new flow direction.
A packet of air which ends up below the wing will experience less uplift and acceleration, and in the convex part of highly cambered airfoils it will experience a compression. It also has to change its flow path, because the cambered and/or inclined wing will push the air below it downwards, creating more pressure and more bouncing from above for our packet below the wing. When both packets arrive at the trailing edge, they will have picked up some downward speed.
Behind the wing, both packets will continue along their downward path for a while due to inertia and push other air below them down and sideways. Above them, this air, having been pushed sideways before, will now fill the space above our two packets. Macroscopically, this looks like two big vortices. But the air in these vortices cannot act on the wing anymore, so it will not affect drag or lift. See here for more on that effect, including pretty pictures.
Following the picture of a pressure field outlined above, lift is the difference of pressure between upper and lower surface of the wing. The molecules will bounce against the wing skin more at the lower side than at the upper side, and the difference is lift.
Or you look at the macroscopic picture: A certain mass of air has been accelerated downwards by the wing, and this required a force to act on that air. This force is what keeps the aircraft up in the air: Lift.
Either way, you will arrive at the same result. By the way: Most of the directional change happens in the forward part of the airfoil, not at the trailing edge!
The misconception about those "wingtip vortices" and induced drag is hard to eradicate. Most authors copy what has been written before without clearly understanding the issue. Therefore I repeat it here again: Induced drag is the backward-pointing part of the pressure force vector. The vortices are only a consequence of downwash, which in turn is a consequence of lift creation. At the same speed, more induced drag is indeed linked to more lift, but the causality is different: Lift and induced drag are both part of the pressures acting on the wing. If you add up all the pressure forces acting on a wing, their resulting vector will point slightly backwards. The streamwise component is drag, and the component orthogonal to the direction of movement is lift. This is just a defininion, made for simplicity.
I believe the answer given by Peter Kampf is confusing pressure drag with induced drag.
Pressure drag is, as he said, a result of the pressure force acting on the wing having a horizontal component(drag) as well as a vertical component(lift). Induced drag is primarily down to the effect that the wingtip vortices have on the airflow.
The vortices make the air ahead of it, i.e. the air on top of the wings, increase their velocity downwards. This has the effect of negative lift or downforce. This means for the same amount of lift the pilot must increase the angle of attack.
However, this also means drag has increased as the pilot has increased the angle of attack. This increase in drag is what is called induced drag. The formula for the coefficient of induced drag is $Cl^2/(\pi \cdot AR \cdot e)$ where $Cl$ is the coefficient of lift, $AR$ is the aspect ratio and $e$ is the Oswald efficiency which relates the shape of the wing to how much induced drag it creates.
From here you can see that increasing the speed of the aircraft reduces induced drag, as for the same lift an aircraft travelling faster requires a lower $Cl$. Increasing $AR$ reduces induced drag as having longer wings means the wingtip vortices are further away so less of the wing is being affected strongly by the vortices. And $e$ is just depended on your wing shape. The optimum value of $e$ is 1 for an elliptically loaded wing.
There are actually two sources of induced drag (see wikipedia article. One source is described well by Peter's post.
But the other source is vortex drag, from the wingtip vortices (turbulence), which depend on the aspect ratio of the wing as well as on wingtip treatments like winglets. In the end the wingtip vortices lower the glide ratio (L/D) of the aircraft. So if they produce a little bit of lift, it doesn't help because the drag has increased by a bigger factor, lowering the L/D.
The induced drag is due to vortex at wing tip and it generates a downwash. Hence, as per above theory it should produce more lift.logical fallacy. lift is produced if THE WING accelerates air downwards. Downwash is not the wing. $\endgroup$