What is irrotational flow? How to judge? For example, when the wing moves horizontally, the direction of fluid flow changes first to upward at the leading edge of the wing and then to downward at the trailing edge. Does it rotate? If the direction of motion of the fluid at the trailing edge changes to horizontal, does this also rotate?
 A: Irrotational flow occurs when the cross gradient of the velocity or shear is zero.
i.e. $\nabla \times \vec {v}=0$
If the air is not deflected downwards by the trailing edge, it wouldn't be able to create a lift. Now, the streamlines should meet (nearly) behind the trailing edge.
An irrotational flow is one where the curl is zero. You can imagine it being like this. If you drop something at a point in the field, would it spin? This can be caused by a difference in magnitude of forces facing the same direction, perpendicular forces on certain faces etc.
If you think about it in terms of a differential element it makes sense. Think of all of the different ways you can make a piece of paper spin. This is where the curl comes from.
EDIT: To sum all of this up and to directly answer your question: yes, wings do have lift in incompressible (and compressible), irrotational, inviscid flow.
A: If your question is about how airplane wings work, they cannot work without inducing a vortex. Read this.
