Exploring the Relationship Between Circulation and Lift in Potential Flow

[hanson]

Hi all.
I am revisiting things about potential flows.
So, I believe the assumptions of potential flows shall be the irrotationality and the ignorance of viscosity.

And I am studying the flow past a cylinder with and without circulation.
Without circulation, there is no lift force but when there is circulation, then there is lift force.
The lift force is due to the inclusion of circulation, which shall be equivlent to the inclusion of vorticity, right?
For cylinder with circulation, the vorticity is concentration in the center of the FREE vortex introduced, right?
Would this sort of contradicts the assumption of irrotational flow?

And it seems to me that this example (cylinder with circulation) is applied to study real flows in daily life, right? Since life force is always observed e.g in airfoils and spinning of gold ball (sudden fall) etc. So there shall be some relationship between the real life examples and this cylinder with circulation, isn't it?

For the case of airfoil, while the cylinder with circulation study ignores viscosity, but the circulation needed for the airfoil to generate a lift force is actually caused by the viscosity in the boudary layer, am I correct? So, I understand this as follows:
Yes, the potential flows ingores the viscosity but it still generate lift force because vorticity is added deliberately. (Somehow vorticity is added) But in real life, this vorticity is caused by the viscous effect in the boundary layer, which genrates some vortex and hence vorticity is introduced. Am I thinking correctly?

And for the case of spinning ball, it is the viscoity in the boundary layer that drags the air around the ball and hence vorticity is diffused. And the study of potential flows on cylinder with circulation can be applied to qualitative explain the life force because there is some circulation which is deliberately added around the cylinder by adding a free vortex, right?

So, if the fluids in the world is really of zero viscosity, then the spinning ball and rotating cylinder and airfoil cannot produce lift, right? Since there is actully no circulation and source of circulation, what do you think?

 

[AlephZero]

Real fluids are not zero viscosity (possibly with a few very exotic exceptions).

In reality, an aerofoil like a plane wing creates a vortex because of viscosity in the air. This is shed from the wing tip and leaves a line vortex in the air behind the aircraft. If an aircraft is flying at low altitude with high wing loading (e.g. a military jet flying a tight turn), this is sometimes visible as a "line" of water condensation trailing from the wing tips (this has nothing to do with the contrails from engine exhaust at high altitude)

The assumption of zero viscosity, plus a circulation to generate lift, is an approximation which simplifies calculations. The magnitude of the vortex relative to the irrotational flow can be set using the "trailing edge condition" that the flow at the trailing edge of the aerofoil is continuous (i.e. the velocity on the top and bottom surfaces at the trailing edge is equal).

This theory doesn't attempt to explain how the vortex forms. It's a simple approximate way to calculate the lift etc for steady flow over a body. The vortex forms (and changes strength) while the flow is unsteady (for example while an aircraft is accelerating or decelerating).

 

[skr777]

OK. Zombie thread, I know. What AlephZero says is true, but it doesn't really answer the OP's question. Maybe I can shed some light.

The easiest analogy I can think of is that in an irrotational flow, a fluid "particle" can't rotate about itself (or rather, the concept of self-rotation is non-existent), but that doesn't mean it can't rotate about other objects. Mathematically, it means that the potential function is a conservative vector field, which is (almost) the same as saying the field is a solution to Laplace's equation (its curl is zero everywhere).

Maybe this will just confuse things, but note also that vortices (and sources/sinks) in a potential flow field are singularities, where the field is not conservative (i.e. not necessarily irrotational).

 

[Mech_Engineer]

If you know it's a zombie thread, why bother bringing it back to life? Especially since the OP hasn't been active for a good while.