Circulation Around an Airfoil and Starting Vortex

[Plasmosis1]

Homework Statement

Using Kelvin’s circulation theorem, find the qualitative and quantitative relation between the circulation around an airfoil and the circulation of the starting vortex.

Homework Equations

Kelvin's circulation theorem: DΓ/Dt=0

The Attempt at a Solution

I don't really know where to begin because I'm not given any information to work with. I don't even know what kind of airfoil or any flow conditions.

 

[JBA]

The only information I can give you is that to the degree that I understand it, Kevin's theorem basically states that for any given curve the properties of the fluid traveling along that curve will be constant at all points along that curve.

Beyond that, I hope that someone on our forum more knowledgeable on the subject can expand on how that relates to the problem given to you.

 

[stockzahn]

Homework Statement

Using Kelvin’s circulation theorem, find the qualitative and quantitative relation between the circulation around an airfoil and the circulation of the starting vortex.

Homework Equations

Kelvin's circulation theorem: DΓ/Dt=0

The Attempt at a Solution

I don't really know where to begin because I'm not given any information to work with. I don't even know what kind of airfoil or any flow conditions.

With the circulation Γ the lift of an airfoil (or of a spinning cylinder, etc.) can be calculated. It is defined as the closed-loop integral of the velocity at the surface of the airfoil. Due to friction and inertia the fluid is deflected at the airfoil and will leave the surface parallel to its rearward edge (with no seperation/Kutta-Joukowski condition). Calculating the circulation of this situation Γ≠0 - that means it is a difference for a particle wheter it flows on the one side or on the side of the airfoil, so what happens is dependent on the chosen way: The velocity field around the airfoil is not nonrotational, so a vortex is created. As Kelvin’s circulation theorem states that dΓ / dt = 0 and with starting (or variable) air flow the circulation around the airfoil will change (let's assume in the time Δt the change of the circulation was +ΔΓ), there also has to be a circulation -ΔΓ to fulfill Kelvin's theorem. That's what happens during the start of an airplane. When it accelerates the circulation around the wings changes and to "compensate" it another vortex rotating in the opposite direction has to be created - the starting vortex. In steady state flying mode the circulation around the wings is supposed to be constant, as no change occurs no "counter-vortices" are generated.

Γ = (closed-loop)∫ v^→ ⋅ dx^→ (around the airfoil surface)

The questions seems to be rather general, so the answer must be quite general too.