Thin Aerofoil Theory & Vortex Panel Method

Question 1

How is a vortex sheet constructed? How are they used?

Answer 1

• Straight lines with line vortices of constant strength
• place an infinite number side by side to create a vortex sheet with strength γ(s)
• Each infinitesimal vortex filament will induce infinitesimal velocity at a given point - use velocity potential instead as it’s a scalar
• thin aerofoil theory: use on chord line
• vortex panel: use on surface of aerofoil
  Can integrate over vortex sheet to get circulation, then lift, etc.

Question 2

What condition is placed on flow over an aerofoil in both thin aerofoil theory and vortex panel method?

Answer 2

Kutta condition:
- for steady flow over aerofoil not near stall
- low to medium AoA
- at a given angle of attack a circulation is adopted such that the flow leaves the trailing edge smoothly
- Finite TE: velocity is zero -> stagnation point
- Cusped TE: Velocity is finite but equal for upper and lower surface
- vortex sheet strength is zero at TE
- nature uses friction to enforce this

Question 3

Why is it dangerous to take off right after another aircraft?

Answer 3

• tip vortices
• starting vortex:
• as aerofoil starts to move through fluid, large vorticity produced at trialing edge which curls into a CCW vortex
• at steady flow vorticity is no longer produced and vortex trails into wake
• from Kelvin’s circulation theorem we know the circulation is the same as at rest - the circulation about the aerofoil in steady flight is zero so the starting vortex has induced circulation about the aerofoil equal in magnitude but opposite in direction
• viscous phenomenon

Question 4

Summarize the thin aerofoil method.

Answer 4

• start with thin aerofoil, approximate as its camber
• approximate camber as a streamline of the flow and replace with vortex sheet
• approximate vortex sheet on the chord line (γ(x))
• find γ(x) such that the normal component of velocity at the camber line (induced and free stream) is zero - make the camber line a streamline of the flow
• fundamental eq of thin aerofoil theory: sum of the induced and free stream velocity at the camber line is zero

Question 5

What are some key findings of thin aerofoil theory?

Answer 5

Lift slope of 2π rad-1
Moment about quarter chord independent of AoA (aerodynamic centre)
For a symmetric aerofoil the centre of pressure is zero

Question 6

Describe the implementation of the vortex panel method.

Answer 6

• numerical method to predict performance of thick aerofoils and other bodies with high angles of attack
• approximate body with a set of straight panels with different lengths and a unique but constant strength
• solve the strength of n panels such that the normal component of velocity at each control point is zero
• drop one panel equation to implement Kutta condition
• can use strengths to find tangential velocity and pressure distribution over aerofoil surface
• can’t solve stall, depends on paneling, accuracy depends on which panel is dropped

Question 7

What are the benefits of using XFOIL?

Answer 7

• viscous silver (bls, stall, drag)
• compressibility effects
• transition can be specified