Lifting-Line Theory

Question 1

What is the basis for the lifting line theory (what makes up the horseshoe vortex)?

Answer 1

• Curved vortex filament
• 3D vortex filament
• Use Biot Savard law to find induced velocity
• Semi-infinite filament starts at bounds of fluid and end in infinity
• Helmholtz Vortex theorems:
  
  • Vortex filament has constant strength along its length
  • vortex filament cannot end in a fluid - must extend to boundaries (incl infinity) or form a closed path

Question 2

How was lifting-line theory developed? How is it implemented?

Answer 2

• started using single horseshoe vortex to model finite wing
• Downwash is induced by tip vortices along bound vortex - bound vortex can’t induce velocity on itself
• single vortex produces unrealistic downwash (infinite at wing tips)
• extend concept to infinite horseshoe vortices placed on lifting-line
• series of trailing vortices in wake forming vortex sheet where strength equal to change in circulation
• fundamental eq: geo AoA = eff AoA + AoA induced
• solve n coefficients at n locations along lifting line
• use KJ theory to find lift (integral of circulation)

Question 3

What are some key findings of lifting line theory?

Answer 3

For an elliptical lift distribution with no twist:
- planform is elliptical
- elliptical circulation distribution
- constant downwash

Span efficiency factor and induced drag factor are dependent on circulation distribution
Can’t be used for swept wings!