Lecture on Prandtl's Lifting Line Theory

Introduction

• Previous discussion on finite wings and their real-world losses.
• Introduction to tip vortex, downwash, and added drag.
• Use of Biot-Savart law for estimating downwash in a vortex.

Prandtl’s Lifting Line Theory

• Developed by Prandtl around World War I (started in 1911).
• A predictive model for finite wing performance losses.
• Three major outcomes:
  • Lift distribution: How lift varies along the wing span.
  • Total lift: Total upward force the wing produces.
  • Induced drag: Efficiency of the wing in moving forward.

Basic Concepts

• Avoidance of surfaces, using a combination of:
  • Biot-Savart law.
  • Elementary flows.
  • Kutta-Joukowski theorem linking circulation to lift.

Horseshoe Vortex Model

• Model wing tip vortices as a semi-infinite line vortex.
• Connection with bound vortex replacing the surface.
• Tip vortices as free trailing vortices.
• Concept of a horseshoe vortex with two trailing vortices inducing a velocity field in the z-direction.

Challenges and Adaptations

• Problem: Vertical velocity blowing up at the edges.
• Solution: Employ multiple horseshoe vortices.
• Result: More accurate representation of trailing vortex sheet reality.
• Implementation of a series of horseshoe vortices to manage vortex strength at the bound vortex.

Derivation and Calculation

• Use of calculus to deal with infinite number of small horseshoe vortices.
• Describing induced velocity at the bound vortex with gamma as a function of z.
• Introduction of effective angle of attack combining set angle and induced downwash.

Thin Airfoil Theory Connection

• Lift slope of dCl/dAlpha = 2π for thin foils.
• Expression for Cl as a function of alpha and camber offset.

Solving for Gamma Distribution

• Fundamental equation for Prandtl’s Lifting Line Theory relating angle of attack to gamma.
• Known spanwise functions: alpha, chord, zero-lift angle of attack.
• Integration to get lift distribution, total lift, and induced drag.

Practical Applications

• Elliptical gamma distribution and its efficiency.
• Introduction to theta space transformation for easier computations.
• Induced drag related to lift and aspect ratio.
• Efficient flight design driven by elliptical gamma distribution.

Limitations and Modern Extensions

• Lifting Line Theory works well for:
  • Straight wings.
  • Moderate to high angles of attack.
  • High aspect ratio designs.
• Limitations for low aspect ratio foils, highly swept wings, and specific aggressive aircraft.
• Extension to vortex lattice methods for complex scenarios.

Importance in Aerodynamics

• Lifting Line Theory as a foundational design tool.
• Drives optimization for high aspect ratio and specific planform shapes (elliptic, tapered).
• Structural benefits of efficient designs.

Conclusion

• Recap of developing and solving the lifting line theory.
• Efficiency and application in modern aerodynamic design.
• Iterative approach for solving arbitrary gamma distributions.