Aerodynamics Lecture: Rotating Cylinder and the Kutta-Joukowski Theorem

Introduction

• Overview of previous lecture on elementary flow fields:
  • Uniform flow
  • Source and sink
  • Doublet
  • Vortex
• Complex flows: semi-infinite body, Rankine oval, stationary cylinder, rotating cylinder.
• Today's focus: Rotating cylinder and Kutta-Joukowski Theorem.

Rotating Cylinder

• Stationary Cylinder:
  • Symmetric flow about vertical plane → No drag.
  • Symmetric flow about horizontal plane → No lift.
• Rotating Cylinder:
  • Symmetric about vertical plane → No drag.
  • Asymmetric in horizontal plane → Non-zero lift.
• Key Difference:
  • Addition of point vortex to stationary cylinder introduces circulation leading to lift.

Kutta-Joukowski Theorem

• Relates lift per unit span in a flow field to circulation, flow velocity, and density.
• Discovered independently by Kutta (German mathematician) and Joukowski (Russian physicist).
• Objective: Calculate lift force from velocity field of rotating cylinder.

Derivation Process

Step 1: Pressure from Velocity

• Use Bernoulli's equation (incompressible, inviscid flow) to relate pressure to velocity.
• Pressure coefficient: Change in pressure relative to dynamic pressure (1/2 ρu²).

Step 2: Pressure Distribution to Force

• Calculate normal and axial forces from pressure distribution.
• Non-dimensionalize pressure into pressure coefficient.
• Transform Cartesian to cylindrical coordinates.
• Integrate pressure around entire cylinder surface (0 to 2π).

Drag Force Calculation

• Confirmed analytically zero drag due to symmetrical properties of sine and cosine integrals.

Lift Force Calculation

• Integrate pressure coefficient and find lift coefficient related to circulation, cylinder radius, and flow velocity.
• Lift derived from flow circulation.

Application of Kutta-Joukowski Theorem

• Works for flows over bodies with arbitrary shapes, beyond rotating cylinders.
• Enclose the viscous boundary layer to apply theorem to airfoils.
• Circulation indicates lift but doesn't cause it; lift and drag come from pressure and shear stress distributions.

Practical Use

• Experimental setup (e.g., Boeing wind tunnel tests) measures circulation to find lift characteristics.
• Simulation techniques like vortex panel method use circulation to determine lift.

Conclusion

• Recap of understanding the rotating cylinder and the vortex's role in lift.
• Verification of lift related to circulation using the Kutta-Joukowski theorem.
• Importance in aerodynamics: theoretical and practical applications.

Final Remarks

• The lecture highlights the physical insights and mathematical tools used in aerodynamics, emphasizing the widespread relevance and application of the Kutta-Joukowski theorem.