Elementary Flows

Dec 2, 2024

Lecture 9: Aerodynamics - Incompressible and Inviscid Flows

Review from Last Lecture

  • Explored incompressible and inviscid flows.
  • Discussed Bernoulli’s equation for rotational flow and irrotational flows.
  • Velocity potential & stream function satisfying Laplace equation.

Today's Focus

  • Continue exploring incompressible and inviscid flows.
  • Introduction to elementary flows:
    • Uniform flow
    • Source and sink
    • Doublet
    • Vortex
  • Use of elementary flows as building blocks for complex aerodynamic flows.

Key Concepts

Superposition of Solutions

  • Laplace Equation allows superposition of solutions.
  • Can add stream functions and velocity potentials for different flows.
  • Useful for modeling flow over objects (e.g., recreate flow over an oval).

Inviscid Flow

  • No boundary layers on surfaces.
  • Fluid can move freely without viscosity effects.

Elementary Flows

  • Represent building blocks of aerodynamic flows.
  • Cylindrical Coordinates are predominantly used.

Uniform Flow

  • Constant velocity field.
  • Velocity: u = dψ/dy (constant u infinity), v = dψ/dx (zero).
  • Stream function & velocity potential derived from velocity field.

Source and Sink

  • Streamlines emit outwards/inwards from a point.
  • Characterized by volumetric flow rate per unit span (λ).
  • Radial velocity (u_r) = λ / (2πr), azimuthal velocity (u_θ) = 0.
  • Stream function and velocity potential defined.

Doublet

  • Combination of source and sink close together.
  • Strength defined by constant λL.
  • Stream function and velocity potential obtained through limits.

Vortex

  • Fluid orbits around a central point.
  • Characterized by no radial velocity and constant angular velocity.
  • Stream function and velocity potential derived.

Building Complex Flows

  • Semi-infinite Body: Combination of uniform flow and source.
  • Rankine Oval: Uniform flow plus source and sink of equal/opposite strength.
  • Stationary Cylinder: Uniform flow plus doublet.
  • Rotating Cylinder: Adding vortex to stationary cylinder.

Practical Applications

  • Use in Computational Fluid Dynamics (CFD).
  • Techniques like source-panel and vortex-panel methods.

Summary

  • Explored addition of potentials and stream functions.
  • Introduced four elementary flows and used them to construct complex flows.
  • Practical relevance in modern CFD techniques.

Conclusion

  • Emphasized practical application in flow solvers.
  • Encouragement to continue exploring aerodynamic principles.