Concise - Self Test 3

Question 1

Compare and contrast the stream function and the velocity potential.

Answer 1

• Stream Function
  
  • Defined only for 2D flows
  • Represents mass continuity
  • Applicable to rotational flows
• Velocity Potential
  
  • Defined for 3D flows
  • Represents irrotationality
  • Does not exist for rotational flows
• Both
  
  • When both exist, they are orthogonal

Question 2

What is the significance of the curl of the velocity field in an incompressible flow?

Answer 2

• The curl of the velocity field is called vorticity
• Vorticity is twice the angular momentum

Question 3

What is D’Alembert’s paradox? State what is the implication for 2D aerofoil theory based on potential flow.

Answer 3

• D’Alembert’s Paradox
  
  • Potential flow analysis predicts zero drag
  • Viscosity is neglected, so no skin friction or boundary layer separation
• Implications for 2D Aerofoil Theory
  
  • Cannot predict drag
  • Cannot predict the onset of stall

Question 4

What is the Kutta-Joukowski theorem?

Answer 4

• Kutta-Joukowski Theorem
  
  • Lift per unit length (L’) is proportional to circulation (Γ)
  • Formula: L’ = ρUΓ

Question 5

In a real fluid flow (i.e. with non-zero viscosity), where is vorticity generated and destroyed?

Answer 5

• Vorticity Generation
  
  • Created by shear forces in the boundary layer
• Vorticity Destruction
  
  • Diffused and dissipated by viscosity as it convects downstream

Question 6

Define a streamline, clearly stating the key characteristics.

Answer 6

• Streamline
  
  • A curve tangential to the local instantaneous velocity at every point
  • No normal velocity across a streamline
  • Solid bodies are streamlines in a body-fixed frame
  • In steady flows, streamlines coincide with pathlines and streaklines