VII. Analysis of Potential Flows

Contents 1. Preservation of Irrotationality 2. Description of 2D Potential Flows 3. Fundamental Solutions 4. Superposition

1. Preservation of Irrotationality

Stokes Theorem S C Vorticity Circulation

In the flow of an ideal fluid with constant density, circulation along a fluid line is invariant if body force is conservative Kelvin ’ s Theorem

fluid is always irrotational if it is initially irrotational A piece of fluid is always irrotational if it is initially irrotational

2. Description of 2D Potential Flows

2D Flow in x-y plane

Basic Equations for 2D Potential Flows

Velocity Potential

Irrotational flow Definition of Velocity Potential

Continuity Equation

Stream Function

Incompressible fluid Definition of Stream Function

Irrotational condition

 = constant represents a streamline Properties of Stream Function

Streamlines and equipotential lines are always perpendicular to each other Properties of Stream Function

Along a streamline

Along an equipotential line

Complex Potential

Cauchy-Riemann Condition

Analytic Function

3. Fundamental Solutions

a. Uniform flow b. Source and sink c. Vortex d. Doublet

a. Uniform Flow

U

b. Source and Sink

Source In polar coordinates

Discharge

Sink

Source or Sink at (x 0,y 0 )

c. Vortex

Vortex In polar coordinates

Circulation

Clockwise Vortex

Vortex centered at (x 0,y 0 )

c. Doublet

Velocity Potential

Stream Function

Streamlines

4. Superposition

a. Circular Cylinder without Circulation

Uniform Flow Doublet

On surface of cylinder Velocity

2U Stagnation Point

Pressure

D’Alembert Paradox

Drag due to viscosity ► Skin friction ► Form drag

b. Circular Cylinder with Circulation

Uniform Flow DoubletVortex

On surface of the cylinder

Stagnation point on cylinder

Pressure

Lift