1. Potential Flow
for inviscid, irrotational flows
velocity potential
can only exist for irrotational flows
curl of gradient = 0
In Cartesian coordinates:

2. In Cartesian coordinates:
Incompressible flows:
continuity in terms of the velocity potential:
For incompressible, irrotational flows, the governing equation is:
Laplace’s equation

3. Potential Flows
Although incompressibility is not required for a velocity potential to exist, only incompressible, irrotational flows are called Potential Flows
The advantage of using a velocity potential, instead of a velocity vector is that one scalar function can contain all three components of velocity vector
Potential function -
Points of different clusters fall in separate valleys

4. The stream function ψ is another scalar function that contains all velocity components
For 2-D incompressible flows:
Continuity equation:
is satisfied if Ψ is
differentiable
For 2-D irrotational flows:
For incompressible & irrotational 2-D flows, ψ satisfies Laplace’s eq.
Therefore, for potential flows, both Φ and ψ satisfy Laplace’s eq.

5. Potential Flows
Partial differential equation -- elliptic
Boundary conditions need to be specified around the entire domain
a) Specify at boundaries : Dirichlet boundary condition
b) Specify at normal to boundaries : Neumann boundary condition
c) Specify at (any linear combination of a) and b)):
Robin boundary condition

6. Examples of Potential Flows
Uniform Flow y u x

7. [x,y]=meshgrid(0:.05:2,0:.05:1);
fi=0.5*x;
contourf(x,y,fi)
xlabel('x','FontSize',14);
ylabel('y','FontSize',14);
title({'Potential Function for ux'},'FontSize',14);
colorbar('FontSize',14);

8. figure
psi=-.5*y;
contourf(x,y,psi)
xlabel('x','FontSize',14);
ylabel('y','FontSize',14);
title({'Stream Function for -uy'},'FontSize',14);
colorbar('FontSize',14);