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

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

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 http://today.slac.stanford.edu/images/2009/data-mining.jpg Potential function - Points of different clusters fall in separate valleys

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.

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

Examples of Potential Flows Uniform Flow y u x

[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);

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);