### 8. Potential Flows

```Potential Flow

u  
for inviscid,
irrotational flows

 x, t 
velocity potential



can
only
exist
for
irrotational
flows
 x, t 
   u  0

    u      0

In Cartesian coordinates:



u
; v
; w
x
y
z
In Cartesian coordinates:



u
; v
; w
x
y
z
Incompressible flows:

 u  0
u v w


0
x y z
           

 
continuity in terms of the



  0    
x 
x y  y  z 
z
velocity potential:


u
For incompressible,
irrotational flows, the
governing equation is:
v
w
 2  2  2
2




 0
2
2
2
x
y
z
Laplace’s equation
 2  0 Potential Flows
Although incompressibility is not required for a velocity potential to
exist, only incompressible, irrotational flows are called Potential
Flows
vector is that one scalar function can contain all three components
of velocity vector
Potential function -

u  
http://today.slac.stanford.edu/images/2009/data-mining.jpg
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:


x
 

 y
u v

0
x y

u
y

v
x
    
 2
 2
 

0

0 
x y x y
 y  x 
v u

0
For 2-D irrotational flows:
x y
is satisfied if Ψ is
differentiable
 2  2


0
2
2
x
y
For incompressible & irrotational 2-D flows, ψ satisfies Laplace’s eq.
Therefore, for potential flows, both Φ and ψ satisfy Laplace’s eq.
 2   2  0 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 n normal to boundaries : Neumann boundary
condition
c) Specify
 at 

(any linear combination of a) and b)):
n
Robin boundary condition
 2  0
Examples of Potential Flows

u  
Uniform Flow

u
   ux  c
x
y
u
x
  ux

0
   f2  x   c
y
  ux
[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);

u
   u y  c2
y
  u y
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);

v 0
   f3 y   c2
x
```