Potential Function (Velocity Potential)
In a Nut Shell: For
steady, inviscid, incompressible, irrotational fluid flow the vorticity
vector, ζ
, is
zero. Then there exists a velocity potential, φ, that automatically provides
zero vorticity where u
= ∂φ/∂x, v =
∂φ/∂y, w =
∂φ/∂z . Recall ζ = curl V = (∂w/∂y - ∂v/∂z) i +
(∂u/∂z - ∂w/∂x) j + (∂v/∂x -
∂u/∂y) k Substitution
of the velocity components into the above expression for vorticity
confirms that it
sums to zero. Substitution of u
= ∂φ/∂x, v =
∂φ/∂y, w
= ∂φ/∂z into the continuity equation
∂u/∂x + ∂φ/∂y +
∂w/∂z = 0
(continuity equation) yields ∂2φ/∂x2 +
∂2φ/∂y2 +
∂2φ/∂z2 = 0 ( Laplace equation ) In
elementary fluid mechanics emphasis is on two-dimensional fluid flow in the
x-y plane. So
∂2φ/∂x2 +
∂2φ/∂y2 = 0 Note
that this equation is linear. So that
if both φ1(x,y) and φ2(x,y) satisfy this equation so does φ1(x,y) +
φ2(x,y) which gives a convenient way to
superimpose potential functions to
simulate different kinds of fluid flows.
An example might be simulating fluid flow around a
circular cylinder where you superimpose the potential function for uniform
flow plus the potential function
for a fluid source as depicted in the figure below. Click
here for a table of potential functions. Click here for examples. |
All rights reserved.