Basic 2-D Potential Theory
We outline here the way in which the "known"
solutions used in panel methods can be generated and obtain some useful
solutions to some fundamental fluid flow problems. Often the known solutions
just come out of thin air and can be applied, but sometimes other approaches
The simplest case, two-dimensional potential flow illustrates this process.
We shall discuss 2-D incompressible potential flow and just mention the
extension to linearized compressible flow.
For this case the relevant equation is Laplace's equation:
There are several ways of generating fundamental solutions to this linear,
homogeneous, second order differential equation with constant coefficients.
Two methods are particularly useful: Separation of variables and the use
of complex variables.
Complex variables are especially useful in solving Laplace's equation because
of the following:
We know, from the theory of complex variables, that in a region where a
function of the complex variable z = x + iy is analytic, the derivative
with respect to z is the same in any direction.
This leads to the famous Cauchy-Riemann conditions for an analytic function
in the complex plane.
Consider the complex function: W = φ+ i ψ
The Cauchy-Riemann conditions are:
Differentiating the first equation with respect to x and the second with
respect to y and adding gives:
Thus, analytic function of a complex variable is a solution to Laplace's
equation and may be used as part of a more general solution.
W = φ+ i ψ is called the complex potential.
It consists of the usual velocity potential as the real part and the stream
function as its imaginary part.
The flow velocities can be then be written as a single complex number:
dW/dz = u - iv (Try deriving this.)
We consider some simple analytic functions for W that are of great use in