It is a method to generate the potential function (velocity field and the stream function) for more complex flows. It is used when we have walls as boundaries. For example, the case of a circle in a free stream close to a wall, with the velocity vector of the free-stream parallel to the wall. The potential function for the circle of radius $a$ with center at $(0,b)$ in a free stream of speed $U$ is


\begin{displaymath}\phi = Ux\left(1+\frac{a^{2}}{x^{2}+(y-b)^{2}}\right) \notag
\end{displaymath}  

Now, we want to put a wall at $y = 0$. To obtain the potential function of the velocity field which respects the no flux boundary condition at the wall, we superimpose a dipole located at the mirror image (located at $(0,-b)$) of the dipole (located at $(0,b)$) representing the actual circle, so we have


\begin{displaymath}\phi = Ux\left(1+\frac{a^{2}}{x^{2}+(y-b)^{2}}+\frac{a^{2}}{x^{2}+(y+b)^{2}}\right). \notag
\end{displaymath}  



Timothy S Choe
2003-03-15