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Conformal Mapping & the Complex Potential

Intuition

In two dimensions, potential flow has a superpower: the potential \(\phi\) and stream function \(\psi\) pair up into a single analytic complex function. Anything complex analysis can do — and it can do a lot — becomes a fluid-mechanics tool. In particular: if you can solve the flow around a circle, a clever change of complex variable hands you the flow around an airfoil for free.

Formal definition

Define the complex potential on \(z = x + iy\):

\[F(z) = \phi(x, y) + i\,\psi(x, y)\]

Because \(\phi, \psi\) satisfy the Cauchy–Riemann equations, \(F\) is analytic, and its derivative packages the velocity:

\[\frac{dF}{dz} = u - iv\]

A conformal map is an analytic function \(w = f(z)\) carrying one flow domain to another. It preserves angles (hence "conformal"), and — crucially — it preserves irrotationality and incompressibility, so solutions map to solutions.

The Joukowski transformation

\[\boxed{\,w = z + \frac{c^2}{z}\,}\]

maps a circle in the \(z\)-plane into an airfoil-like shape in the \(w\)-plane (an offset circle gives a cambered airfoil with a sharp trailing edge). Since the flow past a cylinder (uniform + doublet + vortex) is known exactly, the flow past a Joukowski airfoil follows by substitution — including the circulation and hence the lift (Kutta–Joukowski theorem: \(L' = \rho U \Gamma\)).

Riemann mapping theorem

Any simply connected region of the plane (other than the whole plane) can be conformally mapped onto the unit disk.

For fluid mechanics this is an existence guarantee: every 2-D irrotational incompressible flow in a simply connected domain is, up to a map, the flow in a disk. Solving one canonical problem solves them all in principle; the craft is finding the map.

Method of images (boundary conditions for free)

Solid boundaries demand zero normal velocity. Instead of solving a boundary-value problem, place fictitious singularities so symmetry enforces the condition:

  • Wall: source \(Q\) at \((a,0)\) + image source \(Q\) at \((-a,0)\) ⇒ the \(x\)-axis becomes a streamline.
  • Cylinder: the Milne-Thomson circle theorem inserts an image system inside the cylinder so its surface is a streamline.

Common mistakes

  • Using a non-analytic "map". Only analytic (holomorphic, \(f'(z)\neq 0\)) functions preserve the flow equations; \(|z|\) or \(\bar z\) tricks break everything.
  • Forgetting the velocity transforms too: velocities in the mapped plane pick up a factor \(1/|f'(z)|\); speeds are not copied verbatim.
  • Ignoring the trailing-edge (Kutta) condition when computing airfoil lift — the physical flow selects the circulation that leaves the sharp edge smoothly.

Knowledge graph position

Prerequisites: Potential flow, complex analysis (from PHY622-level math methods). Leads to: airfoil theory, advanced hydrodynamics.

Quiz

Q1 (conceptual). Why does the Joukowski map produce a sharp trailing edge from a smooth circle?

Answer

The map \(w = z + c^2/z\) has \(f'(z) = 1 - c^2/z^2 = 0\) at \(z = \pm c\). Conformality fails exactly at those critical points — angles are not preserved there, and the smooth circle acquires a cusp.

Q2 (multiple choice). The complex velocity is:

  • (a) \(u + iv\) (b) \(u - iv\) (c) \(v + iu\) (d) \(|\mathbf{v}|e^{i\theta}\)
Answer

(b). \(dF/dz = \phi_x + i\psi_x = u - iv\) by Cauchy–Riemann.

Q3 (computational). A source of strength \(Q\) sits at distance \(a\) from a rigid wall. What is the fluid speed at the wall point nearest the source?

Answer

Source + image each contribute \(\frac{Q}{2\pi a}\)... their normal components cancel, tangential components cancel at the foot point by symmetry — the nearest wall point is a stagnation point with speed \(0\). (Maximum wall speed occurs at \(x = \pm a\) along the wall, \(|u| = \frac{Q}{2\pi a}\).)