MATH529 L25: Harmonic functions from conformal mapping

Overview

• Potential flow

• Flow around a circle

• Flow around Zhukovsky (Joukowsky) airfoil

• Honors, course capstone project

Potential flow

• Flow of an incompressible, inviscid fluid is described by the PDEs

  ${\displaystyle \nabla \cdot 𝒗=0,{𝒗}_t+(𝒗\cdot \nabla )𝒗=-\nabla p}$

  expressing conservation of mass (continuity equation), momentum (Euler equation, a continuum formulation of Newton's law $d\left(m𝒗\right)/dt=\sum 𝒇$).

• The continuity equation implies $\exists \phi$ such that $𝒗=\nabla \phi$, and ${\nabla }^2\phi =0$, i.e., the hydrodynamic potential $\phi$ is harmonic

• Complex formulation:

  • Conformal map from domain $Z=X+iY$ to $z=x+iy$, e.g., $z=Z+1/Z$

  • Complex potential $F\left(Z\right)=\Phi (X,Y)+i\Psi (X,Y)$, $f\left(z\right)=F\left(Z\left(z\right)\right)$

  • Complex velocity $W=dF/dZ$, $w=df/dz$

    ${\displaystyle w\left(z\right)=\frac{dF\left(Z\left(z\right)\right)}{dz}=\frac{dF}{dZ}\frac{dZ}{dz}=\frac{\frac{dF}{dZ}}{\frac{dz}{dZ}}}$

Flow around a circle

• Potential in $Z$ plane around a circle $F\left(Z\right)=U_{\infty }(e^{i\alpha }Z+e^{-i\alpha }/Z)$

  • $F=\Phi +i\Psi$, $\Phi$ real velocity potential, $\Psi$ real streamline function

  • $F\left(e^{i\theta }\right)=U_{\infty }(e^{i\alpha }{e}^{i\theta }+e^{-i\alpha }{e}^{-i\theta })=2U_{\infty }\cos (\alpha +\theta )\in ℝ\Rightarrow \Psi =0=\mathrm{constant}$, the circle $Z=e^{i\theta }$ is a streamline

• Potential with circulation $F\left(Z\right)=U_{\infty }(e^{i\alpha }Z+e^{-i\alpha }/Z)+i\Gamma /Z$, the circulation $\Gamma$ introduces asymmetry between velocity fields on circle top/bottom, a mathematical model of lift.

Joukowski conformal map

• Conformal maps can be found between simple domains (e.g., a circle or the upper half plane) and shapes of practical interest, such as airfoils

• The Joukowski transform maps a circle onto air and hydrofoil shapes

  ${\displaystyle z=Z+\frac{1}{Z}}$

  • $Z=e^{i\Theta }\Rightarrow z=e^{i\Theta }+e^{-i\Theta }=2\cos \Theta$, e.g., a flate plate

  • $Z=e^{i\Theta }+i\delta ,\delta \in (0,1/2)$ a cambered, thick airfoil, symmetric fore-aft

  • $Z=e^{i\Theta }+\rho e^{i\varphi },\rho \in (0.2,0.4)$, $\varphi \in (-\pi /2,-\pi /4)$ cambered, thick, unsymmetrical airfoils, used in aircraft design 1920's

  Figure 1. $\rho =0.3,\varphi =-1.38$ airfoil from Joukowski map