MATH529 L25: Conformal mappings

Overview

• Complex mappings

• Conformal mappings

• Zhukovsky (Joukowsky) and Karman-Trefftz mappings

Complex functions map $ℂ$ onto itself

• $w=f\left(z\right)$ maps from $(x,y)$ to $(u,v)$. Examples:

  • $f\left(z\right)=z+z_0$ is a translation by $z_0$

  • $f\left(z\right)=e^{i\theta }z$ is a rotation by angle $\theta$

  • $f\left(z\right)=e^z$ maps the strip $x\in ℝ$, $0⩽y⩽\pi$

  • $f\left(z\right)=1/z$ maps lines parallel to $x,y$ axes onto circles

  • $f\left(z\right)=z^{1/n}$ maps the upper half plane onto a wedge of angle $\pi /n$

• Mappings can be composed, e.g. $h\left(z\right)=(g\circ f)\left(z\right)=g\left(f\left(z\right)\right)$

  • $f\left(z\right)=e^{i\theta }z$, $g\left(z\right)=z+z_0$ is a rotation followed by a translation

Conformal mappings

• A mapping is said to be conformal if it preserves angles between curves

• Complex mappings $w=u+iv=f\left(z\right)=f(x+iy)$ where $f$ is analytic are conformal

Theorem. Conformal mappings $f:D\to D^{\text{'}}$, $D,D^{\text{'}}\subseteq ℂ$, preserve harmonic functions, i.e., if $G$ is harmonic in $D^{\text{'}},$ $G_{uu}+G_{vv}=0$, then $g(x,y)=(U\circ f)\left(z\right)$ is harmonic, $g_{xx}+g_{yy}=0$.

• Knowledge of a harmonic function in some domain can be used to find a harmonic function in a mapped domain (See Lesson25.nb)

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

Karman-Trefftz transform

• In the 1930's better experimental arodynamics led to study of airfoils from

  ${\displaystyle z=nb\frac{{(Z+b)}^n+{(Z-b)}^n}{{(Z+b)}^n-{(Z-b)}^n},}$

  known as the Karman-Trefftz transform

  Figure 2. Karman-Trefftz airfoil with $b=1.18,n=1.92$, the image of circle $Z=e^{i\Theta }-0.14e^{-0.785i}$.