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We have an asymmetric potential flow past the cylinder (i.e. 2D circle) of radius $R$ of well-known complex velocity $W$:

$$ \tilde{W} = v_\infty e^{-i\alpha} + i\frac{\Gamma}{2\pi(\zeta - \mu)} - \frac{v_\infty R^2 e^{i\alpha}}{(\zeta - \mu)^2} $$

where $\mu = \mu_x + i\mu_y$ is the complex coordinate of cylinder axis (circle center) and the rest is just as usual (e.g. the wikipedia article). Circulation $\Gamma$ satisfies the Kutta condition.

How do I calculate parameters of the airfoil and streamlines?

The transformed velocity should be:

$$ W = \frac{\tilde{W}}{\frac{dz}{d\zeta}} = \frac{\tilde{W}}{1-\frac{\ell}{\zeta^2}}. $$

And the airfoil? How do I properly transform the circle defined in a plane corresponding to $\tilde{W}$?

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  • $\begingroup$ I appreciate that the links below are basic stuff and well known to you, this is just for future related questions from other people: grc.nasa.gov/WWW/K-12/airplane/map.html and physics.stackexchange.com/questions/4847/… $\endgroup$ – user108787 Jul 26 '16 at 17:58
  • $\begingroup$ Integrate the velocity field to the complex potential $\Omega(z)$. Then $\mathrm{Re}(\Omega = const$ are the equipotentials, amongst which the aerofoil itself and $\mathrm{Im}(\Omega)=const$ are the streamlines. My answer here works this problem out in detail. $\endgroup$ – WetSavannaAnimal Nov 24 '17 at 6:42
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Here is a Python code for generating the streamlines of the flow past a Joukowski airfoil (static plot and animated streamlines, asociated to a rotating airfoil):

http://nbviewer.jupyter.org/github/empet/Math/blob/master/Joukowski-airfoil.ipynb http://nbviewer.jupyter.org/github/empet/Math/blob/master/Joukowski-airfoil-rotation-online.ipynb

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