Joukowski Airfoil

These animations were created using a conformal mapping technique called the Joukowski Transformation. A Joukowski airfoil can be thought of as a modified Rankine oval. It assumes inviscid incompressible potential flow (irrotational). Potential flow can account for lift on the airfoil but it cannot account for drag because it does not account for the viscous boundary layer (D’Alembert’s paradox). In these animations, red represents regions of low pressure. The left animation shows what the surrounding fluid looks like when the Kutta condition is applied. Notice that the fluid separates smoothly at the trailing edge of the airfoil and a low pressure region is produced on the upper surface of the wing, resulting in lift. The lift is proportional to the circulation around the airfoil. The right animation shows what the surrounding fluid looks like when there is no circulation around the airfoil (stall). Notice the sharp singularity at the trailing edge of the airfoil.

Joukowski Airfoil – C++, 11/8/07

Here is an animation that shows how the streamlines change when you increase the circulation around the airfoil. (Please note: The background fluid motion in this animation is just for effect and is not accurate!) Here is some Mathematica code to plot the streamlines and pressure using Bernoulli’s equation:
(* runtime: 13 seconds *)
U = rho = 1; chord = 4; thk = 0.5; alpha = Pi/9; y0 = 0.2; x0 = -thk/5.2; L = chord/4; a = Sqrt[y0^2 + L^2]; gamma = 4Pi a U Sin[alpha + ArcCos[L/a]];
w[z_, sign_] := Module[{zeta = (z + sign Sqrt[z^2 - 4 L^2])/2}, zeta = (zeta - x0 - I y0)Exp[-I alpha]/Sqrt[(1 - x0)^2 + y0^2]; U(zeta + a^2/zeta) + I gamma Log[zeta]/(2Pi)];
sign[z_] := Sign[Re[z]]If[Abs[Re[z]] < chord/2 && 0 < Im[z] < 2y0(1 - (2Re[z]/chord)^2), -1, 1];w[z_] := w[z, sign[z]]; V[z_] = D[w[z, sign], z] /. sign -> sign[z];
<< Graphics`Master`;
DisplayTogether[DensityPlot[-0.5rho Abs[V[(x + I y)Exp[I alpha]]]^2, {x, -3, 3}, {y, -3, 3}, PlotPoints -> 275, Mesh -> False, Frame -> False, ColorFunction -> (If[# == 1, Hue[0, 0, 0], Hue[(5# - 1)/6]] &)],ContourPlot[Im[w[(x + I y)Exp[I alpha]]], {x, -3, 3}, {y, -3, 3}, Contours -> Table[x^3 + 0.0208, {x, -2, 2, 0.1}], PlotPoints -> 100, ContourShading -> False], AspectRatio -> Automatic];



8 Responses to “Joukowski Airfoil”

  1. November 14, 2008 at 9:12 am

    I think your comment on the picture about stall is not correct since conformal mapping is only for inviscid flow, which cannot capture viscous phenomena such as stall

  2. 3 Jim
    April 17, 2009 at 9:30 pm

    Very interesting site. I’ll explore more when time allows.

  3. August 27, 2009 at 12:02 pm

    May I use an image from your Joukowski Airfoil website in my high school math classroom? I am a teacher and I am making a poster to show some examples of use of complex variables. Thanks.

  4. December 29, 2009 at 1:54 pm

    A very interesting website you made. Such studies could possibly lead to unification. The
    slopes of perpendicular lines ar the negative reciprocal of each other. I have the MSEE plus 6
    courses. The JK airfoil possibly leads to electromanetism popping out. It also could explain
    the strong force. I am mostly reviewing these things right now.
    I am having trouble uploading images from my computer to the website. Obviously you
    accomplished this. See the evidence for the existence of a 120 million year old drawing when
    it is finished. It is also about out of body imagination which needs a physics explanaion.

  5. September 24, 2010 at 2:11 pm

    Still I am mostly reviewing this math. My website is now completed but it needs more resolution. You might reach it by clicking 120 mil…etc. but the URL is lizardofozma.net thanks Clair

  6. 7 REJISH J
    November 16, 2011 at 6:45 pm

    super animations………

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