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.

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];

#### Links

- Joukowski Animation – nice animation showing how the fluid moves
- Joukowski Airfoil – nice Java applet by NASA
- Nikolai Joukowski – used this technique to find the lift on an airfoil in 1906, long before modern computers

nice

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

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

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.

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.

Clair

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

super animations………