Services & ResourcesWolfram ForumsMathGroup Archive

[Date Index] [Thread Index] [Author Index]

smooth Jowkouski Julia of a sinod wing shape

• To: mathgroup at smc.vnet.net
• Subject: [mg103042] smooth Jowkouski Julia of a sinod wing shape
• From: Roger Bagula <roger.bagula at gmail.com>
• Date: Sat, 5 Sep 2009 05:35:59 -0400 (EDT)

I developed this wing like shape in Mathematica
using my sinoid type of function:
Clear[x, y, t]
x = Exp[1 - Cos[t]]
y = Cos[t + 2*Pi/3]
ParametricPlot[{x, y}, {t, -Pi, Pi}, AspectRatio -> Automatic]
r = {{Cos[Pi/15], Sin[Pi/15]}, {-Sin[Pi/15], Cos[Pi/15]}}
{x1, y1} = r.{x, y}
ParametricPlot[{x1, y1}, {t, -Pi, Pi}, AspectRatio -> Automatic]
http://www.flickr.com/photos/fractalmusic/3887559038/

I wondered what the nonlinear flow in terms of a Jowkowski transform
with circulation might be:
Clear[f, x, nz]
(*rotated diaxial sinoid ( wing shape)*)
f[x_] = Exp[I*Pi/15]*(Exp[1 - Cos[Arg[x]]] + I*Cos[Arg[x] + 2*Pi/3])
3D and plane Julia Joukowski with circulation*)
(* SQRT(x^2 + y^2) limited measure*)
(*by R. L. BAGULA 3 Sept. 2009=A9 =A9 *)
numberOfz2ToEscape[z_] := Block[
	{escapeCount, nz = N[z], nzold = 0},
	For[
		escapeCount = 0,
		(Sqrt[Re[nz]^2 + Im[nz]^2] <
        16) && (escapeCount < 256) && (Abs[nz - nzold] > .5*10^(-3)),
		nzold = nz;
		nz = Abs[nz]*((f[nz] + 1/f[nz])/2 + Log[Arg[nz]]/(2*Pi));
		++escapeCount
	];
	escapeCount
]
FractalPureM[{{ReMin_, ReMax_, ReSteps_},
			 {ImMin_, ImMax_, ImSteps_}}] :=
		Table[
			numberOfz2ToEscape[x + y I],
			{y, ImMin, ImMax, (ImMax - ImMin)/ImSteps},
			{x, ReMin, ReMax, (ReMax - ReMin)/ReSteps}
		]

arraym = FractalPureM[{{-3, 3, 300}, {-3, 3, 300}}];

	gr = ListPlot3D[-arraym, Mesh -> False, AspectRatio -> Automatic,
Boxed ->
        False, Axes -> False];

ListDensityPlot[arraym,
                          Mesh -> False,
		                  AspectRatio -> Automatic,
		                  ColorFunction -> Hue];

http://www.flickr.com/photos/fractalmusic/3886762965/

The puzzle here is what is the shock like Arnold tongue nearly up the
y axis?

Back in the late 90's I thought this had the potential of being
a wind tunnel type simulation method,
 ( would be a neat application for this type of fractal iteration0
but I never could raise any interest in it from the engineers.
At the time I was using piece-wise elliptical wing shapes
that I constructed and programming in True basic.
This result is much smoother than my 90's result.
Roger Bagula