3D Modelling

Siluro II Tear Drop

To calculate the perfect teardrop shape we used the Van de Vooren symmetrical airfoil teardrop from http://www.mh-aerotools.de/airfoils/javafoil.htm. The picture above shows the resulting teardrop overlayed over the original Siluro, with the result being a pretty good match. The dotted lines are the Van de Vooren symetrical airfoil output of Javafoil.

Van de Vooren Airfoil

Here is some information on the maths: Low Speed Aerodynamics by Katz and Plotkin 

The simplified version (my understanding) is we transform a circle into an aerodynamic shape (sort of how a rain drop forms its shape). We use complex numbers to represent points on the initial circle. Complex numbers can be thought of a number with an "x" part (also called "real") and a "y" part* (also called "imaginary", "i" or "j") and these parts are treated separately. The "x" and "y" parts can be plotted on a graph and therefore we can use complex numbers to define a circle (i.e. calculate x and y from an angle and radius).

Using "some" formula we can transform a set of complex number points into a new set of complex number points (e.g. a circle into our aerodynamic shape or a squiggle). There are some rules on how to deal with arithmetic of complex numbers, however we can bypass the complexity of complex numbers by using a C++ library or Excel engineering functions then we can treat the arithmetic like normal. The only trick is to turn normal numbers into complex numbers when needed (e.g. to do the inverse of a complex number we divide by complex(1,0) instead of just 1).

So the formula in programming terms would look like...

newpoint = ((originalpoint - radius)^trailing_egde_angle) / (originalpoint - radius * thickness_percentage)^(trailing_egde_angle-1)) + length

where:
newpoint (Y) is a complex number
originalpoint (f) is a complex number of a point on a predefined circle
radius (a) is a complex number of the radius of the predifined circle (where the imaginary part is just set to zero)
trailing_egde_angle (k) is a normal number (I just used one that made the shape fit the Siluro design)
length (l) is just a number (I don't actual know why it is there as it would only change the position of the shape?)

Here is a Excel plot which shows that it works:

Here is the same visualised through OpenGL/C++

6.6 Airfoil with Finite Trailling-Edge Angle

*actually it's not called the "y" part but think of it that way as it gets mapped to "y" in a typical 3D space (or "z")