Kootz airfoil

The equations sets are accurate (to a point) however the remainder of the page is no longer current
Kootz airfoils are several series of airfoil designs developed by A. Kootz, an aeronautical engineer. They are designed to cover most common variants of airfoil types using as simple an equation as possible. There are three sets of airfoils, the earliest set was developed using trig functions, the second was improved and simplified through the use of sequential exponents, the third is designed for functional use.

[Page in progress]-pay no attention to the scientist behind the curtain. Thanks, AK.

Early Equations
The airfoils are based off of the parametric equation: r(t)= [ t^2, sin(t) ] : -pi < t < pi.

Most sections are created from the thinner section given by: r(t)= [ t^2, sin(t)/2 ] : -pi < t < pi.

A camber can be given to the equation several ways:

1) a using a cosine function to adjust the value of the sin(t) term.

2) adding a sine(t^2)/4 term to create a preset camber

3) adding a t^2/pi^2-t^4/pi^4 term

4) a recursive camber can be given by -sin(t+pi/2*t^2/pi^2-pi/2*t^4/pi^4)/2

All the above methods can also be used in the inverse by subtracting the them from -sin(t).

Improved equations
The airfoils is solved more simply by replacing the trig functions with exponential functions.

The bounds of 't' are reduced from -pi<t<pi to -1<t<1.

The fairing function changes as sin(t) is replaced by t-t^3.

The camber function thus changes from sin(t^2) to t^2-t^4

This produces a much broader fairing/higher camber thickness ~77% of unit chord. So the function is given a two user defined variables 'a' which controls the fairing width, and 'b' which controls the height of the camber line.

Here is the finished equation: r(t)=[ t^2, (t-t^3+(t^2-t^4)/b)/a ]

Bezier Curve Airfoils
Once the evaluation of the Improved Airfoil equations were completed, a method of creating airfoils through the use of 5 control-point bezier curves began development.

The x(t) equation has two variants, one provides camber location control, and the other provides control over the thickness location.

x(t)=(1-t)^4+16*(1-t)^3*t*(CamberLocation/100-1/2)+ (1-t)^2*t^2*(32*(1/2-CamberLocation/100)-2)+ 16*(1-t)*t^3*(CamberLocation/100-1/2)+t^4

x(t)=(1-t)^4+18*(1-t)^3*t*(ThicknessLocation/100-1/2)+ (1-t)^2*t^2*(36*(1/3-ThicknessLocation/100)-2)+ 18*(1-t)*t^3*(ThicknessLocation/100-1/2)+t^4

y(t)=4*(1-t)^3*t*(Camber/25+Thickness/77)+ 8*(1-t)^2*t^2*(-Camber/25)+ 4*(1-t)*t^3*(Camber/25-Thickness/77)

0<=t<=1

The constants within the equations are determined experimentally and have an output accuracy to within +- 0.0001 of input values.

Alterations
Several series of alterations to the base function have been examined.

A supercritical airfoil can be obtained through conversion of the Cos(t/(n)) function from t/(n) to t^k/(n*(2*pi)^(k-1)). This pushes the effect of the camber function from the upper surface of the airfoil (the start of the iteration) and further back along the lower surface (toward the end of the iteration). This effect can be seen here with t/4 on the inside as a reference, with graphs of k values of 2, 4, and 8. image

A more general alteration is the addition or subtraction of another Sin(t-pi)term. The 'y' equation becomes sin(t)*cos(t/4) - k*sin(t). The results are a thinning or thickening of the airfoil without affecting the camber. This can be observed in this graph of k=-.25(inside),0 (middle),+.25(outside). image

Notable values of 'N'
n = 2; 'thin' airfoil it constitutes the inner limit of the camber function. image

n = 3; this produces a concave airfoil where the bottom surface crosses the chord line at the aerodynamic center. image

n = 4; this gives an airfoil similar to the Clark'Y' that has the bottom trailing edge along the chord. image

n = 6; common NACA airfoil imitation, biconvex with camber. image

n >> inf; becomes symmetric airfoil, can be graphed by replacing cos(t/(n)) with value '1'. image