MATLAB: Centroid of an airfoil

clc;clear all;close all
%I separated the data points of the top and the bottom of the airfoil so I
%can use interp1 separately for both
%Data points of the top part of the airfoil
x=[1 0.8518 0.7040 0.5536 0.3988 0.2454 0.0937 0.0199 0.0015 0];
y=[0 0.0355 0.0819 0.1206 0.1347 0.1200 0.0777 0.0363 0.0162 0];
%Data points of the bottom part of the airfoil
x2=[0 0.0169 0.0812 0.2054 0.3525 0.4979 0.6457 0.7974 0.9497 1];
y2=[0 -0.0197 -0.0428 -0.0645 -0.0749 -0.0701 -0.0506 -0.0249 -0.0026 0];
%Smoothing both the top and bottom part of the airfoil using
%interp1('pchip')
xx=linspace(1,0);
yy=interp1(x,y,xx,'pchip');
xx2=linspace(0,1);
yy2=interp1(x2,y2,xx2,'pchip');
%Plotting both my smoothed top and bottom data points into the same plot
figure
plot(xx,yy,'b',xx2,yy2,'b')
grid on

This is what I have come up with so far. How do I find the centroid of this airfoil

Best Answer

If you have the image processing toolbox, you can use the xx, yy, xx2, yy2 coordinates as inputs to poly2mask(). Then you can use regionprops() to find the centroid of the mask.

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MATLAB: Aerofoil graph interpolation/extrapolation

You lack sufficient information to construct a meaningful surface, since you have only 2 pieces of information in terms of Reynolds number.

First, plot what you have. A 3-d plot is effectively useless here, and makes it far more difficult to understand what you do have.

plot(Cl1,Cd1,'-or',Cl2,Cd2,'-*b')
legend('RE 400K','RE 500K')
xlabel 'Cl'
ylabel 'Cd'

So we see two curves that are not terribly similar in shape. There is a bit of noise seen. And the abscissa for these two curves are not the same, so you could not just use a tool like interp2 to interpolate. Extrapolating those curves, or for Reynold's numbers that fall (significantly) outside the interval [4e5,5e5] would seem to be a foolish (risky) task, due to the high amount of curvature of the respective curves, and due to the noise in the data.

If you merely wish to predict a point for some other Reynolds number however, you could simply use a spline interpolant, a linear interpolant, or perhaps a smoothing spline for each curve. Then, since you have only two distinct Reynolds numbers and thus only two distinct curves, just use linear interpolation between the two. For example, if you have the curve fitting toolbox, I might do this:

CF1 = fit(Cl1,Cd1,'smoothingspline');
CF2 = fit(Cl2,Cd2,'smoothingspline');
CFinterp = @(Cl,Re) (5e5 - Re)/(5e5-4e5)*CF1(Cl) + (Re - 4e5)/(5e5-4e5)*CF2(Cl);
Cl = linspace(0,1,50);
Reinterp = 4.5e5;
hold on
plot(Cl,CFinterp(Cl,Reinterp),'g-')
legend('RE 400K','RE 500K','RE 450K')

I hard coded the Reynolds numbers in the interpolant there, but it is easy enough to fix that.

MATLAB: Finding the area under a polynomial equation in MATLAB

How do you doit? Basic calculus?

X=[0   0.0300   0.0333   0.0469   0.0575   0.0649   0.0960   0.1262   0.1610   0.1717   0.2017   0.2411   0.2711   0.3000   0.3035   0.3368   0.3711   0.4068   0.4411   0.4758   0.5253   0.5553   0.5853   0.6000];
Y=[0 126.2766 140.0068  190.5906  211.6557  219.3601  236.8709  253.2271  266.4663  269.3085  273.6913  279.3095  283.5479  286.1191  286.2725  282.7424  277.6845  272.1010  266.4556  260.1458  248.5416  241.1381  233.2857  230.3781];
xinterp = linspace(min(X),max(X),100);
P = polyfit(X,Y,6);
plot(X,Y,'ro',xinterp,polyval(P,xinterp),'b-')

As models go, I'd call it your basic piece of crapola. And DON'T use a higher order polynomial, thinking it will do better. Just use a spline instead, if you ABSOLUTELY insist on a model. Actually, pchip (shape preserving interpolant) would be my choice here.

If I had to guess, that looks like essentially a piecewise linear curve, with 5 linear segments. But why do you need to integrate a poorly fit polynomial at all?

The area under the curve is...

trapz(X,Y)
ans =
     149.16

In fact, this is arguably a better estimator of the area than the integral of that crappy polynomial model.

What do I get for the pchip integral? My SLM toolbox has a nice little utility to integrate a spline. But it is easily enough done without that tool.

pp = pchip(X,Y);
integral(@(x) ppval(pp,x),min(X),max(X))
ans =
       149.28

That is not too far off from what trapz gave. Bother are about as good as you can get.

Can we integrate the polynomial model? Again, of course. As I said, basic calc suffices. (Or, you could use polyint.)

P = polyfit(X,Y,6);
Pint = [P./(7:-1:1),0];
diff(polyval(Pint,[min(X),max(X)]))
ans =
     149.78

Or if you are feeling lazy,

integral(@(x) polyval(P,x),min(X),max(X))
ans =
     149.78

In any event, I'd use what pchip or what trapz gave here as better estimators.

Best Answer

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MATLAB: Aerofoil graph interpolat​ion/extrap​olation

MATLAB: Finding the area under a polynomial equation in MATLAB

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MATLAB: Aerofoil graph interpolation/extrapolation