MATLAB: Aerofoil graph interpolat​ion/extrap​olation

aerofoilsairfoilsextrapolationinterpolationsurface

I have a bunch of data for a NACA 63-412 aerofoil which I want to use in my software. I have two data sets, one for reynolds number 400000 and one for 5000000. I need to interpolate/extrapolate the data so that I can calculate with other reynolds numbers. The furthest I have gotten now is this, I have drawn two lines from this data and now I need to connect those into i suppose a surface but really wouldn't know how? any help would be greatly appreciated!
Cd1 = [0.0088
0.0089
0.0089
0.009
0.009
0.009
0.009
0.009
0.0091
0.0091
0.0092
0.0093
0.0094
0.0096
0.0098
0.0101
0.0106
0.0116
0.0133];
Cl1= [-0.0053
0.0523
0.1096
0.1675
0.2248
0.2823
0.3392
0.3952
0.452
0.5089
0.5647
0.6206
0.6767
0.7321
0.7858
0.836
0.885
0.928
0.9606];
RE1= [400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000
400000];
Cd2 = [0.008
0.008
0.008
0.0081
0.0081
0.0081
0.0081
0.0082
0.0082
0.0083
0.0084
0.0085
0.0087
0.0089
0.0093
0.0101
0.0112
0.0127
0.0141];
Cl2 = [-0.0048
0.053
0.1108
0.1689
0.2267
0.2837
0.3406
0.3982
0.455
0.5117
0.5687
0.6249
0.681
0.7356
0.7879
0.8352
0.8793
0.9172
0.9541];
RE2 = [500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000
500000];
figure(1)
hold on
plot3(Cl1,Cd1,RE1);
plot3(Cl2,Cd2,RE2);

Best Answer

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.
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