Fourier Series for thin airfoil theory

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Tyler Reohr
Tyler Reohr 2021년 11월 16일
편집: Paul 2021년 11월 16일
I'm trying to use a Fourier Series Approximation, specifically that specified in thin airfoil theory, to write a function that will do this more universally than hard coding it. So far what I have is the following code. It runs, and my value for is correct, however none of my other values are even remotely close to correct. I believe the problem is coming from the way that I find dydx, but I don't understand why the same method works for and no the other A values.
clear all;close all;clc
alph = 0;
[ ~, ~, ~, xcam, ~ ]= NACA4Digit('2412',100,100,100);
[ ~, ~, ~, ~, ycam ]= NACA4Digit('2412',100,100,100);
ncoeff = 15;
% %% Error Messages
% if size(xcam) ~= size(ycam)
% error('Camber line vector sizes do not match.');
% return;
% else
% end
%% Variable Initialization
iter = length(xcam);
Uinf = 1; % Freestream Velocity, value is a placeholder as the final
% results are non-dimensionalized.
c = iter; % Chord Length, value is a placeholder as the final results are non-
% dimensionalized.
alpha = deg2rad(alph); % Converts the angle of attack to rad
%% Constant Functions
DF = gradient([xcam;ycam]); % Takes the gradient at the given points
[~,fore] = max(ycam); % Finds the point within the array where the camber is greatest, important for differentiating
zeroToFore = linspace(0,fore,iter); % Defines Integration Bounds
zeroToFore = acos(1 - ((2*zeroToFore)/c)); % Converts to theta from x
foreToAft = linspace(fore,pi,iter); % Defines Integration Bounds
foreToAft = acos(1 - ((2*foreToAft)/c)); % Converts to theta from x
A = zeros(1,ncoeff-1); % Intializes A
A0 = alpha - (1/pi)*(trapz(zeroToFore,DF(1,:)) + trapz(foreToAft,DF(2,:))); % Defines A_0
U_theta = zeros(1,iter); % Intitalizes U_theta
%% Solve
for i = 1:iter
theta = acos(1-((2*xcam(i))/c)); % Defines Theta
for n = 1:1:ncoeff
trapzDataFore = DF(1,:)*cos(n*theta) % Trapz Data from the given function for thin airfoil theory using Fourier Series Approx, fore
trapzDataAft = DF(1,:)*cos(n*theta) % Trapz Data from the given function for thin airfoil theory using Fouerier Series Approx, aft
An(n) = (2/pi) * (trapz(zeroToFore,trapzDataFore) + trapz(foreToAft,trapzDataAft)) % A_n value for the Fourier Series, there are ncoeff A values
end
Eqn1 = 0 % Intitializes Equation 1
for n = 1:ncoeff
Eqn1 = Eqn1 + An(n)*sin(n*theta) % Sums all of the A_n Values
end
gamma = 2*Uinf*(A0*((1 + cos(theta))/(sin(theta))) + Eqn1) % Eqn for gamma at a given theta
U_theta(i) = gamma/2; % U at theta
end
Cp(1) = 0;
Cp(iter) = 0;
C_p = 1 - ((Uinf+U_theta)/Uinf).^2;
Cl = pi*(2*A0 + A(1));
Cm = (pi/4)*(A(2)-A(1));
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Tyler Reohr
Tyler Reohr 2021년 11월 16일
KSSV,
When I utilize the other output from gradient, I then have 3 DF outputs:
2x100 DFX, which is equal to what I had with just the DF output, and
2x100 DFY, which is 2 equal 1x100 rows
I'm assuming that you're right and I do need to incorporate this data, but I'm not exactly sure how do to so with the way I have the code set up, how would I define the dydx for different iterations in the first nested for loop? Under trapzDataFore and trapzDataAft?
For reference, if you're not familiar with thin airfoil theory, the equation I am going from is derived from:
and can be simplified to:
and trapzDataFore and trapzDataAft are the respective equations within the integral.
Thanks!
Tyler

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Paul
Paul 2021년 11월 16일
편집: Paul 2021년 11월 16일
From the equation it looks to me that y is scalar function and the independent variable x is also a scalar.
Are xcam and ycam both 1-dimensional?
In which case I think this line needs to change:
%DF = gradient([xcam;ycam]); % Takes the gradient at the given points
DF = gradient(ycam,xcam);
I've used that form of the gradient command for a long time, but it doesn't seem to be documented anymore.
  댓글 수: 6
Tyler Reohr
Tyler Reohr 2021년 11월 16일
EDIT:
I misinterpreted the graph, it is in fact, much weirder than a scaled version of the xcam and ycam values:

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