Question

4. The following formulas are commonly used by engineers to predict the lift and drag of an airfoil: L 30,8V2, 3PCSV? D where L and D are the lift and drag forces, V is the airspeed, S is the wing span, p is the air density, and C & Co are the lift and drag coefficients, which are polynomial expressions that depend on the attack angle, a. A wind tunnel experiment for a particular airfoil has generated the following formulas: CL 4.47 x 10-2 + 1.15 x 10-30? +6.66 x 10-20 +1.02 x 10- CD 5.75 x 10-6° +5.09 x 10-4.? + 1.8 x 10-a +1.25 x 10-2 where a is in degrees.

use matlab to solve

Answer 1

% Matlab code to plot a graph of Lift Force v/s Velocity, Drag Force v/s
% Velocity and plot of Cl/Cd vs alpha

%a.
% define the variables
p = 0.002378; % slug/ft^3
S = 36; % ft
a = 1; % degrees

% create vector V containing velocity values from 0 to 150 in steps of 0.01
% in mi/hr and convert them into ft/sec
V = (0:0.01:150).*(5280/3600); % ft/sec

% calculate lift and drag coefficient
Cl = (4.47 * (10^-5) * (a^3))+(1.15 * (10^-3) * (a^2)) + (6.66 * (10 ^ -2 ) * a) + (1.02 * (10 ^-1));
Cd = (5.75 * (10^-6) * (a^3))+(5.09 * (10^-4) * (a^2)) + (1.8 * (10 ^ -4 ) * a) + (1.25 * (10 ^-2));

% calculate the lift force using element by element operation
L = (p.*Cl.*S.*(V.^2))./2;
% calculate the lift force using element by element operation
D = (p.*Cd.*S.*(V.^2))./2;

% plot lift force in y-axis and velocity in x-axis
figure(1);
plot(V,L);
xlabel('Velocity(ft/sec)');
ylabel('Lift Force');
title('Plot of Lift Force v/s Velocity');
grid on;

% plot drag force in y-axis and velocity in x-axis
figure(2);
plot(V,D);
xlabel('Velocity(ft/sec)');
ylabel('Drag Force');
title('Plot of Drag Force v/s Velocity');
grid on;

%b.
% create vector alpha from -2 to 22 degrees in steps of 0.01
a = (-2:0.01:22);

% create vectors for CL and CD of same size as alpha
CL = zeros(size(a));
CD = zeros(size(a));

% create vector of same size as alpha to store the ratio of CL and CD
ratio_L_D = zeros(size(a));

% loop to calculate CL and CD for different values of alpha and calculate
% their ratio
for i=1:length(a)
CL(i) = (4.47 * (10^-5) * (a(i)^3))+(1.15 * (10^-3) * (a(i)^2)) + (6.66 * (10 ^ -2 ) * a(i)) + (1.02 * (10 ^-1));
CD(i) = (5.75 * (10^-6) * (a(i)^3))+(5.09 * (10^-4) * (a(i)^2)) + (1.8 * (10 ^ -4 ) * a(i)) + (1.25 * (10 ^-2));
ratio_L_D(i) = CL(i)/CD(i);
end

figure(3);
plot(a,ratio_L_D);
title('Plot of L/D v/s alpha');
xlabel('alpha');
ylabel('L/D');
grid on;

% determine the maximum value of L/D and the index at which this
% maximum value is found using max function
[ratio_max,I] = max(ratio_L_D);

% display the maximum value of L/D and the angle of attack which maximizes
% the ratio
fprintf('Maximum value of L/D is %f and the angle of attack that maximizes is %f degrees.\n',ratio_max, a(I(1)));

%end of script

Output:

Figure 1 Х Eile Edit View Insert Tools Desktop Window Help Plot of Lift Force v/s Velocity 400 350 300 250 Lift Force 200 150 100 50 0 0 50 200 250 100 150 Velocity (ft/sec)

Figure 2 Eile Edit View Insert Tools Desktop Window Help Plot of Drag Force v/s Velocity 30 25 20 Drag Force 15 10 5 0 0 50 200 250 100 150 Velocity (ft/sec)

Maximum value of L/D is 17.944793 and the angle of attack that maximizes is 3.810000 degrees. Figure 3 Eile Edit View Insert Tools Desktop Window Help 7 Plot of L/D v/s alpha 18 16 14 12 10 L/D 8 6 st N 0 -2 -5 0 5 15 20 25 10 alpha