MCG 4345 – Aerodynamics

Computational Assignment I

Report Presented to

Dr. Stavros Tavoularis

Prepared By

University of Ottawa

November 21st 2014

ii

Table of Contents Table of Contents ..................................................................................................................................................................... ii

List of Figures ........................................................................................................................................................................... iii

List of Tables ............................................................................................................................................................................. iii

Introduction ............................................................................................................................................................................... 1

Airfoil Geometry ....................................................................................................................................................................... 2

Vortex Panel Method Calculations and Discussion .................................................................................................... 4

Pressure Coefficient Distribution ................................................................................................................................. 4

Non-Dimensional Velocity Distribution .................................................................................................................... 6

Lift Coefficient Calculation .............................................................................................................................................. 6

Pitching Moment Calculation ......................................................................................................................................... 8

Conclusion ................................................................................................................................................................................ 10

References ................................................................................................................................................................................ 11

APPENDIX A: MATLAB CODE ........................................................................................................................................... 12

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List of Figures

Figure 1 - NACA 4418 geometry, camber line and maximum camber and thickness locations ............. 2

Figure 2 - Angle of Trailing Edge ....................................................................................................................................... 2

Figure 3 – NACA 4418 Thickness Distribution ............................................................................................................ 3

Figure 4 – NACA 4418 Geometry discretized into panels ....................................................................................... 3

Figure 5 – NACA 4418 Surface Pressure Coefficient Distribution Using Vortex Panel Method with

N=100 Panels ............................................................................................................................................................................ 4

Figure 6 – NACA 4418 Surface Non-Dimensional Velocity Distribution Using Vortex Panel Method

with N=100 Panels .................................................................................................................................................................. 6

Figure 6 – NACA 4418 CL-alpha Curve Using Different Methodologies and Comparison with

Experimental Data for Re = 3e6......................................................................................................................................... 8

Figure 6 – NACA 4418 CM-alpha Curve Using Different Methodologies and Comparison with

Experimental Data for Re = 3e6......................................................................................................................................... 9

List of Tables

Table 1 – Maximum Camber and Thickness for NACA 4418 airfoil ................................................................... 2

Table 2 – Leading Edge Radius of Curvature ............................................................................................................... 2

Table 1 – Angle of Zero-Lift for NACA 4418 Airfoil Calculated Using Different Methods ......................... 7

1

Introduction Numerical methods are now essential in the aerospace industry. Simple methods such as the vortex

panel method are even used in large companies like Bombardier Aerospace, as they offer a rapid and

reliable alternative to more time-consuming Navier-Stokes solvers. This assignment shows us the

basic application of the vortex panel method to a 2D, NACA airfoil. This assignment will concentrate

on 4-digit NACA airfoils, more specifically the NACA 4418. Firstly, it is important to know that NACA

airfoil geometries can be calculated using a simple polynomial function. For an arbitrary chord-

normalized x position, the chord-normalized y-position can be found by equations 6.2 and 6.4 of the

family of Wing Sections document [1]. The half-thickness for a given x-position is presented in equation 6.2, whereas the camber line position is presented in equation 6.4.

Using these two equations, the y-position of the airfoil boundary is given by equation 𝐴.

𝒚𝒕𝒐𝒑 = 𝒚𝒄 + 𝒚𝒕

𝒚𝒃𝒐𝒕 = 𝒚𝒄 − 𝒚𝒕

𝑨

The radius of curvature at the leading edge is determined by equation 6.3 of the family of Wing Sections document [1].

2

Airfoil Geometry Using the points provided in the NACA technical report [2], the geometry has been plotted and is shown in Figure 1.

Figure 1 - NACA 4418 geometry, camber line and maximum camber and thickness locations

Finding the maximum of equations 6.2 and 6.4 gives the maximum thickness and maximum camber

line values. In the case of maximum thickness, it is twice the value given by equation 6.2. The x-

position of the maximum camber line and thickness is simply the x-position corresponding to this maximum value in the equation. These values are presented in Table 1.

Table 1 – Maximum Camber and Thickness for NACA 4418 airfoil

Parameter Chord-Normalized x-position Chord-Normalized Value

Maximum Camber 0.4 0.04

Maximum Thickness 0.17 0.29

The chord-normalized radius of curvature of the leading edge is obtained using equation 6.3. It is presented in Table 2.

Table 2 – Leading Edge Radius of Curvature

Radius of Curvature 0.198

The angle of the trailing edge is obtained by calculating the angle between the trailing edge point on the airfoil and the second-last points on the top and bottom surfaces. It is shown in Figure 2.

𝛾 = 22.9°

Figure 2 - Angle of Trailing Edge

-0.1

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z/c

x/c

Maximum Camber NACA 4418 Airfoil Camber Line Maximum Thickness

𝛾

3

The chord-normalized thickness distribution of the Airfoil, calculated using twice the value provided by equation 6.2 is plotted in Figure 3.

Figure 3 – NACA 4418 Thickness Distribution

The airfoil must now be discretized into panels. Figure 4 shows the NACA 4418 geometry with 100 panels

shown with dashed lines. As we can see, using 100 panels approximates the airfoil geometry quite well.

Figure 4 – NACA 4418 Geometry discretized into panels

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

t/c

x/cAirfoil Thickness

-0.1

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

z/c

x/c

Airfoil Contour Panels

4

Vortex Panel Method Calculations and Discussion Flow about the NACA 4418 airfoil was calculated using a vortex panel method potential flow solver.

Results are shown in the following section. Calculations have been performed for angles of attack ranging from -10 to 20 degrees inclusively, in increments of 5 degrees.

Pressure Coefficient Distribution

The results for the pressure coefficient distribution are shown in Figure 5 and Figure 6.

Figure 5 – NACA 4418 Surface Pressure Coefficient Distribution for Angles of Attack of -5 to 20 Degrees Using Vortex Panel Method with N=100 Panels

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Figure 6 – NACA 4418 Surface Pressure Coefficient Distribution for Angles of Attack of -5 to 0 Degrees Using Vortex Panel Method with N=100 Panels

The pressure coefficient distribution with respect to the angle of attack (AOA) is expected. As the

AOA is increased, one would expect the stagnation point to move in the direction of the lower surface. As we can see in Figure 5, the stagnation point, represented by –𝐶𝑝 = −1, moves further down the

lower surface of the airfoil as the AOA approaches 20°. With an increase in AOA, it can also be seen

that the point of maximum suction (also called suction point: represented by highest point on the curve) becomes larger with an increase in AOA.

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Non-Dimensional Velocity Distribution

The non-dimensional velocity coefficient is shown in Figure 7.

Figure 7 – NACA 4418 Surface Non-Dimensional Velocity Distribution for Angles of Attack of -5 to 20 Degrees Using Vortex Panel Method with N=100 Panels

The vortex panel method results for non-dimensional velocity also concord with expectations. It can

be observed that the highest point of the non-dimensional velocity is located at the same area on the

wing surface as the suction point.

Lift Coefficient Calculation

The lift coefficient was calculated by numerically integrating the surface pressure, and circulation

intensity over each panel. It has also been calculated using thin airfoil theory. Numerical methods

were also used to calculate the lift coefficient using thin airfoil theory. The code used for the

calculations of the lift coefficient using all three methods is presented in Appendix A. The equations

solved numerically for these three methods are equations 𝐵, 𝐶 and 𝐷 for pressure coefficient, circulation and thin airfoil methods respectively.

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𝑪𝑳 = ∫ ∆𝑪𝒑𝒅𝒙

𝑻𝑬

𝑳𝑬

(𝑩)

𝑪𝑳 = ∫ 𝚪𝒅𝒔

𝑻𝑬

𝑳𝑬

(𝑪)

𝑪𝑳 = 𝟐𝝅(𝜶 − 𝜶𝟎), 𝜶𝟎 = ∫𝒅𝒛

𝒅𝒙

𝝅

𝟎

(𝟏 − 𝒄𝒐𝒔𝜽)𝒅𝜽 (𝑫)

The angle of zero lift for numerical and experimental results have been obtained using linear interpolation to find a zero lift coefficient within the data calculated or provided.

The angles of zero lift predicted by each method are shown in Table 3.

Table 3 – Angle of Zero-Lift for NACA 4418 Airfoil Calculated Using Different Methods

Method Angle of Zero-Lift

[°] Error [%]

Vortex Panel Method (𝐶𝑝 distribution) -4.36 3.3

Vortex Panel Method (Circulation distribution) -4.36 3.3 Thin Airfoil Theory -4.13 2.1

Experimental (Re = 106) -4.22 - The results obtained numerically for the angle of zero-lift are in excellent accordance with the experimental value, with a maximum error of only 3.3%. To verify the validity of these methods for angles of attack near the angle of zero-lift, we must refer to the results for the lift coefficient for these different methods, shown in Figure 8. As it can be observed, the values of lift coefficient can be assumed valid for angles of attack approaching the angle of zero-lift

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Figure 8 – NACA 4418 CL-alpha Curve Using Different Methodologies and Comparison with Experimental Data for Re = 3e6

The results shown in Figure 8 show that the initial slope and order of magnitude coincide quite well

for low angles of attack. However, as the angle of attack increases and goes further away from the

angle of zero-lift, the results for the numerical methods used become invalid. The vortex panel

method and thin airfoil theory are incapable of predicting the stall angle obtained experimentally, as

they can only predict linear increases in Lift with respect to the angle of attack. This is why these

methods are valid for only small angles of attack. When comparing the three theoretical methods

used to compute the lift coefficient, it can be observed that they are in accordance. Since these three

methods consider the flow to be inviscid, it is expected that they cannot perfectly match the

experimental curve or predict the stall angle.

Pitching Moment Calculation

The pitching moment about the leading edge has been calculated using the vortex panel method

pressure distribution along the surface of the airfoil as well as thin airfoil theory. Equations E and F are used to compute this coefficient for the vortex panel and thin airfoil methods respectively.

𝑪𝑴 = − ∫ 𝑪𝒑𝒙 𝒅𝒙

𝑻𝑬

𝑳𝑬

(𝑬)

𝑪𝑴 = 𝝅

𝟐(𝑨𝟎 + 𝑨𝟏 −

𝑨𝟐

𝟐) (𝑭)

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𝑨𝟎 = 𝜶 −𝟏

𝝅 ∫

𝒅𝒛

𝒅𝒙𝒅𝜽

𝝅

𝟎

𝑨𝒏 = 𝟐

𝝅 ∫

𝒅𝒛

𝒅𝒙𝒄𝒐𝒔(𝒏𝜽)𝒅𝜽

𝝅

𝟎

The results obtained for the pitching moment about the leading edge are shown in Figure 9 and

compared to experimental results for Re = 3e6.

Figure 9 – NACA 4418 CM-alpha Curve Using Different Methodologies and Comparison with Experimental Data for Re = 3e6

The pitching moment predictions using thin airfoil theory and the vortex panel method are very far

off from the experimental values for Re = 106. Even though these methods are in accordance with

each other, and predict values in the same order of magnitude as experimental results, the variation

of pitching moment is very far from experimental results. Once again, the fact that these methods do not consider viscous effects can relate the different values obtain numerically and experimentally.

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Conclusion In summary, this work has helped to understand the validity of thin airfoil theory and vortex panel

methods when it comes to computing the lift coefficient, pitching moment coefficient about the

leading edge, and angle of zero-lift. These methods are quite valid to solve for the lift at angles of

attack approaching the angle of zero-lift, but fail to predict the stall angle as they can only

approximate the lift to vary linearly with the angle of attack. For the pitching moment, the predictions

had the same order of magnitude and increasing slope with respect to the angle of attack, but failed

to efficiently predict the values. To predict the lift and pitching moment at higher angles of attack,

other methods, such as Navier-Stokes solver using finite element methods should be used, as they can adequately simulate the effects of skin and pressure drag.

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References 1. IRA H. ABBOTT, ALBERT E. VON DOENHOFF , Theory of Wing Sections, DOVER

PUBLICATIONS, INC., NEW YORK, 1959

2. IRA H. ABBOTT, ALBERT E. VON DOENHOFF , NATIONAL ADVISORY COMITEE FOR

AERONATICS, REPORT #824, Langley Field

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APPENDIX A: MATLAB CODE The following appendix show the MATLAB code used to compute and plot the lift coefficient for each of the methods.

% computational assignment 1

clear all

close all

%% Thin Airfoil Theory

% Using the equation for the camber line for a NACA 4412 Airfoil

n = 10000;

m = 0.04; % For NACA 4418

p = 0.4; % For NACA 4418

x = linspace(0,1,n);

z = zeros(1,n);

thetamid = zeros(1,n-1);

for i = 1:n

if x(i) < p

z(i) = m/p^2*(2*p*x(i) - x(i)^2);

else

z(i) = m/(1-p)^2*(1 - 2*p + 2*p*x(i) - x(i)^2);

end

end

% Find value of theta for change in camber line equation

for i = 1:n-1

thetamid(i) = acos(1-(2*(mean([x(i) x(i+1)]))));

end

dzdx = diff(z)./diff(x);

func = dzdx.*(1-cos(thetamid));

% Angle of zero lift

alpha0 = 1/pi*trapz(thetamid,func);

% Pitching Moment

j = 1;

for aoa = -10:5:20

A0(j) = aoa*pi/180 - 1/pi*trapz(thetamid,dzdx);

A1(j) = 2/pi * trapz(thetamid,dzdx.*cos(thetamid));

A2(j) = 2/pi * trapz(thetamid,dzdx.*cos(2*thetamid));

j = j + 1;

end

CMta = 0.5*pi*(A0 + A1 - A2/2);

clearvars thetamid theta z x func

%% Experimental CL-alpha & CM-alpha curve for Re = 3*10^6

expdat = load('naca4418_cl_alpha.txt');

CMexpdat = load('naca4418_cm_alpha.txt');

alphaexp = expdat(:,1);

CLexp = expdat(:,2);

CMalphaexp = CMexpdat(:,1);

CMexp = CMexpdat(:,2);

j = 1;

%% Compute data from program

for aoa = -10:5:20

alpha(j) = aoa;

filename = sprintf('aoa%i.csv',aoa);

data = csvread(filename);

xp = data(:,2);

yp = data(:,3);

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xc = data(1:end-1,4);

yc = data(1:end-1,5);

t = data(1:end-1,6);

L = data(1:end-1,7);

g = data(:,8);

cv = data(1:end-1,9);

cp = data(1:end-1,10);

cl = data(end,10);

clearvars data

CP(:,j) = cp;

CV(:,j) = cv;

% Number of control points

N = numel(xp);

% Get top and bottom Cp and V distribution

k = 0;

l = 0;

for i = 1:N-1

if yc(i) >= 0

k = k + 1;

xu(k,j) = xc(i);

cpu(k,j) = cp(i);

cvu(k,j) = cv(i);

else

l = l + 1;

xl(l,j) = xc(i);

cpl(l,j) = cp(i);

cvl(l,j) = cv(i);

end

end

xl = fliplr(xl);

cpl = fliplr(cpl);

%% Get lift coefficient from Cp distribution

CLcp(j) = sum(-cp.*L.*cos(t-aoa*pi/180));

%% Get Pitching Moment Coefficient from Cp distribution

CM(j) = -trapz(xc,cp.*xc);

% Get lift coefficient from gamma distribtution

% First, we must find the average circulation on each panel

gamma = g(1:end-1) + diff(g)/2;

GAMMA = sum(L.*gamma);

CLg(j) = 4*pi*GAMMA;

%% Lift Coefficient calculated by program

CL(j) = cl;

%% Thin airfoil theory lift

CLta(j) = 2*pi*(alpha(j)*pi/180-alpha0);

j = j+1;

end

figure

hold on

plot(alpha,CLcp,'kx','MarkerSize',10)

plot(alpha,CLta,'kd','MarkerSize',10)

plot(alpha,CLg,'ko','MarkerSize',10)

plot(alphaexp,CLexp,'k--','LineWidth',1)

title('CL-alpha curve for NACA 4418 Airfoil Using Different Methods');

legend('Cp','Circulation','Thin Airfoil','Re 3*10^6')

xlabel('Alpha');

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ylabel('CL');

figure

hold on

plot(alpha,CM,'kx','MarkerSize',10)

plot(alpha,CMta,'ko','MarkerSize',10)

plot(CMalphaexp,CMexp,'k--','LineWidth',1)

title('CM-alpha curve for NACA 4418 Airfoil Using Vortex Panel Method');

legend('Cp','Thin Airfoil','Re 3*10^6')

xlabel('Alpha');

ylabel('CM');

alpha0cp = interp1(CLcp,alpha,0);

alpha0g = interp1(CLg,alpha,0);

alpha0ta = alpha0*180/pi;

fprintf('\nUsing thin airfoil theory, alpha0 is %.2f',alpha0ta);

fprintf('\nUsing pressure distribution, alpha0 is %.2f',alpha0cp);

fprintf('\nUsing circulation, alpha0 is %.2f\n',alpha0g);

figure

hold on

color{1} = 'k-';color{2} = 'b-';color{3} = 'g-';color{4} = 'r-';

color{5} = 'k--';color{6} = 'b--';color{7} = 'g--';

for i = 1:j-1

plot(xc,CV(:,i),color{i});

leg{i} = cat(2,'aoa ',num2str(alpha(i)));

end

legend(leg)

title('Non-Dimensional Velocity on Surface of NACA 4418 Airfoil');

xlabel('x/c');

ylabel('Cv');

figure

hold on

for i = 1:j-1

plot(xc,-CP(:,i),color{i});

end

legend(leg)

title('Pressure Coefficient on Surface of NACA 4418 Airfoil');

xlabel('x/c');

ylabel('-Cp');