MAE 150B AerodynamicsCFD Simulations in COMSOL over a Circle and NACA 4410 and 0010 Airfoils

Tara MellorUID: 904347586

Abstract— Computational fluid dynamics (CFD) is used tosolve complex fluids flows around aerodynamic bodies. Thisstudy uses COMSOL 5.4, a commercial CFD software, tocompute 2D pressure and velocity distributions around a 0.1mdiameter circle, a NACA 4410 airfoil, and a NACA 0010 airfoil.The convergence of different mesh types and the relationbetween angle of attack and drag and lift coefficients on theairfoils was investigated. Results from COMSOL simulationsfor the lift coefficient were compared to predictions from thinairfoil theory. The effect on the drag coefficient from changingthe magnitude of the uniform velocity over the circle wasalso tested. Streamlines, velocity contours, velocity vectors, andpressure contours were plotted in COMSOL for the NACA 0010and 4410 airfoils at a zero angle of attack.

I. MODEL AND THEORY

The shape of a symmetric NACA airfoil is give by:

yt =ct

0.2(0.2969

√x

c−0.1260(

x

c)−0.3516(

x

c)2+0.2843(

x

c)3−0.1015(

x

c)4)

(1)

Where c is the chord length, t is the thickness given bythe last two digits of a four digit NACA number, and x isthe position along the chord from 0 to c.

To calculate the shape of a cambered airfoil, additionalinformation must be known:

yc =

{m x

p2 (2p− xc ) for 0 ≤ x ≤ pc

m c−x(1−p)2 (1 +

xc − 2p) for pc ≤ x ≤ c

(2)

The Cartesian coordinates of the upper surface of the airfoilare then given by Xu = x − y sin θ and Yu = yc + cos θ.Where θ = tan−1(dyc

dx ) Along the lower surface, theCartesian coordinates are given by Xl = x + y sin θ andYl = yc − cos θ.

II. METHODS

Stationary (time-independent), 2D, laminar flow studieswere conducted in COMSOL 5.4 over the three differentaerodynamic shapes. The airfoil shapes were computed usingthe model above in a MATLAB program (see Appendix) andloaded into COMSOL. All airfoil simulations were carriedout using an extremely fine mesh. The ’no-slip’ boundarycondition was applied to the surface of the aerodynamicbodies. The dynamic viscosity (ηo) was a constant 0.003kg/ms and the density was a constant 1 kg/m3 throughout allthe simulations. Unlesss otherwise noted, the inlet velocity(Uo) is 3m/s. This results in a Reynold’s number of 100.

III. RESULTS AND DISCUSSION

A. Normal, finer, and extremely fine mesh around a circle

Fig. 1: From top to bottom: normal mesh, finer mesh, and extremelyfine mesh. All meshes were generated on a 2m x 4m Cartesian grid

In CFD simulations, finer meshes lead to greater accuracyin results. Figure 1 displays three different mesh types:normal, finer, and extremely fine from top to bottom. As canbe see in table 1 below, the COMSOL computed value forthe drag coefficient converges on the experimental solutionfor drag over a cylinder as mesh type becomes finer (SeeAppendix A for experimental results). All simulations forthe NACA 0010 and 4410 airfoils used an extremely finemesh for this reason. The main drawback to using a finermesh is increased computation time.

Mesh Type CD CD (exp) % Difference

Normal 1.340 1.2 11Finer 1.2674 1.2 5.5

Extremely Fine 1.2304 1.2 2.5

TABLE I: Computed drag coefficient of different mesh types forflow over a 0.1 m diameter circle.

B. Effect of changing the magnitude of uniform velocity onlift and drag coefficients

Uo (m/s) CL CD CD(exp) % Difference

0.06 -4.812 E-04 8.622 8.0 7.50.15 -1.441 E-03 4.764 4.0 360.3 -2.823 E-03 3.237 3.2 1.10.6 -5.672 E-02 2.306 2.3 0.261.5 -1.044 E-02 1.571 1.4 123.0 -1.638 E-02 1.230 1.2 2.5

TABLE II: Effect on lift and drag of increasing uniformvelocity magnitude for flow over a 0.1m circle. See Appendixfor experimental CD coefficients. Percent difference is betweenexperimental and COMSOL CD values.

As uniform velocity increases, the drag on the cylinderappears to decrease. Overall, COMSOL’s results fordrag coefficient resemble that obtained from experiment(Appendix A). The lift increases.

C. Velocity contour, pressure contour, streamline, andvelocity vector field plots over a 0.1m diameter circle

Fig. 2: Uniform flow over a cylinder velocity vector field

Fig. 3: Uniform flow over a circle velocity contours

Fig. 4: Uniform flow over a circle streamlines

Fig. 5: Uniform flow over a circle pressure contours

The circular shape is not an optimal aerodynamic body. Ithas a much higher drag coefficient than both NACA airfoilsat Re = 100. Also, the lift is negligible compared to the dragat all magnitudes of uniform velocity tested. It should alsobe noted that the streamline plot in fig. 4 shows a region ofturbulence just behind the body.

D. Velocity contour, pressure contour, streamline, andvelocity vector field plots over a NACA 0010 airfoil

Fig. 6: Extremely fine mesh around NACA 0010 airfoil

Fig. 7: Velocity contour plot over NACA 0010 airfoil

Fig. 8: Uniformly distributed streamlines over a NACA 0010 airfoil

Fig. 9: Pressure contour plot over NACA 0010 airfoil

E. Velocity contour, pressure contour, streamline, andvelocity vector field plots over a NACA 4410 airfoil

Fig. 10: Velocity vector field over a NACA 4410 airfoil

Fig. 11: Velocity contour plot over NACA 4410 airfoil

Fig. 12: Uniformly distributed streamlines over NACA 4410 airfoil

Fig. 13: Pressure contours over NACA 4410 airfoil

F. Angle of attack vs. drag coefficient

Fig. 14: Drag vs. angle of attack over a 0010 airfoil

Fig. 15: Drag vs. angle of attack over a 4410 airfoil

Drag increases with increased angle of attack for both thesymmetric 0010 airfoil and the cambered 4410 airfoil. TheCambered airfoil has a higher drag coefficient at all angle ofattack than the symmetric airfoil.

G. Angle of attack vs. lift coefficient

Fig. 16: Lift vs. angle of attack over a 0010 airfoil

α (deg) CD CL CL (Theory) % Difference0 0.1287 3.232 E-04 0 - -2 0.1308 0.1042 0.2193 714 0.1362 0.2146 0.4386 406 0.1447 0.3162 0.6580 708 0.1562 0.3952 0.8773 76

10 0.1703 0.4387 1.097 86

TABLE III: Lift and Drag Coefficient at different angles of attackover a NACA 0010 airfoil. Theory refers to thin airfoil their whichpredicts that CL = 2πα

Fig. 17: Lift vs. angle of attack over a 0010 airfoil

α (deg) CD CL

0 0.2682 0.16112 0.2776 0.37614 0.2928 0.58926 0.3111 0.77918 0.3349 0.934310 0.3627 1.054

TABLE IV: Lift and Drag Coefficient at different angles of attackover a NACA 4410 airfoil. Theory refers to thin airfoil their whichpredicts that CL = 2πα

The lift coefficients predicted from thin airfoil theory andthose calculated in COMSOL for the 0010 airfoil differ

significantly as seen in table III and fig. 16. This implies thatthin airfoil theory is not an accurate method of predicting thelift coefficient if the body of interest has thickness.

The lift increases with angle of attack as seen in fig. 16, fig.17, table III, and table IV. The cambered airfoil has a higherlift coefficient at all angles of attack than the symmetricairfoil. The 4410 airfoil is the most aerodynamic of all threebodies tested. At all angles of attack the lift coefficient ishigher than the drag coefficient which is desired.

APPENDIX

A. Experimental Results for Drag on a Cylinder vs.Reynold’s Number

Fig. 18: Drag on a Cylinder vs. Reynold’s Number. From JohnAnderson’s Aerodynamics, 6th edition.

B. Velocity Surface Plots on Airfoils

Fig. 19: Velocity surface profile of NACA 4410 airfoil

Fig. 20: Velocity surface profile of NACA 0010 airfoil

C. MATLAB Airfoil Calculations Source Code

Contents

• user inputs.• number of grid points• constants• calculations• plot• Setting up deliminited text file for use in COMSOL.

% Coordinates for NACA Airfoil

user inputs.

%type of airfoiltypeNACA = ’4410’;

number of grid points

% # points on EACH surface of airfoil.grid_points = 101;

%extract values from inputM_initial = str2double(typeNACA(1));P_initial = str2double(typeNACA(2));T_initial = str2double(typeNACA(3:4));M = M_initial/100;P = P_initial/10;T = T_initial/100;

constants

a0 = 0.2969;a1 = -0.1260;a2 = -0.3516;a3 = 0.2843;a4 = -0.1015;%Note: this constant can change...%depending on airfoil type!

calculations

% derivative, theta, ycx = linspace(0,1,grid_points)’;yc = ones(grid_points,1);d_yc = ones(grid_points,1);theta = ones(grid_points,1);for i = 1:grid_points

if x(i) >= 0 && x(i) <= Pyc(i) = (M/Pˆ2).*(2*P*x(i) - x(i)ˆ2);d_yc(i) = ((2*M)/Pˆ2).*(P-x(i));

elseif x(i) >P && x(i) <=1yc(i) = (M/(1-P)ˆ2)*(1-2*P+2*P*x(i) - x(i)ˆ2);d_yc(i) = ((2*M)/(1-P)ˆ2).*(P-x(i));

endtheta(i) = atan(d_yc(i));

end

% thickness distributionyt = ones(grid_points,1);term0 = a0.*sqrt(x);term1 = a1.*x;term2 = a2.*x.ˆ2;

term3 = a3.*x.ˆ3;term4 = a4.*x.ˆ4;yt = 5.*T.*(term0 + term1 + ...term2 + term3 + term4);

% Upper Surface Pointsxu = ones(grid_points,1);yu = ones(grid_points,1);for i = 1:grid_points

xu(i) = x(i) - yt(i)*sin(theta(i)) ;yu(i) = yc(i) + yt(i)*cos(theta(i));

end% Lower Surface Pointsxl = ones(grid_points,1);yl = ones(grid_points,1);for i = 1:grid_points

xl(i) = x(i) + yt(i)*sin(theta(i)) ;yl(i) = yc(i) - yt(i)*cos(theta(i));

end

plot

f1 = figure(1);hold on; grid on;axis equal ;plot(xu,yu)plot(xl,yl);

Setting up deliminited text file for use in COMSOL.

xu_flip = flip(xu) ;yu_flip = flip(yu) ;X = [xu_flip(1:end-1);xl];Y = [yu_flip(1:end-1);yl] ;Airfoil_Coordinates = [X,Y];% EXPORT : Airfoil_Coordinates to excel!