Mae 360 Hwk3

J.Murray Hwk 3 MAE 360 Question 1: Find the dimensionless velocities at at least 50 points along the airfoil surface. To find these velocities the equation for the u values is (1) for this equation t is the panel distribution according to the cosine distribution. X is the distance from the origin along the cord from the leading edge to the trailing edge. Z is the location of the point on the upper surface of the airfoil at a distance x along the airfoil.

 −1 t (i +1) − x j ti − x j  A b −  2 2  (t ( i +1) − x j ) + Z 2j (t i − x j ) + Z 2j   t ( i +1) − x j  t − xj   − tan −1  i  tan −1   Z   Z  j j     (this is a matrix) A= Zj

(

 1 u = 1+   2π

 1 w = −  2π

Zj Zj  −  2 2 2  (t ( i +1) − x j ) + Z j (t i − x j ) + Z 2j

 −1 A b  

(

)

(1)

(2)

)

(3)

Using these equations it is possible to find the velocities of u and w of a point along the upper surface of the airfoil. Question 2: Find and plot the pressure coefficients as a function of the dimensionless x at these points. Figure (1) is the plot that these points provided. Cp Distribution on a NACA 0012 Airfoil at 0 Degree Angle of Attack -0.5

Cp

0

0.5

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Chord (Unitless)

0.7

0.8

0.9

1

Figure 1: MATLab Figure of the airfoil Cp

Question 3: Compare your results with the following. Comment on how well or how poorly the pressures compare. The two plots compare very well. The plots are almost identical expect where the x value approaches the leading and trailing edges. As the x value approaches these edges the equation for the plot causes a discontinuity. This discontinuity can not be plotted accurately using discrete methods. The plots created using this method are very accurate until very close to the stagnation points. As figure (2) and (3) show when the gridlines and ticks are the same, the plots are the same until very close to 0 and 1 on the chord axis. Cp Distribution on a NACA 0012 Airfoil at 0 Degree Angle of Attack -0.5

Cp

0

0.5

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Chord (Unitless)

0.7

0.8

0.9

Figure 2 Plot of the Pressure Coefficient

Figure 3 Homework Assignment 3 Provided Figure

1

The code: function aero=hmk3 clc warning off; global Npanel mu t N =51; Npanel = N-1; % there are N endpoints and Npanel panels % locations of first and last endpoints - varies depending on the number of % panels. Determine using trial and error.xs =.0018; xf =.9985; xs=.009; xf=.9985; c=linspace(xs,xf); % distribute panels according to a cosine distribution ths = pi-acos((.5-xs)/.5); thf = acos((xf-.5)/.5); dth = (thf-ths)/Npanel; th = [ths:dth:thf]; t= 1/2 + 1/2*cos(th); %assign field point locations at the center of the panels for i =1 : Npanel x(i) =(t(i) + t(i+1))/2; end % this is the upper surface of a NACA 0012 airfoil Z=.6.*(.2969.*sqrt(x)-.126.*x-.3516.*x.^2+.2843.*x.^3.1015.*x.^4); ma=max(Z); %find radius of curvature Tau=ma; rho=.5*Tau^2*1; E=rho/2; % creat the A matrix and the rhs b vector for j = 1:Npanel b(j)=2*pi; for i = 1:Npanel A(j,i)=(atan2(t(i+1)-x(j),Z(j))-atan2(t(i)-x(j),Z(j)))/Z(j); end end %solve for the doublet strengths mu = inv(A)*b'; for i=1:Npanel Psi(i)=Z(i)-sum(mu(i)*atan((t(i)-x(i))/Z(i))); for j=1:Npanel U(j,i)=[(t(i+1)-x(j))/((t(i+1)-x(j))^2+(Z(j))^2)-((t(i)x(j))/((t(i)-x(j))^2+(Z(j))^2))]; W(j,i)=[(Z(j)/((t(i+1)-x(j))^2+(Z(j))^2))-(Z(j)/((t(i)x(j))^2+(Z(j))^2))]; end end u=1+(1/(2*pi)).*U*mu; w=(-1/(2*pi)).*W*mu; for i=1:Npanel Cp(i)=1-(u(i))^2-(w(i))^2; end figure(2) plot(x,Cp); set(gca,'YDir','reverse');