MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Homework #4 Solution: Questions (1) – (4)

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

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QUESTION 1: SAMPLE MATLAB SCRIPTclose all;clear all; % Question 1a=1;lambda=1;[x,y]=meshgrid(-5:0.01:5,-5:0.01:5);psi=lambda./2./pi.*atan(2.*x.*y./(x.^2-y.^2+a.^2)); % Now I am going to set some fixed values of the streamfunction psi to% plot. I am setting up a vector V which will range from the minimum value% of psi to the maximum value. I use the double command min(min(psi)) since% psi is two-dimensional, and the first min will find the minimum row and% the second the minimum of that row, or a single value. Same idea with% max(max(psi)). Since I wanted to plot 10+1 values of constant psi, I% arranged them to be equally spaced between the minimum and maximum value. V=[min(min(psi)):(max(max(psi))-min(min(psi)))/10:max(max(psi))];cs=contour(x,y,psi,V); % Note the use of the contour function here. To use the streamline command we could have% found the the u and v components of the velocity, which would have added some complexity% to this problem. Since we are only interested in sketching some of the streamlines,% I simply plotted a constant values of psi without any regard for the exact values.% Note that this would involve iteration to solve the resulting equation for either x or y given% a constant value of psi. xlabel('x-direction'); ylabel('y-direction'); title('Homework #4: Question 1'); grid; % clabel(cs,V);% If you uncomment the above line, MATLAB will label all the contours for% you as well. It is a little messy so use the zooming tool to see what is% really going on, especially in Quesiton #2.

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QUESTION 1: STREAMLINE PLOT

• Find the stream function and plot some streamlines for the combination of a line source, , at (x,y)=(0,a) and an equal line source located at (0,-a)

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QUESTION 1: EXAMPLE OF LABELED CONTOURS

• Find the stream function and plot some streamlines for the combination of a line source, , at (x,y)=(0,a) and an equal line source located at (0,-a)

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QUESTION 2: SAMPLE MATLAB SCRIPT

% Question 2

gamma=-1;

[x,y]=meshgrid(-2.001:0.01:2.001,-2:0.01:2);

% Notice the choice on the starting and ending bounds that I have selected.

% Since the natural log of 0 is negative infinity MATLAB has problems

% plotting streamlines with this value. Therefore I selected some starting

% and ending points that eliminate the argument of the natural log from

% ever being zero.

psi=gamma./2./pi.*log(sqrt((x-a).^2+y.^2))+gamma./2./pi.*log(sqrt((x+a).^2+y.^2));

figure(2)

V=[min(min(psi)):(max(max(psi))-min(min(psi)))/20:max(max(psi))];

cs=contour(x,y,psi,V);

clabel(cs,V);

xlabel('x-direction'); ylabel('y-direction'); title('Homework #4: Question 2'); grid;

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QUESTION 2: STREAMLINE PLOT

• Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex with strength at (x,y)=(a,0) and an equal line vortex placed at (-a,0).

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QUESTION 2: EXAMPLE OF LABELED CONTOURS

• Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex with strength at (x,y)=(a,0) and an equal line vortex placed at (-a,0).

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QUESTION 3: SAMPLE MATLAB SCRIPT, POLAR AND CARTESIAN

% Question 3 in Polar Coordinateslambda=-1000;gamma=1600; [psi,r]=meshgrid(0:100:1000,0:0.01:10);theta=(psi-gamma./2./pi.*log(r))./(lambda./2./pi); % Note that here we were able to solve for theta explicitly, unlike either% x or y in Question (1) or Question (2). Also, I am not sure if MATLAB can% do contour plots in polar coordinates. figure(3)polar(theta,r);title('Homework #4: Question 3, Polar Approach'); grid; % Question 3 in Cartesian Coordinateslambda=-1000;gamma=1600; [x,y]=meshgrid(-10:0.1:10,-10:0.1:10);psi=lambda./2./pi.*atan(y./x)+gamma./2./pi.*log(sqrt(x.^2+y.^2));figure(4)V=[min(min(psi)):(max(max(psi))-min(min(psi)))/20:max(max(psi))];cs=contour(x,y,psi,V);xlabel('x-direction'); ylabel('y-direction'); title('Homework #4: Question 3, Cartesian Approach'); grid;

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QUESTION 3: POLAR COORDINATES

• A tornado is simulated by a line sink =-1000 m2/s plus a line vortex =1600 m2/s. Find the angle between any streamline and radial line, and show that it is independent of both r and . If this tornado forms in sea-level standard air, what is the local pressure and velocity 25 meters from the center of the tornado?

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EXTRA CREDIT: LOG SPIRALS AND THE DIVINE RATIO

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QUESTION 3: CARTESIAN COORDINATES

• A tornado is simulated by a line sink =-1000 m2/s plus a line vortex =1600 m2/s. Find the angle between any streamline and radial line, and show that it is independent of both r and . If this tornado forms in sea-level standard air, what is the local pressure and velocity 25 meters from the center of the tornado?

Explain what isgoing on here

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QUESTION 3: INCREASED FIDELITY

Why are these streamlines not continuous?

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QUESTION 4: SAMPLE MATLAB SCRIPT% Question 4 close all; clear all; a=1;gamma=1;[x,y]=meshgrid(-3.001:0.01:3.001,-3:0.01:3);psi=-3.*gamma./2./pi.*log(sqrt(x.^2+(y-a).^2))+1.*gamma./2./pi.*log(sqrt(x.^2+(y+a).^2));V=[min(min(psi)):(max(max(psi))-min(min(psi)))/50:max(max(psi))];figure(1)contour(x,y,psi,V);xlabel('x-direction'); ylabel('y-direction'); title('Homework #4: Question 4'); grid; figure(2)phi=3.*gamma./2./pi.*atan((y-a)./x)-1.*gamma./2./pi.*atan((y+a)./x);Z=[min(min(phi)):(max(max(phi))-min(min(phi)))/50:max(max(phi))];contour(x,y,psi,V);hold on;contour(x,y,phi,Z);xlabel('x-direction'); ylabel('y-direction'); title('Homework #4: Question 4'); grid;

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QUESTION 1: FINDING MINIMUM VELOCITY USING SYMBOLIC MATH% find velocity components using MATLAB's symbolic toolbox syms x y a gamma u v psi=-3.*gamma./2./pi.*log(sqrt(x.^2+(y-1).^2))+1.*gamma./2./pi.*log(sqrt(x.^2+(y+1).^2)); % To find Velocity Components, take derivatives of psi with respect to x and yu=diff(psi,y);v=-diff(psi,x); % clear alla=1;gamma=1;[x,y]=meshgrid(-3:0.05:3,-3:0.05:3); u=-3./4.*gamma./pi./(x.^2+y.^2-2.*y+1).*(2.*y-2)+1./4.*gamma./pi./(x.^2+y.^2+2.*y+1).*(2.*y+2);v=3./2.*gamma./pi./(x.^2+y.^2-2.*y+1).*x-1./2.*gamma./pi./(x.^2+y.^2+2.*y+1).*x; V=sqrt(u.^2+v.^2); Vmin=min(min(V))loc_Vmin=find(V==Vmin);xloc_Vmin=x(loc_Vmin);yloc_Vmin=y(loc_Vmin); % The following command will generate a 3-D picture of the velocity fieldfigure(3)surf(x,y,V);hold on;plot3(xloc_Vmin,yloc_Vmin,Vmin,'.r','MarkerSize',20);colorbar;title('Homework #4: Question 4 Velocity Magnitude'); grid;xlabel('x-direction'); ylabel('y-direction'); zlabel('Velocity Magnitude'); % Examine velocity magnitude on constant xx=xloc_Vmin;y=-3:0.01:3;u=-3./4.*gamma./pi./(x.^2+y.^2-2.*y+1).*(2.*y-2)+1./4.*gamma./pi./(x.^2+y.^2+2.*y+1).*(2.*y+2);v=3./2.*gamma./pi./(x.^2+y.^2-2.*y+1).*x-1./2.*gamma./pi./(x.^2+y.^2+2.*y+1).*x;V=sqrt(u.^2+v.^2);figure(4)plot(y,V,yloc_Vmin,Vmin,'ro'); grid;legend('Velocity Magnitude on x=0','Location of Minimum Velocity','Location','NorthWest');xlabel('y-direction'); ylabel('Magnitude of Velocity'); title('Plot of Velocity Profile between Vorticies');

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QUESTION 4: PLOT OF • A counterclockwise line vortex of strength 3 at (x,y)=(0,a) is combined with a

clockwise vortex at (0,-a). Plot the streamline and potential-line pattern, and find the point of minimum velocity between the two vorticies.

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QUESTION 4: PLOT OF AND

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QUESTION 4: VELOCITY MAGNITUDE (SYMBOLIC MATH)

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QUESTION 4: MAGNITUDE AND LOCATION OF VMIN