gibson:teaching:spring-2016:math445:lecture:cylinderflow
====== Differences ====== This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
gibson:teaching:spring-2016:math445:lecture:cylinderflow [2016/03/31 09:55] gibson |
gibson:teaching:spring-2016:math445:lecture:cylinderflow [2016/03/31 09:59] (current) gibson |
||
|---|---|---|---|
| Line 4: | Line 4: | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | v_x &= v_0 \left(1 - \frac{a^2}{r^2} \cos(2 \theta) \\ | + | v_x &= v_0 \left(1 - \frac{a^2}{r^2} \cos 2 \theta \right) \\ |
| - | v_y &= -v_0 \frac{a^2}{r^2} \sin(2 \theta) \\ | + | v_y &= -v_0 \left(\frac{a^2}{r^2} \sin 2 \theta \right) \\ |
| - | p &= 2 \frac{a^2}{r^2} \cos(2 \theta) - \frac{a^4}{r^4} | + | p &= 2 \frac{a^2}{r^2} \cos 2 \theta - \frac{a^4}{r^4} |
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | The following Matlab code plots the velocity as a quiver plot and the pressure with contours. | ||
| + | <code matlab> | ||
| + | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||
| + | % Inviscid, irrotational flow around a cylinder | ||
| + | % Make quiver plot of velocity and contour plot of pressure for | ||
| + | % Vx(r,theta) = V0 (1 - a^2/R^2 cos(2 theta)) | ||
| + | % Vy(r,theta) = -V0 a^2/r^2 sin(2 theta) | ||
| + | % p(r,theta) = 2 a^2/r^2 cos(2 theta) - a^4/r^4; | ||
| + | |||
| + | % for a=1, V0=1, on domain -4 <= x <= 4, -4 <= x <= 4 | ||
| + | |||
| + | %%%%%%%%% Define constants %%%%%%%%% | ||
| + | V0 = 1; | ||
| + | a = 1; | ||
| + | |||
| + | %%%%%%%%% First draw the cylinder %%%%%%%%% | ||
| + | |||
| + | theta = linspace(0, 2*pi, 100); | ||
| + | plot(a*cos(theta), a*sin(theta), 'k-') | ||
| + | hold on | ||
| + | |||
| + | |||
| + | %%%%%%%%% Second make the contour plot of pressure %%%%%%%%% | ||
| + | |||
| + | % define grid | ||
| + | x = linspace(-4,4,201); % contour plots need fine grids, so use lots of points | ||
| + | y = linspace(-4,4,201); | ||
| + | [X,Y] = meshgrid(x,y); | ||
| + | |||
| + | % compute polar coords r, theta on the x,y gridpoints | ||
| + | R = sqrt(X.^2 + Y.^2); | ||
| + | Theta = atan2(Y,X); % use atan2(y,x) to get correct quadrant for theta | ||
| + | |||
| + | % evaluate the formula for the pressure field | ||
| + | P = (2*a^2)./R.^2 .* cos(2*Theta) - a.^4./R.^4; | ||
| + | |||
| + | % set P to zero inside the cylinder | ||
| + | P = (R > a) .* P; | ||
| + | |||
| + | % draw the contour plot, using ten contour lines | ||
| + | contour(x,y,P, 10); | ||
| + | |||
| + | colorbar | ||
| + | colormap jet | ||
| + | |||
| + | %%%%%%%%% Third make the quiver plot of velocity %%%%%%%%% | ||
| + | |||
| + | x = linspace(-4,4,21); % quiver plots work better on coarse grids | ||
| + | y = linspace(-4,4,21); | ||
| + | [X,Y] = meshgrid(x,y); | ||
| + | |||
| + | R = sqrt(X.^2 + Y.^2); | ||
| + | Theta = atan2(Y,X); | ||
| + | |||
| + | % evaluate the formula for the velocity field | ||
| + | Vx = V0*(1 - a^2./R.^2 .* cos(2*Theta)); | ||
| + | Vy = - V0*a^2./R.^2 .* sin(2*Theta); | ||
| + | |||
| + | % set Vx, Vy to zero inside the cylinder | ||
| + | Vx = (R > a) .* Vx; | ||
| + | Vy = (R > a) .* Vy; | ||
| + | |||
| + | % draw the quiver plot | ||
| + | quiver(x,y, Vx, Vy); | ||
| + | |||
| + | hold off | ||
| + | |||
| + | axis equal | ||
| + | axis tight | ||
| + | |||
| + | xlabel('x'); | ||
| + | ylabel('y'); | ||
| + | title('inviscid 2D flow around a cylinder') | ||
| + | |||
| + | </code> | ||
| + | |||
| + | {{ :gibson:teaching:spring-2016:math445:lecture:cylinderflow.png?direct&500 |}} | ||
gibson/teaching/spring-2016/math445/lecture/cylinderflow.1459443301.txt.gz · Last modified: 2016/03/31 09:55 by gibson
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
