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--- gibson:teaching:spring-2016:math445:lecture:cylinderflow [2016/03/31 09:54]
gibson created
+++ gibson:teaching:spring-2016:math445:lecture:cylinderflow [2016/03/31 09:59] (current)
gibson
@@ Line 4: / Line 4: @@
 \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}
-\endPeqnarray*}
+\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 |}}

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