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====== Math 445 lecture 22: inviscid flow past a cylinder ======

The velocity and pressure fields of inviscid, irrotational flow past a cylinder are given by 

\begin{eqnarray*}
 v_x &=  v_0 \left(1 - \frac{a^2}{r^2} \cos 2 \theta \right) \\
 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}
\end{eqnarray*}
where $v_0$ is the speed of the incoming fluid, $a$ is the radius of the cylinder centered at the origin, and $r,\theta$ are the polar coordinates of point $x,y$. 
These functions are valid only for points outside the cylinder $(r>a)$.

The following Matlab code plots the velocity as a quiver plot and the pressure with contours. 

<file matlab plotcylinderflow.m>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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')

</file>

{{ :gibson:teaching:spring-2016:math445:lecture:cylinderflow.png?direct&500 |}}
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