Streamlines , Magnus and Cp aroud a cylinder section

Plot stream line and vectors around a circle.

Description

Plot streamlines and pressure coefficent of an unviscious, uncomprimibile,irrotational flow around a cylinder section (r=1) that spins around the z axis (coming out of the xy plane). This result is achieved by superimposition of elementary solution of the potential function PHI, where [Ux , Uy] = GRAD(PHI) which comprehend Uniform Stream,Doublet, Vortex.
This case rappresents a good example of Magnus Effect,that is the reason why spinning balls have an effected trajectory.And rappresent the basis of Kutta-Joukowski theory.

• MATLAB release: MATLAB 5.2 (R10)

[prabhakar subrahmanyam]

Good job

[imran sajid]

 

[Guy Incognito]

Your a life saver buddy

[C Szlaie]

best program ive seen

[CJ Rocha]

Good, just missing read data file function to be used with any geometry.

[reza maghsoodi]

hello

[Serkan türkmen]

Could you check here:
cp = 1 - 4*sin(t).^2 + 2* G / (pi*a*V_i) *sin(t) - (2* G/ (pi*a*V_i) )^2 ;
my opinion it must be like that.
cp = 1 - 4*sin(t).^2 + 2* G / (pi*a*V_i) *sin(t) - ( G/ (2*pi*a*V_i) )^2 ;
btw your code is prety good.

[Yogesh PARTE]

I just modified and updated the code without for loops and better graphics. Here is the code

% Plot streamlines and pressure coefficent of an unviscious,uncomprimibile,irrotational
% flow around a cylinder section (radius = 1) that spins around the z axis (coming out
% of the xy plane).
% This result is achieved by superimposition of elementary solution of the potential
% function PHI, where [Ux , Uy] = GRAD(PHI) which comprehend Uniform Stream,Doublet,Vortex.
% This case rappresents a good example of Magnus Effect,that is the reason why spinning
% balls have an effected trajectory.And rappresent the basis of Kutta-Joukowski theory.
%
%
% INPUT
% V_i = Asymptotic Speed
% G = Angular Speed (positive anti-clockwise)
%
% -----EXAMPLE------
% V_i = 20
% G = 50
%
% Created by: Dario Isola
% DATA : 24/03/2004
% Politecnico di Milano - Aerospatial Engeeniering Departement
%----------
%Modification log
%Author Date Description
%----------------------------------------------------------------------------------
%Yogesh PARTE,IMT Toulouse 15 Jan 2010 Added Cp distribution plot
% over a cylinder see
% figure(4), removed for loop, added comments
%----------------------------------------------------------------------------------

close all;

clear all;

%% Input section
V_i = input(' Asymptotic speed V_0 [m/s] = ');
G = input(' Circulation Value G [rad/s] [Anti-clockwise] = ');

%% Actual computation
%Radius of the circle
a = 1 ;

c =-a*2;

b =a*2;

% Number of intervals
n =a*50;

[x,y]=meshgrid([c:(b-c)/n:b],[c:(b-c)/n:b]');

warning off;

%%Preliminary DATA & purification

% Set values of X and Y inside the cylinder to zero
[I J]=find( (x.^2+y.^2) < a);

if ~isempty(I)
    x(I,J) = 0;
    y(I,J) = 0;
end

%Definition of polar variables
rho=sqrt(x.^2+y.^2);

theta=atan2(y,x);

% Creation of the streamline function
z=V_i.*sin(theta).*rho.*(1-(a^2./(rho.^2)))-G*log(rho)/(2*pi);

%% Generate unite cicle for plotting
n=100;
r=ones(1,n+1)*a;
t=[0:2*pi/n:2*pi];

Xcircle = r.*cos(t);
Ycircle = r.*sin(t);
%% Plot the data

% Streamline Plot
figure(1);
contour(x,y,z,25);
colorbar;
hold on;
fill(Xcircle,Ycircle,'k');
title('Stream Lines');
xlabel('x \rightarrow');
ylabel('y \rightarrow');
axis square;

%% SECOND PART
% Reproduce streamlines
figure(2);
contour(x,y,z,15);
hold on;
fill(Xcircle,Ycircle,'k');
%Compute velocity at new X,Y coordinates for better viewing
%Creation of vectors around the circle
x=[-a*2:a/3:a*2];
[x]=meshgrid(x);
y=x';

% Set values of X and Y inside the cylinder to zero
[I J]=find( (x.^2+y.^2) < a);

if ~isempty(I)
    x(I,J) = 0;
    y(I,J) = 0;
end

r=sqrt(x.^2+y.^2);
theta=atan2(y,x);
ur=V_i*cos(theta).*(1-a^2./(r.^2));
ut=-V_i*sin(theta).*(1+a^2./(r.^2))+G./(2*pi*r);
u=ur.*cos(theta)-ut.*sin(theta);
v=ur.*sin(theta)+ut.*cos(theta);

%Vectors and Filled Circle plotting
hold on;
quiver(x,y,u,v);
title('Speed Vectors')
xlabel('x \rightarrow');
ylabel('y \rightarrow');
axis square;
grid off;

% Analytical solution for Cp distribution
t=0:.1:2*pi;

cp = 1 - 4*sin(t).^2 + 2* G / (pi*a*V_i) *sin(t) - (2* G/ (pi*a*V_i) )^2 ;

% Non lifting solution
cp_sim = 1 - 4*sin(t).^2 ;

% Lift from Kutta Jokowaski theorm
L = - 1.225*V_i*G;

L = strcat('Kutta Joukowski Lift: ',num2str(L),' [N]');

% plot cp distribution
figure(3);
plot(t,cp,t,cp_sim,'--r');

axis([0 2*pi min(cp) max(cp_sim)]);

title(strvcat('Pressure coefficient around the surface (standard air density)',L));
xlabel('Theta (angle with horizontal)')
ylabel('C_p')
legend('Lifting solution','Symmetrical solution');
grid on;

% Plot cp distribition on cylinder

figure(4);
%filled cirlcle
cpx = cp.*cos(t);
cpy = cp.*sin(t);
fill(Xcircle,Ycircle,'y');
hold on;
scale =2; % used to control length of Cp in quiver plot
quiver(a*cos(t),a*sin(t),cpx,cpy,scale);
title('Pressure coefficient around the cylinder surface (standard air density)');
xlabel('x \rightarrow');
ylabel('y \rightarrow');
ylim([-1.5 1.5]);
axis square;