integral of contour plot

I would like to compute the circulation from a velocity field. From my understanding the circulation can be computed from surface integral of vorticity plot (curl).

My input parameters are be X,Y,U,V which are all 2D matrices.

I think I found a solution for a retangular surface https://uk.mathworks.com/matlabcentral/answers/36218-surface-integral-of-discrete-data but I would like to carry out an integral over a contour plot.

I basically have the same problem as previously posted https://uk.mathworks.com/matlabcentral/answers/307767-how-to-integrate-the-data-from-contour-over-a-surface but I couldn't find the solution.

I am very poor with coding and some line of codes would be much appreciated.

Thanks

Answer 1

[Nathaniel H Werner]

You can calculate the circulation using the bounds of a contour in 2018b.

I completed this on cylindrical slices for my purposes, but I'm sure you can adjust this. You will need velocity data, not vorticity to use the code I have provided but the results will be equivalent.

R = 1;
Nt = floor(2.1*50*R);
[Xc,Zc,Yc] = cylinder(R*ones(1,50),Nt);
surf(Xc,Yc,Zc)
hsli = contourslice(Xw,Yw,Zw,D_ta,Xc,Yc,Zc,[-3,-3],'linear');
% D_ta is the name of my vorticity field in this case.
% You can always use other terms to define your contour, or just choose an arbitrary shape.
if ~isempty(hsli)
    hsli.LineWidth = 2;
    hsli.EdgeColor = [144,144,144]/255;
    axis equal
    Xcs = hsli.XData; Xcs = Xcs(1:end-1);
    Ycs = hsli.YData; Ycs = Ycs(1:end-1);
    Zcs = hsli.ZData; Zcs = Zcs(1:end-1);
            
    ux_loc = interp3(Xw,Yw,Zw,Ux,Xcs,Ycs,Zcs);
    uy_loc = interp3(Xw,Yw,Zw,Uy,Xcs,Ycs,Zcs);
    uz_loc = interp3(Xw,Yw,Zw,Uz,Xcs,Ycs,Zcs);
    GammaLoc = zeros(1,length(Xcs));
    
    for g = 1:length(Xcs)-1
        dl = [Xcs(g+1)-Xcs(g),...
             Ycs(g+1)-Ycs(g),...
             Zcs(g+1)-Zcs(g)];
                
        GammaLoc(g) = dot([ux_loc(g),uy_loc(g),uz_loc(g)],...
        dl);
     end
     Gamma = sum(GammaLoc);
end