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Homework Help: Modelling a flow past a symmetric body

  1. Oct 18, 2008 #1
    1. The problem statement, all variables and given/known data
    I need to model the streamlines for an incompressible, irrotational flow about various symmetric bodies, starting with an ellipse


    2. Relevant equations
    for a uniform flow in the positive x direction, psi =y
    for a source/sink of strength K at (xn,yn)
    psi = Karctan((y-yn)/(x-xn))
    at any point along the body, there must be no flow across it, and hence psi = 0


    3. The attempt at a solution

    the stream function for the whole flow = Uy+ SUM Knarctan(y/(x-xn)) (no ydisplacement as all sources are on axis)
    this = 0 at each of the control points y(m) (and everywhere else along the surface of the sphere, but a number of control points = to the number of sources should be appropriate)

    hence
    Knpsin(m) - Uy(m) = 0

    and so I should be able to get the strengths from the following matlab code (taking U =1)


    Code (Text):

    n = 10;
    L = 1;
    D = 0.5;
    dx = 2*L/(n+1);
    xs = -L+dx:dx:L-dx;   % uniformly distributed
    xb = -L+dx:dx:L-dx   % uniformly distributed
    %
    % definition of body geometry
    %
       yb = D*sqrt(1-xb.^2./L^2); % ellipse
    %
    % plot control points
    %
    figure
    plot([-L xb L],[0 yb 0],[-L xb L],[0 -yb 0])
    axis equal
    %
    % compute source strengths
    %
    % K = strength
    %
    for i=1:n
        for j=1:n
            a(i,j) = atan(yb(i)./(xb(i)-xs(j)));
        end
        b(i,1) = -yb(i);
    end
    K = a\b
     
    However this, results in all negative values for K, meaning a huge total removal of volume from the system instead of the expected no (or approximately no) change, and hence a set of funky streamlines that look nothing at all like a flow around an ellipse.

    EDIT: realised I made a mistake, should be b(i,1) = yb(i) (not -yb(i)) however this just results in all strengths being positive, so I get what looks like a funky superposition of flows around multiple semi-infinite bodies.
     
    Last edited: Oct 18, 2008
  2. jcsd
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