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In this worksheet, we take a look at potential flow around a cylinder with circulation. We study the effect of varying the circulation on the topology of the streamlines. The standard reference for this system is Batchelor, paragraph 6.6.
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\r\n\r\nIn this worksheet, we take a look at potential flow around a cylinder with circulation. We study the effect of varying the circulation on the topology of the streamlines. The standard reference for this system is Batchelor, paragraph 6.6.
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x, y, U, Gamma, R = var('x, y, U, Gamma, R')
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Sage has some capabilities to calculate the stream function directly from the complex potential. However, in the interest of speed it is best to define psi directly.
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psi(x, y, U, Gamma, R) = U*(1 - R^2/(x^2 + y^2))*y - Gamma/(4*pi)*log(x^2 + y^2)
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import matplotlib.cm
from sage.ext.fast_eval import fast_float
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# We now fix U = 1.0 and R = 1.0 and let Gamma be a free parameter
#
# (The 'fast_float' command below optimizes the stream function for fast evaluation)
psi_fast = fast_float(psi(x, y, 1.0, Gamma, 1.0), 'x', 'y', 'Gamma')
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Note that psi becomes unbounded when we approach the origin. If we were to feed psi to the contour_plot command directly, this singularity would obscure all other features of the plot. To avoid this, we define a new function which returns infinity when evaluated on a point inside the circle. These points will be left out of the contour_plot.
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Note that psi becomes unbounded when we approach the origin. If we were to feed psi to the contour_plot command directly, this singularity would obscure all other features of the plot. To avoid this, we define a new function which returns infinity when evaluated on a point inside the circle. These points will be left out of the contour_plot.
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# The following is a helper function which does all the dirty work for us:
def plot_body_flow(Gamma):
psi_body = lambda x, y: psi_fast(x, y, Gamma) if x^2 + y^2 > 1.0 else oo
contour_plot(psi_body, (x, -2, 2), (y, -2, 2), cmap='jet', contours=50, aspect_ratio=1).show()
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# For Gamma = 0.0, we clearly observe steady flow around a circular body
plot_body_flow(0.0)
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# We clearly see the effect of increasing the circulation:
plot_body_flow(8.0)
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# When Gamma = 4*pi, the solution changes quantitatively: the two stagnation points on the boundary of the body coalesce.
plot_body_flow(4*pi)
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# When Gamma is increased even further, the stagnation point moves off the boundary of the rigid body and into the flow.
# The streamlines close to the rigid body are closed curves, while far away they are unbounded.
plot_body_flow(4*pi + 0.5)
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