Problem 60749. Compute the dispersion coefficient

A contaminant dumped or spilled into a river will move downstream with the flow, but it will also spread in the flow direction because of several mechanisms. One of these mechanisms is shear dispersion: the spreading results because the velocity varies across the cross section, and parcels of the contaminant sample different velocities as eddies transport them across the cross section.
G.I. Taylor showed that the concentration averaged over the cross section evolves according to an advection-diffusion equation, and the dispersion coefficient can be computed with
K = -(1/hD) integral(u' integral(integral(u'(y2),0<=y2<=y1),0<=y1<=y),0<=y<=h)
where h is the width of the stream, D is the transverse mixing coefficient, and u' = u - bar(u) is the deviation of the velocity profile from the cross-sectional average velocity
ubar = (1/h) integral(u(y),0<=y<=h)
Write a function that takes a (normalized) velocity profile u(y) specified at several points and computes the quantity
I = -integral(u' integral(integral(u'(y2),0<=y2<=y1),0<=y1<=y),0<=y<=h)
Solve

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Last Solution submitted on Oct 27, 2024

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