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view/download model file: Dispersion2.nlogo

This is a model of shear dispersion. It simulates the release of a passive tracer into a channel of width B with a longitudinal velocity profile U(y). The individual tracer particles (think of them as molecules) advect with the current at the same time as they undergo a random diffusion process in the y-direction. The combination of transverse diffusion and "differential advection" gives rise to a longitudinal dispersion process that follows Fick's Law. This means that the tracer concentration profile in the longitudinal direction is Gaussian and longitudinal variance grows linearly with time, after an initial transient period.

The user first initiates the trace molecules by pressing the "setup" button. This action releases the molecules in a transverse line at x = 0. Pressing the "Go" button then starts the simulation. Prior to starting the simulation, the user may use the slider buttons to vary the values of U, the longitudinal velocity along the channel centerline, and Dy, the trasverse diffusion constant. The transverse-dependence of the velocity, U(y) is visually represented by the color variation of the channel (dark blue is the maximum velocity, near the channel centerline, while white corresponds to zero velocity at the edges). The red lines are the solid channel boundaries, which tracer molecules are prohibited from crossing. The triangles represent the tracer particles.

Once the simulation has begun, the motion of the tracer can be monitored by watching the particles move within the colored channel. In addition, the behavior of the tracer is shown by the three plots entitles "Histogram", "Tracer Concentration Profile vs. Gaussian", and "Growth of Tracer Variance". The histogram plot shows a histogram of the locations of all tracer particles with time. Note that the x axis changes with time, as the tracer becomes more and more spread out. The next plot compares the tracer concentration profile to an ideal Gaussian curve. When the Fickian dispersion regime has been reached, the tracer curve should closely resemble a Gaussian. The last curve shows the growth of the tracer variance with time, and compares it with the prediction based on Fickian dispersion theory. Note that when the slope of the line from the simulation matches the slope of the ideal linear curve, the Fickian regime has been reached.

Notice that the higher the value of Dy, the faster the particles move in the transverse direction.
Note that the time necessary to reach the Fickian dispersion regime depends on the values of U and Dy. See if you can determine this dependence through some combination of scaling and simulation. Try varying the values of Dy and for each run, note the time at which you think Fickian dispersion has begun (Gaussian concentration profile and/or linear growth of tracer variance).

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