Finite Volume Equations

Part 1

A portion of a straight pipe with a circular cross-section is to be modeled with a staggered mesh finite volume method. It is divided into 3 volumes each containing 0.5 cubic meters. The cross-sectional area is 0.5 square meters. Flow velocity is uniform and constant at 2 meters per second directed from volume 1 towards volume 3. Initial densities of a solute are 1 ppm in volumes 1 and 2 and 10 ppm in volume 3. Draw a representation of the finite volume mesh. What is the cell length? What is the cell (pipe) diameter? Use a time step of 0.2 seconds and an explicit difference method to calculate the new time density (density at the end of the time step) in the 2nd volume. In a first calculation  predict the new time density in volume 2 using a donor cell average to obtain the mean density at the cell edges. In a second calculation of new time density, apply a central difference (linear average) for the cell edge densities. I only want results for volume 2 density at the end of a single time step  for each of the 2 edge averaging methods.

Part 2

Starting with the problem in Part 1,  treat volume 1 as a boundary condition that never changes.  Set the initial solute densites to 1 ppm, 1.1 ppm, and 1 ppm in volumes 1 through 3 respectively.   Use an explicit donor cell method to calculate the solute density in the 2nd volume for five separate 10 second transients, using time steps of  0.25 s, 0.5 s, 0.75, 1.0 s, and 1.25 s.  Provide results for each of your calculations (density vs. time) on a separate plot (or table).  Make sure to plot one point for every time step.  How do these results relate to the discussion of numerical stability in class?

Show all work and feel free to ask me for clarifications.

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