# 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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