Solve the same problem as in 11.20 except have the concentration = 1 in the lower half of the inlet.

Solve the same problem as in 11.20 except
have the concentration = 1 in the lower half of the inlet and 0 in the top
half.

Problem 11.20

Consider flow into a 3D channel that is one
unit high, eight units across, and ten units long for a Reynolds number of 1.0.
Solve for the flow coming in the 1 × 8 cross section. Use “P1 + P1”
discretization for flow. First, use a uniform velocity in, to check for any
errors in your representation. Next, apply linear coupling (see Chapter 10) to
have a fully developed flow coming in. Then solve the convective diffusion equation
with the concentration = 1 on the right-hand side of the inlet and zero on the
left-hand side of the inlet. Use linear discretization for concentration.
Compute the variance at the outlet for Pe = 100, 200, 500, 1000. Discuss the
accuracy of your calculations and indicate steps that could be taken to improve
the accuracy. Also solve the problem when the velocity profile is the fully
developed velocity in a channel, which is equivalent to having slip on the side
walls. To achieve this, just take the velocity in the x direction as 6∗z∗(1−z). Although the problem with slip
could be solved in two dimensions, solve it in the 3D geometry with the same
mesh used with no-slip.