FLOWLAB SOLUTION
8.125 Use the FlowLab template pipe_el to investigate the flow field in the entrance
region for pipe flow. Set the pipe geometry to a diameter of D = 0.1 m and length of l =
20 m. Use the default air properties for the fluid, set the Reynolds number to Re = 400,
and set the grid to medium resolution. After running the simulation to convergence,
discuss the following concepts concerning the entrance length of the pipe. (a) Note the
centerline velocity profile as a function of distance along the pipe and explain why its
value exhibits a sharp increase in magnitude and why it eventually levels off. (b) By
plotting velocity vectors, demonstrate the inviscid core in the entrance region. (c)
Compare the difference in the velocity profiles in the entrance region and the fullydeveloped region.
Problem Setup
The pipe geometry was set according to the problem statement shown above. Note that
the problem statement specifies a diameter whereas the FlowLab input requires the radius
of the pipe. The material properties and boundary conditions were set to produce a
Reynolds number of 400 (see below).
For this particular problem, the flow was set to laminar and the heat transfer was turned
off by de-selecting the “Heat Transfer” button, as shown in the figure below.
The grid resolution was set to “medium” for this problem. The students could also run
this problem with the “fine” grid resolution and compare the results, but the simulation
time is increased slightly with the additional grid points. A portion of the medium-density
grid is shown below. The problem is solved as an axisymmetric flow field so only the
two-dimensional geometry (length of pipe and distance from centerline to pipe wall) is
needed.
For the solver settings, the default values for the number of iterations and the
convergence limit were selected. Five locations along the pipe length were chosen to
output radial velocity profiles, as shown below.
The values of x/D were selected to have several profiles in the entrance region, one
profile just prior to the analytic value of the entrance length (see Eqn. 8.1 in the text), and
one profile just past the analytic value. Profiles at the inlet and outlet are automatically
included in this plot by FlowLab.
Answer
The following figure displays the convergence history for the problem setup. The
simulation converged in approximately 250 iterations. Very often the continuity variable
is the last one to reach convergence.
Looking at the problem statement, the students are to note the centerline velocity as a
function of distance along the pipe and explain the increase in magnitude and eventual
leveling off of the values.
The plot below shows the velocity magnitude (m/s) at the centerline of the pipe as a
function of distance (m) along the pipe. On a conceptual level, the students should
discuss the reasons why the flow is accelerating in the entrance region at the centerline
and why the flow eventually levels off to a constant value.
The velocity vectors can be plotted from the post-processing section of FlowLab. A
vector plot for a portion of the entrance region in shown in the figure below. There is a
clear change in velocity moving out from the wall, but a constant velocity region
(inviscid core) is also visible. As you move down the pipe length, the velocity vectors
show this region to become progressively smaller.
The final part of the problem asks the student to compare axial velocity profiles in the
entrance region and the fully-developed region. Looking at Eqn. 8.1 from the textbook,
the non-dimensional entrance length, le D , should be 24 for a Reynolds number of 400.
As noted above, output profile locations were selected at various locations, including one
before and one after the analytic location.
The figure below shows axial velocity profiles at various locations along the pipe length
(including inlet and outlet by default). The x/D value of 20 is quite close to the fullydeveloped output profile as predicted by Eqn. 8.1. The x/D = 30 profile is an excellent
match for the outlet profile, which is fully-developed.