FLOWLAB SOLUTION
8.128 Use the pipe_el FlowLab template to investigate how the Reynolds number
affects the entrance region of a pipe flow. Alter the Reynolds number of the problem by
varying the fluid properties and the inlet velocity, U. Obtain data for Re = 100, 500, and
1000. Demonstrate that the selected pipe diameter, D, and pipe length, l, are adequate for
studying entrance region flows (hint: the pipe exit should be a sufficient distance
downstream of the anticipated end of the entrance region). Plot the static pressure along
the pipe for each of the Reynolds number cases and calculate the increased pressure drop
in the entrance region of the pipe. Determine the entrance length, le , from your
calculations and compare them with those given in Equation 8.1 by plotting le /D as a
function of Re. Provide several conclusion statements from this investigation.
Problem Setup
If the pipe diameter is held constant, the case with the highest Re value will produce the
largest entrance length. Therefore, for Re = 1000 and D = 0.2 m, the predicted entrance
length is 12 m. For this problem, the resulting pipe geometry to be used is D = 0.2 m and
L = 30 m.
The Reynolds number was varied by changing the inlet velocity of the Boundary
Conditions in the Physics section of the problem setup. The student could also change
fluid properties (density and viscosity) under the Materials section. The flow was set to
Laminar with no Heat Transfer.
A medium grid density was used for the three Reynolds number simulations for this
problem. Implementing the fine-grid option showed no appreciable difference in the
results for the Reynolds numbers used in this problem. A sample portion of the grid is
shown below.
Answer
A sample convergence history plot is shown below for Re = 1000. A convergence limit of
1x10-5 was used for all simulations.
For this particular problem, it was decided to use the centerline pressure distribution to
determine the entrance length of the pipe since part of the problem is also to determine
the entrance region pressure drop. As shown in Figure 8.6 of the textbook, the fully
developed region of the pipe produces a constant pressure gradient, whereas the pressure
values show a nonlinear variation with pipe position in the entrance region. Of course
other methods to determine the entrance length include examination of axial velocity
profiles along the pipe and the centerline velocity, both of which are briefly discussed at
the end of this solution.
The plot below shows a sample of results from one of the simulations, which gives static
pressure as a function of position along the pipe. The two curves represent the CFD
results and a linear fit of data for the downstream half of the pipe – i.e. a linear fit for data
that should be fully developed and linear. As shown, the additional pressure drop in the
entrance region as well as the entrance region length can be determined from the plot.
There is a bit of subjectivity in determining the entrance length from this plot, but if you
zoom in an adequate amount, a reasonable approximation can be made.
Entrance pressure drop
Static Pressure (Pa)
CFD Results
Linear fit
Entrance flow
Fully developed flow
Position (m)
For generating the plots, the FlowLab pressure data was exported to a file and then
opened into a spreadsheet application, where the data was plotted and a linear fit applied.
Similar plots for the three Reynolds number values are given below. The axis scale for
each plot has been adjusted to aid in determining the pressure drop and entrance length. A
dashed line is provided at the approximate location of fully developed flow.
Re = 1000:
0.035
Static Pressure (Pa)
0.03
CFD Results
Linear fit
0.025
0.02
0.015
0.01
0
2
4
6
8
10
12
14
Position (m)
Re = 500:
0.015
0.014
CFD Results
Linear Fit
Static Pressure (Pa)
0.013
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0
1
2
3
4
Position (m)
5
6
7
8
Re = 100:
0.003
CFD Results
Linear Fit
Static Pressure (Pa)
0.0028
0.0026
0.0024
0.0022
0.002
2
1
0
3
Position (m)
The results are summarized in the following table, which includes entrance pressure drop,
the calculated entrance length and the analytic entrance length.
Re
Entrance
Calculated
le (m)
Pressure Drop
le (m)
Eqn. 8.1
1000
0.0031696
11.5
12
500
0.0007643
5.4
6
100
2.8E-05
1.05
1.2
As expected, the increase in Reynolds number increases the pressure drop in the entrance
region. Also, the entrance length increases with Reynolds number, as predicted by
Equation 8.1 of the text. This of course is for laminar flow conditions. Turbulent flows
can develop quickly, so the entrance length is shorter than for laminar flow (see Eqn. 8.1
and Eqn. 8.2 of the text). The calculated and predicted values for non-dimensional
entrance length as a function of pipe Reynolds number are shown in the figure below.
The results are in fairly good agreement.
100
90
80
70
l e/D
60
50
40
30
CFD Results
20
Eqn. 8.1
10
0
0
200
400
600
800
1000
1200
Pipe Reynolds Number
Additional Material
As mentioned above, a determination of the entrance length can also be made from the
centerline velocity and/or axial velocity profiles. Though not required for the problem,
these methods were applied to the Re = 100 and Re = 1000 cases, respectively. The
results are shown below.
The first figure shows the centerline velocity (Re =100) as a function of pipe length.
Once the flow has become fully developed, the centerline velocity should reach a
constant value. The second figure shows a close-up of the region where the velocity
reaches a constant. A horizontal white line is added to the plot to help determine the
length of the entrance region. According to this plot, the entrance length is approximately
1.5 m, where Eqn. 8.1 gives a value of 1.2 m.
The second set of plots show radial profiles of axial velocity (Re = 1000) at upstream and
downstream locations of the predicted entrance length. FlowLab automatically includes
the inlet and outlet profiles; the entrance length can be determined when a radial profile
matches the outlet profile signifying fully developed flow. The last plot is a close-up of
near-wall values for the various profile locations. From this plot, the profile at x = 65*d is
in good agreement with the outlet profile. This translates to an entrance length of 13 m,
while Eqn. 8.1 predicts a value of 12 m. Refining the location of these radial profiles may
help to improve the comparison.