FLOWLAB SOLUTION
9.115 (Note: This problem requires running the unsteady portion of FlowLab. It
therefore requires more time and computer resources than other FlowLab problems.) Use
the same template as in Problem 9.114 to make a comparison between the predicted and
the theoretical drag coefficient, CD, for flow past a cylinder. Set the physics of FlowLab
to allow for viscous flow and run simulations for Reynolds numbers of Re = 50 and Re =
200. Note: for Re > 40, it is advised to use the unsteady flow solver. It is also advisable
to use 50 iterations per time step and the default convergence limit for these unsteady
calculations. FlowLab provides default values for the number of time steps and the time
step size. Compare your computed drag coefficients to the values obtained from Eqn. (2)
of Example 9.9. Also comment on the trend of the drag coefficient as the Reynolds
number is increased.
Warning: the simulations outlined below take considerably more computer time than
other FlowLab problems in the text. On a typical PC, a simulation for one Reynolds
number case may take up to ½ hour or more, depending on the specific computer used.
Caution is also needed in the output of results. Since these problems are unsteady in
nature, FlowLab will output data at a specified frequency. See below for more details.
Problem Setup
The default cylinder radius was used for this problem:
For the Reynolds number values of this problem, the Laminar condition was selected for
the viscous simulation. For Re > 40, it is advised to use the unsteady flow solver, as
shown below.
For this simulation, the Boundary Condition and Materials were altered to give Re = 50
or 200. For example:
For both simulations, the fine grid resolution was selected for the grid around the
cylinder, which is shown in the following figure. For the lower Reynolds number case, it
is probably acceptable to use the medium resolution grid.
A close-up of the grid region surrounding the cylinder is shown in the figure below.
Since these simulations are for unsteady flow, the Solve portion of FlowLab has
additional parameters such as Timesteps, Timestep Size, Iterations/Timestep, and
Autosave Frequency (as shown below).
As stated in the FlowLab template for flow past a cylinder, it is recommended to use
between 20 and 50 time iterations for each cycle of the vortex shedding process. This is
based on the non-dimensional Strouhal number, which increases with Reynolds number.
Given the Strouhal number, you can find the frequency of the shedding, then the resultant
time for one cycle. FlowLab automatically calculates an appropriate time step value
given the Reynolds number. For these simulations, the default time steps were t = 20 s
and t = 5 s for Re = 50 and 200, respectively.
The default number of time steps is set to 100. As shown below, the convergence of the
drag coefficient was used to determine when the solution had converged. Therefore, the
number of time steps had to be increased several times until convergence was reached.
Once the simulation has reached the maximum number of time steps, you can continue
the simulation by clicking the Restart button in the Solve window.
For all simulations, the iterations per time step was left as the default value of 50, though
the solution typically converged prior to this iteration level at each physical time step.
The Autosave Frequency represents the number of time steps when FlowLab should
output results. This was usually set to either 50 or 100. If the value is set lower, be
careful of available disk space.
Answer
For both simulations, the default convergence limit of 1x10-4 was used. To achieve a
converged solution, the drag coefficient needed to asymptote to a constant value. A plot
of the convergence of the drag coefficient for Re = 50 is shown below. Note that the
simulation was run out to 8000 s (with t = 20 s).
Re = 50
The final drag coefficient for Re = 50 was obtained from the Reports window shown in
the following figure.
The same two figures are shown below for the Re = 200 case.
Re = 200
Note the oscillations in the drag coefficient plot. At this Reynolds number, there is
periodic vortex shedding from the cylinder as depicted in Fig. 9.21b of the text, which
produces the oscillations in the drag coefficient. See the additional material below for a
visualization of the vortex shedding phenomenon.
The students are required to compare the drag coefficient values with the analytic values
from Eq. (2) of Example 9.9 of the text. The results of this comparison are shown in the
following table.
Reynolds No. CD – FlowLab CD – Eqn (2)
Ex. 9.9
50
1.50
2.01
200
1.27
1.59
FlowLab underpredicts the drag coefficients as compared to the analytic solutions, but
both show the correct trend as documented in Fig. 9.21a of the text. It is noted in
Example 9.8 of the text that this analytic expression is only valid for sufficiently large
Reynolds numbers where there is actual boundary layer structure to the flow – Re > 100.
The student should also comment on the trend of the drag coefficient as Reynolds number
is increased. As discussed in Sec. 9.3 of the text, the drag coefficient is proportional to
1/Re at low Reynolds number values and relatively constant at moderate Reynolds
numbers. The Re values in this problem fall between these two regimes. The decrease in
the drag coefficient is also shown in Fig. 9.21a.
Additional Material
The following figures are additional material to help visualize the flow past the cylinder.
Note that some of this information is required for Problem 9.116. The first plot is the
velocity contours at Re = 50 (taken at t = 4000 s). At this Reynolds number, the wake
was starting to show some preliminary signs of oscillation.
The next two plots are the velocity contours (at t = 3250 s) and streamlines for Re = 200.
Depending on the setting for the Autosave Frequency, there will be multiple output files
contained in the working directory – output at time step intervals. To look at the different
results, select your postprocessing choice from the Post window (in this case, contours of
velocity). Click on the Activate button to show the results. Then click on the Modify
button to bring up the Modify Simulation Object window shown below.
To look at results at different times, click on the Edit button associated with the Contour:
velocity magnitude. This will open the Specify Contour Attributes window as shown in
the following figure.
From this window, click on the menu button next to the Time Step. This will give you a
listing of output data files at the various time steps. In the snapshot above, an output file
at 650 time steps was selected which corresponds to 3250 s. The results are shown
below, which is the velocity contours at this physical time. You can clearly see the
vortex shedding pattern.
The last figure is a plot of the streamlines for Re = 200.