Proceedings of PACAM XII
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
EXPERIMENTAL RESULTS AND SIMULATION OF FLOW
INSTABILITIES IN A ROTATING LID-DRIVEN CYLINDER
Zhao C. Kong, chad.kong@rice.edu
Duncan Eddy, duncan.eddy@rice.edu
Nathan K. Martin, nkm1@rice.edu
Brent C. Houchens, houchens@rice.edu
Rice University, Dept. of Mechanical Engineering and Materials Science - MS 321, 6100 Main Street, Houston, TX 77005, USA
Abstract. The axisymmetric base flow and instabilities in a rotating lid-driven cylinder are investigated using experimental and computational methods. The flow is governed by two parameters: the Reynolds number Re (based
on the angular velocity of the top lid, the cylinder radius, and kinematic viscosity) and the aspect ratio γ (cylinder
height/radius). Instabilities are explored using ultrasonic Doppler velocimetry in a mixture of glycerin and water. A
detailed velocity profile in the cylinder is constructed using data collected by the velocimeter for γ = 2.5 and 1000
≤ Re ≤ 3000 in increments of 500. The results are compared to computational solutions from a spectral element
model of the flow, as well as previously published numerical linear stability analysis and laser Doppler velocimetry
findings. The base flow velocity profiles measured by ultrasonic Doppler velocimetry are in good agreement with
the numerical results below the critical Reynolds number. However, the first instability was sometimes observed
below the expected critical Re, and always dissipated upon reaching the predicted critical value. The most probable
explanation is transition to localized turbulence which cannot be resolved by the velocimeter.
Keywords: ultrasonic Doppler velocimetry, instabilities
1. INTRODUCTION
Flow in a rotating lid-driven cylinder is investigated experimentally through the use of ultrasonic Doppler velocimetry
(UDV). The configuration was selected to determine the effectiveness of UDV as the flow and associated instabilities are
well-known. Experimental measurements are compared with existing work and new computations.
The long-term motivation for this research stems from difficulties in growing large, single crystals of alloyed ternary
semiconductors with uniform composition. In many crystal growth techniques a feed rod is drawn through a heater where
it is melted then re-solidified into a single crystal. Controlling the flow dynamics of the melt region is a desirable goal
as this influences the final microstructure and composition distribution of the crystal. Such control would be facilitated
with real-time knowledge of the flow field. However, most common visualization techniques require a transparent liquid
medium, and thus are unable to investigate flows with opaque fluids such as molten semiconductors.
The preferred flow within the melt is dependent on the type of crystal being grown. Ternary semiconductor alloy
growth, for example, could benefit from controlled, periodic, three-dimensional flow that maintains a well-mixed melt
and ensures uniform chemical composition throughout the crystal. Such flow could be controlled through a combination
of crystal or feed-rod rotation and magnetic stirring. However, to optimize flow-based control in crystal growth the flow
dynamics within the melt region must be understood. It is difficult to characterize the flow in real-time due to the small
size of the melt region, the optical opacity of the material, and proximity of the heater, typically operating at hundreds of
degrees Celsius. Given these complications, experimentalists can benefit from analysis of idealized models of the flow
field, such as the lid-driven cylinder presented in Figure 1.
In this initial work, transparent water-glycerin mixtures are used as these are
well-benchmarked and allow for direct comparison of UDV results to laser Doppler
velocimetry studies and numerical simulations. This serves as a sensitivity study
on UDV based flow measurement. As a result, tolerances and limitations are better
understood for future measurements to be performed in opaque media.
The first investigations of flows in rotating lid-driven cylinders were carried out
by Vogel [1968] and expanded upon by Ronnenberg [1977]. The observed vortex breakdown bubbles were later confirmed by Escudier [1984], who also identified Reynolds number and aspect ratio combinations where the flow transitions
from steady to unsteady. Numerical linear stability analysis identified the critical
Reynolds numbers and frequencies of the instabilities, including those of relevance
for this investigation [Gelfgat et al., 2001]. Sørensen et al. [2006] performed an experimental investigation of periodic instabilities using laser Doppler anemometry
Figure 1. Idealized rotating lidand confirmed existence of the critical modes predicted by linear stability theory
driven cylinder enclosure
[Sørensen et al., 2009].
Proceedings of PACAM XII
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
This paper is organized as follows. The problem formulation is described in Sect. 2 and the experimental set-up and
UDV flow field measurements are discussed in Sect. 3. Sect. 4 presents the numerical modeling and visualization. Results
of experiments and simulations are compared in Sect. 5, and these are also compared to previous investigations of flow in
rotating lid-driven cylinders.
2. PROBLEM FORMULATION
There are no-slip boundary conditions on all walls of the cylinder and the lid rotates with a constant angular velocity
Ω. The non-dimensional parameters are the Reynolds number Re = ΩR2 /ν and aspect ratio γ = H/R, where ν is the
kinematic viscosity of the fluid, R is the radius of the cylinder, and H is the height of the cylinder. The rotating lid acts
like a viscous pump. The rotation accelerates the nearby fluid outward, away from the axis. As this fluid reaches the
cylinder wall it is forced downward, while retaining an azimuthal circulation. Incompressible continuity requires fluid to
be pulled upwards along the cylinder axis to replace that accelerated outward by the lid, forming the viscous pump.
3. EXPERIMENTAL METHODS
The experimental setup is composed of three components: the cylindrical fluid container, the motor and lid assembly,
and the UDV. The cylinder is 51.2 ± 0.08 mm in inner-radius and can be adjusted to a maximum height of 177.8 mm.
The cylninder is permenantly affixed to a larger flat bottom disk. Six metal struts raise the cylinder, allowing UDV probes
to be mounted normal to the bottom. The lid is a removable acrylic plate to which the motor is bolted. There is a hole
in the center of the plate through which a smaller acryllic disk (101.35 mm in diameter) is connected by a metal rod to
the motor. By varying the length of the metal rod, the aspect ratio γ can be varied from 0 to 3.5. For all results presented
here the aspect ratio was held constant at 2.5. A 20% by volume glycerin-water mixture was used as the working fluid.
Copolyamide seed particles with diameter distribution of 60%–50 µm and 40%–80 µm were mixed into the working fluid
at a concentration of 0.1 g/L. These particles produce the Doppler echo for the UDV measurements. The viscosity of the
glycerin-water mixture was calculated to be 1.72 cS at 24◦ C based on the fit by Cheng [2008].
3.1 Experimental Setup
The time-dependent evolution of the velocity profile was measured with ultrasonic Doppler velocimetry. UDV is
non-invasive as the acoustic waves and seed particles do not impose any significant influence on the velocity field. The
experimental setup is shown in Figure 2a,b with components labeled by numbers. The velocity profile in a cone of fluid
(1) is collected by a 4 MHz, 8 mm ultrasonic transducer (2) mounted normal to the bottom of the cylinder. The signal
is processed and conditioned by a Signal-Processing SA UDOP 3010 UDV (3). The rotating lid of the cylinder (4) is
actuated by a 24 volt, 1 amp brushed DC Maxon A-max32 motor (236655), a Maxon GP32A gearbox (166161) with a
2-stage, 23:1 reduction and a Maxon ENC HEDS 5540 encoder (110513) with 500 counts per turn (5). The motor is
controlled by an NI cRIO-9014 CompactRIO Real-Time Controller with an NI 9505 module (6). A PC (7) operates the
motor controller via NI LabView. The PC also controls and collects data from the UDOP 3010 using UDOP software.
Figure 2. left: Photo of experimental setup. right: Cross-section of experimental cylinder in the r-z plane.
The ultrasonic transducer acts as both emitter and receiver for the ultrasonic waves. The transducer measures only
the component of velocity along the axis of the probe. As only the flow field in a relatively small cone is measured, the
Proceedings of PACAM XII
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
position of the probe is varied along the bottom (in successive experiments) to map the entire flow profile. The quality of
the data from the UDV is governed by parameters that control the resolution, range, noise and accuracy, as well as other
conditions. The settings used are listed in Table 1. The only value that was varied is the pulsed resonance frequency, PRF.
The PRF determines the maximum recordable velocity before aliasing occurs. As the UDOP 3010 can only resolve 256
distinct velocities the PRF was optimized for each Reynolds number to achieve the best resolution. At each Reynolds
number velocity profiles were collected every 2.5 seconds for an average total time of 10 minutes.
Table 1. Settings and parameters for UDV operation
US Frequency (kHz): 4000
Number of Gates: 500
First Gate Depth (mm): 5
PRF (µs): Variable
Resolution (mm): 0.264
Velocity Scale Factor: 1.0
Doppler Angle: 0 deg
Burst Length: 4
Emitting Power: High
Emissions/Profile: 150
Sound Speed (m/s): 1582
Sensitivity: Medium
TGC (dB): 40
Sampling Volume (mm): 0.949
Duration (s): 2.5
3.2 Uncertainty and Experimental Error
The aspect ratio is determined by the height at which the upper rotating lid is set. With the cylinder radius of 51.2
± 0.08 mm and the lid placed at a height of 128 ± 0.8 mm, the overall error in the aspect ratio was less than 0.8%.
The top disk is connected directly to a brushed DC motor controlled by a simple PID control scheme. For Re = 3000
the corresponding motor rotation rate is 440 rpm. Since the viscosity of glycerin-water mixtures is sensitive to slight
temperature variations, experiments were carried out in a room held at 23.1 ◦ C with variations measured to be ± 0.1 ◦ C,
leading to a maximum error of 0.27% in Re.
3.3 Data Processing
The UDV measured profiles were plotted in a color map in time. A Fast Fourier Transform was performed to search
for dominant frequency modes. In cases with no periodic phenomena the velocity profiles were averaged to reduce signal
noise inherent in the UDV. Both time-dependent and steady, time-averaged profiles are also compared to corresponding
numerical solutions.
z
4. COMPUTATIONAL METHODS
Numerical simulations for the flow field were calculated using SEMTEX, an open-source CFD spectral element code [Blackburn and Sherwin, 2004]. SEMTEX uses spectral elements in the r − z plane and a
Fourier expansion in the azimuthal direction to calculate the flow field.
Each simulation requires specification of only a 2-D mesh in the r − z
plane and the number of terms for the expansion.
Re was adjusted numerically by modifying the kinematic viscosity ν
of the fluid. Simulations were then re-dimensionalized for comparison
to experimental results. The flow velocity is given by v = (vr , vθ , vz ),
where vr , vθ and vz are the radial, azimuthal, and axial velocities, respectively. The incompressible Navier-Stokes and continuity equations
were solved throughout the domain by time-stepping to either a converged steady or periodic solution at a desired Re.
4.1 Visualization
Numerical simulations were visualized in Paraview [Yoo et al., 2002].
To compare numerical simulations to experimental results, the component of the velocity along the lines measured by the UDV probes was
extracted. Furthermore, it is known that the beam of the probe diverges
as it propagates through the fluid medium. Therefore velocity profiles
were extracted along lines that represent the maximum possible beam divergence as shown in Figure 3. The green and red lines will be referred to
as ’outer’ and ’inner’ respectively, indicating their proximity to the axis.
The time evolution of the numerical velocity profiles were post-processed
in the same manner as the experimental measurements.
r
r 0.5
r 0.75
r 0
r 0.25
Figure 3. Beam divergence superimposed on
streamlines. Green ’outer’ and red ’inner’
lines indicate maximum beam divergence.
Proceedings of PACAM XII
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
5. RESULTS AND DISCUSSION
Velocity profiles were collected experimentally and numerically from Re = 1000 to 3000 in increments of 500. The
experimental measurements were collected from the bottom face at a range of radial offsets from the axis. Roughly 200
velocity profiles per data set were collected with an acquisition time of about 2.5 seconds per profile. Numerical solutions
were sampled along along lines corresponding to three beam paths shown in Figure 3: the middle of the UDV beam
(blue lines), the maximum beam divergence toward the center of the domain (red ’inner’ lines) and the maximum beam
divergence line toward the outer wall (green ’outer’ lines). These were compared to experimentally measured velocities.
Although multiple data sets were collected, only the results from the 0.25 inch (6.35 mm) and 0.75 inch (19.05 mm) offsets
are presented here. The onset of periodic instabilities is expected at a critical Reynolds number Recr = 2600 [Gelfgat et al.,
2001], [Sørensen et al., 2006, 2009]. Thus, results below and above Recr are addressed separately.
5.1 Results for 1000 ≤ Re ≤ 2500
Figures 4 and 5 contain plots of the time-averaged axial velocity profiles for Re from 1000 to 2500 at radial offsets of
0.25 and 0.75 inches, respectively. The solid black line represents vz measured experimentally and time-averaged. The
dashed lines represent velocity components calculated numerically. The dashed blue line was taken along the same line
as the UDV transducer axis. The dashed red and green lines align with the inner and outer paths of the ultrasonic beam
divergence, respectively. The time-averaged velocity components from UDV measurements and SEMTEX simulations
are in good agreement. The best agreement is with the velocity extracted from simulations along the transducer axis, as
expected. These results indicate that the UDV can accurately measure the steady flow below the critical Reynolds number.
Figure 4. Time-averaged velocity along a line offset by r = 0.25 inches from the centerline
It should be noted that, when making the measurements using newly mixed glycerin and water, periodicity was observed below the expected critical Reynolds number. This transient phenomena was noted on three separate occasions in
experiments conducted in the first 24 hours after the initial mixing. Mixing methods from gentle stirring for 15 minutes
with a glass rod to vigorous mixing with low heating for 30 minutes with a magnetic stirrer were investigated, as were
increases in Re via smooth ramping and discrete jumps of 500. The observed periodic flow below Recr was repeatable
for new batches of glycerin and water, independent of both the method of mixing the solution and the smoothness with
which Re was increased. If the solution was allowed to sit for 24 hours, this periodic behavior was never again observed.
These observed periodic flows were incredibly similar to that expected at or just above Recr . It is possible that
nonlinear perturbations in the system excited instabilities below Recr , though this should have been repeatable at any
time and the resulting periodic state should have been stable. It is also possible that the chemistry of the solution changed
over a 24 hour window due to incomplete initial mixing, though further work is required to verify this explanation.
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
Proceedings of PACAM XII
Figure 5. Time-averaged velocity along a line offset by r = 0.75 inches from the centerline
5.2 Results at Re = 3000
Figure 6 contains plots of the measured axial velocity profiles at Re = 3000 versus time, mapped to a color scale
(left). It is evident from the colormaps that periodicity was not observed experimentally. Time-averaged axial velocity
profiles are also presented (right), where the black line represents UDV measurements and the dashed red line represents
the SEMTEX simulations along the same line. Again, the time-averaged velocity profiles from the experiments and
simulations are in good agreement. However, the numerical simulations actually predict periodic flow at Re = 3000. Only
time-averaging the simulation results recovers the apparent steady state observed experimentally.
Figure 6. Colormap and time-averaged velocity profiles for Re = 3000.
Proceedings of PACAM XII
12th Pan-American Congress of Applied Mechanics - PACAM XII
January 02-06, 2012, Port of Spain, Trinidad
Linear stability theory simulations and laser Doppler velocimetry experiments predict that this flow should be oscillatory, with a period of approximately 18 seconds [Gelfgat et al., 2001], [Sørensen et al., 2006, 2009]. This same periodicity
was observed in the numerical simulations at Re = 3000, but not experiments. As the UDV acquisition time was on average 2.5 seconds per profile, the periodic motion should have been observed and should have repeated every 7 profiles. A
very similar oscillatory flow was observed below Recr , but only for newly mixed solutions, as discussed above. It is hypothesized that the experimental flow at Re = 3000 transitioned to localized turbulence and the UDV therefore measured
the equivalent of Reynolds Averaged Navier-Stokes flow. The good agreement between the time-averaged numerical data
and the experimental findings supports this theory.
6. CONCLUSION
The axisymmetric base flow and periodic instabilities in a rotating lid-driven cylinder were investigated experimentally
and numerically for an aspect ratio of 2.5. Ultrasonic Dopper velocimetry was used to investigate the base flow and to
search for the onset of periodic instabilities in a 20% glycerin, 80% water solution by volume. This non-intrusive technique
is suitable for measuring velocity profiles of the axisymmetric base flow. Good agreement with numerical simulations
was observed.
Above the critical Reynolds number the expected periodicity was not observed experimentally, but time-averaged
experimental results agreed well with time-averaged simulations. This indicates that the UDV is able to resolve timeaveraged flows. Observation of periodicity below the critical Reynolds number is thought to be due to instabilities that
grow from nonlinear perturbations, though the reason that this is not repeatable after the solution sits for 24 hours is not
understood.
7. ACKNOWLEDGMENTS
This research was supported in part by the United States Air Force Office of Scientific Research and by the Brown
Undergraduate Research Internship Program.
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