Student(s):
Ho Yun Bobby Chan
Dates of Involvement:
Summer 2008- Present
Faculty Advisor(s):
Professor Keith Julien, Professor Emeritus Patrick Weidman, Professional Research Assistant Alan Scott Kittelman
Graduate Mentor:
Experimental Study of Thermal Convection Driven by Centrifugal Buoyancy in a Rapidly Rotating Annulus
INTRODUCTION
In 1976, Busse proposed a simplified model that explains the dynamics of Jovian planet’s atmosphere [3]. His model holds under the assumption and conditions:
1.) fluid is ideal
2.) fluid under test is rapidly rotating
3.) columns are formed in a band parallel to the axis of rotation
4.) the tangents is small at the end of the columns
5.) weak variation in the vertical
6.) temperature gradient between concentric cylindrical boundaries
7.) columns forms in the region away from the solid core
Compared to Earth, Jovian planets are massive and rapidly rotating. The radius of Jupiter is 11 times the radius of the Earth. Due to the massive size of Jupiter, heat is driven by internal thermal forces. One revolution of Earth is equal to 9.9 days on Jupiter. It is assumed that the in Jovian planet has a relatively small solid core [2].
Figure 1: Convective columns in slices within a sphere.
Convection occurs as a result of the size, rapid rotation, and internal thermal forces on the planet. In effect, convective columns are believed to form and be responsible for bands or zones on Jupiter. Figure 1 gives an illustration of the convective columns that form the zones on Jupiter [4]. Figure 2 shows Jupiter’s south pole with the zones and bands alternating from the pole to the equatorial zone [5].
Figure 2: Cassani flyby shot of Jupiter’s south pole with alternating zones and bands.
Just over a decade ago at the University of Colorado at Boulder, John Hart and Scott Kittelman constructed a device to model Busse’s theory. After running a few experiments, Hart was unable to observe anything that fit Busse’s model. He eventually left the experiment and moved on since he was not able to see oscillations, fast precession of regular cells, or jets. This experiment was brought back to life with the help of Scott Kittelman, Professor Patrick Weidman, and Professor Keith Julien with funding from the National Science Foundation under the Mentoring through Critical Transition Points program. After looking at the device that Hart constructed, Professor Julien believes that by changing the parameters of the apparatus and applying his numerical computations, the experiment should fit Busse’s model.
EXPERIMENTAL SETUP
The device consists of an annulus with a flat end cap at one end and frustum of a cone at the other, designed to allow cooling along the axis of rotation and heating outside the annulus to provide a radial temperature gradient. The annulus is filled with a viscous liquid, void of bubbles, and seeded with Kalliroscope [1] and drops of green food coloring for visualization. A radial temperature gradient is applied across the annulus, and the temperature difference (ΔT) is measured. There are six temperature probes protruding into the fluid at the flat end cap and three more along the side of the outer shell of the annulus. Figure 3 shows the most recent setup of the experiment.
Figure 3: Experimental Setup
CONTROL PARAMETERS AND MEASURED QUANTITIES
The observations of the cellular structure and their phase speed will be compared to Professor Julien’s numerical computations made for the exact geometry of the apparatus at measured values of the Reynolds number (proportional to angular rotation Ω) and Rayleigh number (proportional to the radial temperature difference across the annulus) which are respectively
(1)
where d is the gap width and ν is the kinematic viscosity of the fluid, and
(2)
where α is the coefficient of thermal expansion of the fluid and κ is the fluid’s thermal diffusivity.
In order to validate Julien’s computations to determine the onset of instability we need to:
1) identify the expected azimuthally periodic modal structure.
2 ) measure the number (n) of azimuthal cells.
3) determine the direction (prograde or retrograde) of motion of the disturbances.
4) measure the angular phase speed (ω) of the cells relative to the rotation (Ω) of the annulus.
By applying a temperature gradient from the core to the outside shell and rotating the annulus at high speeds, thermal convection will eventually result. In effect, there will be heat and mass transfer in the radial direction. The more dense fluid (relatively cold liquid) should flow outwards to the shell while the less dense fluid (relatively warm liquid) should flow in toward the core. The inflow/outflow boundaries mark the edges of counter-rotating cells that can be seen at the end cap of the annulus. The cells extend from one end of the annulus to the other as columns. The experiments are being conducted at the Geophysical Fluid Dynamics Laboratory (GFDL) in the Duane Physics building.
REGIME DIAGRAM
Three solutions have been made over the course of this experiment. For each solution tested, there was a temperature gradient applied from the core (cooler temperatures) to the outside shell (warmer temperatures) of the annulus as the device rotated at about 500 RPM for each trial. The slope of the conical endcap opposite to the side that was being viewed had an angle of 45°. At higher viscosities, the cells were more defined during convection as opposed to those at lower viscosities. Figure 4 shows a regime diagram of convection occurring with varying kinematic viscosity and varying radial temperature difference.
Figure 4: Regime Diagram of Convection with Varying Kinematic Viscosity
Regular cells were better defined at the higher viscosity with glycerol solution 2 than at the lower viscosities. The three regimes of different solutions are boxed in Figure 4. The higher viscosity convective cells were more stable compared to other trials that were made. At this point, it was decided to extend the duration of the experiments from 3-6 minutes to about 20 minutes. The cells became fully developed after extending the length of the trials.
For the trials that were run over the longer time interval, the cells were more stable in the band when a small temperature gradient was applied between the core and outer shell. When a large temperature gradient was applied, the cells were less stable. Observation showed that the cells were not complete throughout the band and would break apart in a turbulent-like fashion.
INFLUENCE OF GRAVITY ON THE ANNULUS
In order to create an environment similar to the Jovian planets, the experimental apparatus needed to overcome Earth’s gravity. To do that, the apparatus had to be rotated at high speed, which in effect creates its own “artificial radial gravity.” The centrifugal force generated by the rotation was expected to overcome that of gravity. To verify this, I looked at the device’s centripetal acceleration.
Knowing the angular velocity ω of the experiment, the centripetal acceleration of the fluid at radius r is
(3)
Using this equation for the measured value ω = 52.3 rad/sec at radius r = 2.87 cm, I calculated the centripetal acceleration to be about 7878 cm/s2. Looking at average gravity on Earth, which is about 981 cm/s2, the acceleration of the device is about 8 times that of Earth’s gravity. Therefore, the effect of gravity on the fluid in the annulus is small compared to that of the centripetal acceleration.
Flow Patterns of Cells Diametrically Opposite
Possible flow patterns in an annulus can consist of an even or odd number of cell pairs. An example is in Figure 5 where there are a total of eight cells in the annulus which can be divided into an even number of cell pairs. For this example we can say that the white cells flow in the clockwise direction while the darker cells flow in the counter-clockwise direction. For an even number of cell pairs, the cells are correlated because the same type of cell will occur directly opposite of a given cell. When two probes are placed in the cells diametrically opposite of each other, the temperatures should be in-phase.
Figure 5: Even Number Cell Pairs
In Figure 6, there are a total of six cells in the annulus which can be divided into an odd number of cell pairs. Therefore, there are three cells rotating clockwise and three rotating counter-clockwise. For an odd number of cell pairs, the cells are anti-correlated because a cell will be counter-rotating with respect to the cell that is opposite to a given cell. When two probes are opposite of each other as shown in Figure 6, the temperatures should be out-of-phase.
Figure 6: Odd Number Cell Pairs
For the 60 trials run from 3-5 minutes with the three solutions, I was unable to find uniform, regular cells. By looking at Figure 7, there were some regular cells at the top when a temperature gradient of 2.8°C was applied but most of them were unstable. They would eventually break up like the cells on the right of the image.
Figure 7: RRC2009Aug11v1 Frame 1526
It was not until we increased the time to about 20 minutes that we saw uniform, regular cells. Figure 8 shows the uniform, regular cells recorded during the last 7 minutes. There was a 0.5°C temperature gradient applied between the core to the outer shell using the solution that had a high viscosity of about 15 cSt.
Figure 8: RRC2009Oct5v1 Frame 2600 with 7 paired cells
The video for this run showed that the cells were propagating slowly in the retrograde direction with respect to the rotation of the annulus. There were a total of 14 cells, which implies out-of-phase temperatures on opposite sides of the outer shell for the odd number of pairs. The average speed at which a cell is moving over a point is about 0.050 rad/sec.
Figure 9 shows anti-correlation between two diametrically opposite cells as they respectively passed over the thermistor closest to the core (right inner endcap thermistor) and the left shell thermistor.
Figure 9: RRC2009Oct5 Left Shell and Right Inner Endcap Thermistors (respectively top and bottom curves)
In another trial that ran for 25 minutes with a temperature gradient of 0.2°C, we were also able to see uniform, regular cells. By looking over the video that was recorded in the last 8 minutes of the trial, the cells were also propagating in the retrograde direction with respect to the rotation of the annulus.
Figure 10: RRC2009Oct25v1 Frame 2100 with 9 paired cells
Figure 10 shows a total of 18 cells that formed. This implies that temperatures recorded on opposite sides will be out-of-phase for the odd number of pairs. The average speed at which a cell is moving over a point is about 0.037 rad/sec.
Figure 11 shows anti-correlation of the convective cells in the right inner endcap thermistors and cells being out-of-phase.
Figure 11: RRC2009Oct25 Left Shell and Right Inner Endcap Thermistors (respectively top bottom curves)
Flow Patterns along a Convective Column
Probes along the side of the annulus should measure the same changes in temperature as a convective column passes over the thermistors. Figure 12 shows two temperature probes along the side of the annulus where one probe is farther down the length of the cylinder.
Figure 12: Thermistors along the Side of the Annulus
Figure 13 shows when convective cells passed two thermistors that were embedded along the generatrix of the cylinder protruding into the fluid. The two thermistors (left and right shell thermistors) are about 4 cm apart along the 6.33 cm long cylinder. This takes up about a 27% of the length of the cylinder. When facing the endcap, the right shell thermistor is farther down the generatrix.
Figure 13: RRC2009Oct5 Left and Right Shell Thermistors (respectively bottom and top curves)
Figure 14 also shows when a convective cell passed over the thermistors. The video showed three convective columns that passed over the two thermistors that give in-phase temperatures.
Figure 14: RRC2009Oct25 Left and Right Shell Thermistors (respectively bottom and top curves)
CONCLUSION
With all the trials that have been run over the course of the project, we still have not been able to see Busse’s model in action. In particular, his model predicts prograde motion of the cells. The next phase of the experiment is to change the angle of the endcap determined from Professor Julien’s reduced set of equations for the experiment. By changing the angle, we believe we will be able to see fast prograde precessing cells as opposed to the slow retrograde precessing cells from the trials obtained on October 5th and 25th. Figure 15 shows an annulus that is unwound to form a band and time increasing down the bands [6]. In the future, we predict that there will be a greater number of cells in the band.
Figure 15: Band with a Greater Number of Fast Precessing Cells
About Ho Yun Bobby Chan:
Ho Yun Chan is an undergraduate senior in the department of electrical and computer engineering. He recently decided to take on an applied math minor in hopes of advancing his skills in order to better prepare for the future. He has an interest in the field of electromagnetic fields and waves. His future plans are to work in the industry and eventually attend graduate school.
References:
[1] Matisse, Paul. “Kalliroscope.” (19 Dec 2009) Kalliroscope [Online]. Available: http://www.kalliroscope.com.
[2] “Basics of Space Flight Section 1: The Jovian Planets.” (13 Nov 2009) JPL [Online]. Available: http://www2.jpl.nasa.gov/basics/bsf1-2.php.
[3] “Atmosphere of Jupiter” (19 Dec 2009) Wikipedia [Online]. Available: http://en.wikipedia.org/wiki/Atmosphere_of_Jupiter.
[4] “Geophysical Fluid Flow Cell: A Planet in a Test Tube.” (19 Dec 2009) SpaceRef Interactive Inc. [Online]. Available: http://spacescience.spaceref.com/newhome/development/images/gffc/GFFC.pdf.
[5] “Map of Jupiter’s South Pole” (19 Dec 2009) National Geographic [Online]. http://science.nationalgeographic.com/science/wallpaper/map-jupiter.html.
[6] Brummell, Nicolas H. and John Hart. “High Rayleigh Number Beta Convection.” Geophysical Astrophysical Fluid Dynamics Vol. 68 (1993): pg 85-114.
Financial support for this project provided by the NSF's Mentoring Through Critical Transition Points (MCTP) Grant, No DMS-0602284.
