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Hydrodynamic Instabilities in Rotating Magnetohydrodynamic Flows

Elie ABI RAAD 1

Institute of Technical Acoustics, Technical Acoustics Group, RWTH Aachen University, Kopernikusstraße 5, 52074 Aachen

elie.abiraad@akustik.rwth-aachen.de

1 This work was done at the American University of Beirut, Lebanon, during the author’s Bachelor studies.

Abstract. In this project, we modeled Saturn’s hexagonal vortex. While previous research focused on mechanically stirring a fluid to generate the vortices (shear forces), we used electromagnetic forces (body forces), which are closer to the actual physics happening at the vortex.

The experimental setup consisted of a rod in the center of a Plexiglas cylinder filled with a potassium hydroxide solution. Stainless steel coated the inner lining of the cylinder. Current flowed from the rod to the steel through the fluid, giving a varying radial current distribution, and a magnetic field was channeled through the cylinder. This induced a Lorentz force that moved the fluid. The experiments and simulations were done for varying heights of fluids, diameters of cylinders, and current strengths, and the experimental data was captured using PIV. The numerical simulations were done on OpenFOAM and recreated the experiments.

We were able to experimentally recreate Saturn's hexagonal vortex. We believe that the backflow from the circulation around the outer edge of the cylinder and the rotation of the fluid interact to create instabilities at specific points that become the hexagon’s vertices.

Keywords Electromagnetic forcing, turbulence, Particle Image Velocimetry, Computational Fluid Dynamics, OpenFOAM.

1. Introduction In the 1980s, the space probe Voyager took the first

images of Saturn. These images showed a hexagonal vortex that has maintained its magnitude till today. That vortex is shown on Figure 1. Planetary atmospheric studies show that heat is transported from the equator to the poles due to the different temperatures of the air, inducing a temperature-gradient-driven flow of atmospheric air. At the pole, this movement of air is coupled with the Coriolis force from the planet’s rotation and creates a polar vortex.

In Saturn’s case, that vortex is shaped like a hexagon. The current theory is that Saturn’s vortex shape is due to an anisotropic steady-state system which stems from Saturn’s complex weather patterns; in other words, a situation specific to its atmosphere.

Fig. 1. Saturn's North Pole vortex

Flows on Saturn are characterized by a Reynolds number- the ratio of inertial forces to viscous forces- bigger than 1012, and an Ekman number- the ratio of viscous forces to the Coriolis force- smaller than 10-14. These numbers cannot be reproduced under laboratory settings. However, the ratio of inertial forces to the Coriolis force, the Rossby number, can be simulated in the lab. Aguiar et al. show that this is sufficient when modeling the stability of the vortex on Saturn [1]. However, the experiments done until now all involved stirring the fluid using shear forcing mechanisms, which does not replicate the reality of Saturn's vortex.

Experimental and numerical approaches have already been taken to model comparable vortices. In 2009, Kenjereš et al. set up a series of experiments coupled with numerical simulations showing that electromagnetically induced turbulence can be controlled and studied at its fundamental level [2]. In addition, Moubarak and Antar propose an experimental setup to study the effect of a

2 E. ABI RAAD, HYDRODYNAMIC INSTABILITIES IN ROTATING MAGNETOHYDRODYNAMIC FLOWS

strong magnetic field on the flow of a small layer of conducting liquid subject to a potential difference [3]. In 2014, Banerjee and Pandit studied numerically the energy spectrum of a two-dimensional magnetohydrodynamic turbulence, focusing on the inverse cascade regime [4].

In our study, we tried to simulate the “non-invasive” forces found on planets, both numerically and experimentally, by using electromagnetic forcing [3]. This is done by conducting experiments to recreate the hexagonal vortex and by numerical simulations. Our hypothesis is that it is the circulation happening on the edges of the cylinder that cause these instabilities and the anisotropy of the system.

2. Methods This study is comprised of both an experimental and a

numerical part. First the experimental part was done to see whether the hexagonal vortex could be recreated. Then, the numerical simulations attempted to recreate the experimental results so as to give a better understanding of the physics of the phenomenon, since PIV measurements only give information about the surface of the fluid, and not what happens within it.

2.1 Experimental methods The experimental setup, shown in figure 2, consists of

a rod in the center of a Plexiglas cylinder. The cylinder is placed inside a cylindrical magnet, and is filled with a potassium hydroxide (KOH) solution. Stainless steel coats the inner lining of the cylinder. Electrodes attached to the cylinder are plugged into a power generator: the electrode on the edge is plugged to the negative side and the one in the middle to the positive side. Current flows radially from the rod to the steel through the fluid, leading to a varying radial current distribution. The magnetic field and the passing current induce a Lorentz force on the ion particles, and force the fluid to move. The cylinder has an inner diameter D of 19cm, the rod has a diameter d of 5mm, the solution a concentration of 25% and a height h, and the current an intensity I. The strength of the magnetic field was 80mT, and its direction is from the top of the cylinder to the bottom.

It is worth noting that the steel lining of the cylinder is much more conductive than the KOH solution, therefore the time needed for the current to travel along the steel lining is much smaller than the time needed for the current to penetrate the solution. This means that the whole steel lining can be assumed to be a lumped cathode. This has been verified experimentally by the use of a multi-meter; current density was only radially dependent. In addition, the concentration of KOH was chosen to maximize the conductivity of the solution, as seen in figure 3.

Data was acquired using PIV measurements: 325- microns floating beads were placed in the fluid under a UV light and a high-resolution camera provided imaging data

for the PIV. The camera used for the PIV was a Codec Y800 working at 15 frames per second.

The experiments were done in the following way: first, the cylinder was filled with a specific volume of solution to reach the desired height, and the beads were placed on the water. Then, the current was passed in the solution, starting from the lowest current intensity, and then raised each time to the desired intensity. Every time, the fluid was allowed time to reach a steady-state rotation. Each experiment ran for approximately four minutes. The cases are shown in the Results section.

Fig. 2. The experimental setup

Fig. 3. Various Chemical Compounds Concentration vs. Conductivity

2.2 Numerical methods The simulations were based on the code developed by

Habchi and run on OpenFOAM [8]. The code is based on modified Navier-Stokes equations that account for the Lorentz forcing. We assumed a constant current direction, going from the inner positive node to the outer negative node homogeneously such that the current is radially constant and the current density decreases radially.

The code starts by initializing the Lorentz force and the pressure from the previous time step, then proceeds to solve the following equations:

∇u = 0 (1)

∂u∂t+ (u.∇)u = −∇p+

2∇ uRe

+2Ha

Re(−∇φ +u×Ez )×Ez

(2)

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where u is the velocity of the fluid particle in a certain direction, ϕ is the electric potential, or voltage, -∇ϕ is the electric field, and is related to the current, p is the dynamic pressure, Ez is the directional magnetic force, Re is the Reynolds number, and Ha is the dimensionless Hartmann number, defined as

Ha = BLc (σµ)0.5 (3)

where µ is the dynamic viscosity, B is the magnetic field Lc is the characteristic length of the system, and σ is the electrical conductivity.

Equation (1) stems from mass conservation and gives the velocity. It indicates that the fluid is incompressible. Equation (2) is the Navier-Stokes equation modified to include the Lorentz forcing, which is the term at the extreme right of the equation; solving this equation gives the pressure.

The geometry and the values used were chosen to replicate the experimental parameters. The geometry is a cylinder of inner diameter 19cm with a concentric 5mm diameter cylinder inside it. The inner cylinder is the positive node, and the outer cylinder is the negative node, and between them is the fluid. The height of the fluid is either 3mm or 15mm. Current intensity values range between 10mA and 100mA. The magnetic field is of 80mT, the density of the solution 1000 kg/m3, and the kinematic viscosity 10-7m2/s. No-slip boundary conditions were given to the inner wall of the outer cylinder, the outer wall of the inner cylinder, and the bottom surface. The upper surface was given a slip boundary condition, since in the experiment it is in contact with the atmospheric air.

The whole cylinder was divided into four identical blocks, each with a similar mesh. The reason for that is that we expected the vortices to travel across around the center in a sinusoidal fashion and not in a symmetrical fashion. Each block had a mesh of eighty cells in the x-direction, eighty cells in the y-direction, and twenty cells in the z-direction, for a total of 512000 cells. The time step was 0.1s, with 15 iterations per time step.

3. Results

3.1 Experimental results The experiments were done for heights of 3mm, 5mm

and 15mm and for different current intensities. The results are summarized in table 1. As shown in the table, the instabilities only happened for a large height of fluid, and only for currents of 80mA. The hexagonal vortex was successfully created for h=15mm and I=100mA and I=120mA. All the other experiments showed a laminar flow on the surface.

Unfortunately, we were unable to acquire data for the hexagonal case, as the PIV beads stuck together at that

point of the experiments and were not suitable for data acquisition. Instead, figure 4 shows the radial and angular velocities at the surface of the fluid for h=15mm and I=80mA, just before the hexagonal vortex was reached. Each cross indicates a different PIV bead position.

Figure 5 shows our experimental hexagon. The figure shows six distinct points. The side on the upper right of the hexagon is distorted due to the current being turned off at the moment the picture was taken.

It is interesting to note that we were also able to create non-hexagonal vortices, such as a heptagonal and an octagonal vortex, by increasing the current.

h= 3mm h= 5mm h= 15mm

I= 10 mA Laminar flow Laminar flow Laminar flow

I= 40 mA Laminar flow Laminar flow Laminar flow

I= 60 mA Laminar flow Laminar flow NA

I= 80 mA NA NA Laminar flow

I= 100 mA Laminar flow Laminar flow Hexagonal vortex

I= 120 mA NA NA Hexagonal vortex

Tab. 1. Flow characterization of experimental results

Fig. 4. Velocities components along the radius for h=15mm, I=80mA

Fig. 5. Experimental hexagon vortex

4 E. ABI RAAD, HYDRODYNAMIC INSTABILITIES IN ROTATING MAGNETOHYDRODYNAMIC FLOWS

3.2 Numerical results Figures 6 and 7 show plots of the velocity along the

diameter of the cylinder, at the surface of the fluid, at steady-state, and for h=3mm. Figures 8 and 9 show similar plots, but for h=15mm. Ux, in red, is the radial velocity, Uy, in blue, is the angular velocity, and Uz, in green, is the axial velocity. The center of the cylinder is at 0, and the edges of the plots correspond to the edges of the cylinder.

There are no values at the center of the cylinder, because there is no fluid there, only the conducting rod. The symmetry between the positive distance and the negative distance is due to the fact that the force depends on the current concentration, which is radially constant. The forcing, and from there the velocity, are therefore equal at the same distance from the center of the cylinder.

The greater velocities are found next to the inner node, where the current density is highest, and die out as we travel away from the inner node. In all the plots, Uy is the largest. All the velocities increase as the current is increased. In addition, the shape of the plots stays relatively constant. The large drops near the inner and outer nodes are due to the no-slip boundary condition.

The simulations showed no singularities and no hexagonal vortex, even for h=15mm and I=100mA. We believe the aggregate error at this point had become too significant and was perturbing the results.

Fig. 6. Velocity components for h=3mm, I=10mA

Fig. 7. Velocity components for h=3mm, I=100mA

Fig. 8. Velocity components for h=15mm, I=40mA

Fig. 9. Velocity components for h=5mm, I=100mA

4. Analysis The experimental results show that the instabilities

happened only for large heights of fluid and large currents, the main component being the larger height. The larger fluid height means the fluid is less constrained by the top and bottom layers, so there is less viscous damping due to the friction against the wall, which leads to better circulation: figures 6 and 7, for example, show Ux and Uz reaching 0 much more rapidly than figures 8 and 9.

In the numerical simulations, the non-zero radial and angular velocities, Ux and Uy, indicate that the surface of the fluid is rotating. Beneath the surface, the non-zero axial velocity indicates circulation. All numerical cases show circulation around the inner node, where the current density is highest, but only for a larger height of the fluid h do we see circulation around the outer node as well. This indicates that a large enough fluid height is necessary to create a hexagonal vortex. Note that, since our Lorentz force is applied only along the planar direction and not along the axial direction, Uz only exist as a result of circulation.

In addition, the greater velocities are found next to the inner node, where the current density is highest, and die

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out as we travel away from the inner node. All the velocities increase as the current is increased, due to the increased Lorentz forcing. As the current is increased, Ux and Uz increase drastically near the inner node, indicative of the increased circulation. This circulation would allow instabilities at the critical parameter settings to develop into straight edge forcing vortex jets, and small instabilities at critical settings would generate the largely anisotropic flows.

Furthermore, the fact that, for the same height, different polygonal vortices were obtained indicates that, after a critical height, the shape of the vortex can be controlled with the current intensity.

In other words, as the height of the fluid is increased, the fluid is allowed to develop circulation zones at the far ends of the cylinder. The increase in current leads to an increase in the electro-magnetic forcing, which leads to higher velocities in the circulation zones. We suspect that the high-velocity circulation at the far edges of the cylinder create backflow. In the right conditions, these backflow points would interact with the rotating flow of the fluid to create instabilities, which give the hexagonal vortex its shape.

5. Conclusion Our results indicate that a hexagonal vortex is

possible starting a certain height of the fluid, at which circulation along the outer edges is possible. There is therefore a correlation between the presence of circulation and that of the hexagonal vortex. This would indicate that the circulation growth is the cause of our anisotropic flows, and that a similar phenomenon is happening in Saturn’s atmosphere due to the temperate-driven–gradient flow, which is similar to our flowing fluid, and the Coriolis rotation, similar to our circulation-induced backflow.

References [1] AGUIAR, A. C., READ, P. L., WORDSWORTH, R. D., SALTER,

T., & YAMAZAKI, Y. H. A laboratory model of Saturn’s North Polar Hexagon. Icarus, 2010, vol. 206, no. 2, p. 755-763.

[2] KENJEREŠ, S., VERDOOLD, J., TUMMERS, M., HANJALIĆ, K., KLEIJN, C. Numerical and experimental study of electromagnetically driven vortical flows. International Journal of Heat and Fluid Flow, 2009, vol. 30, no.3, p. 494-504.

[3] MOUBARAK L. M., ANTAR G. Y. Dynamics of a two-dimensional flow subject to steady electromagnetic forces. Experimental Fluids, 2012, vol. 53, no. 5, p. 1627–1636.

[4] BANERJEE, D., PANDIT, R. Statistics of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence. Physical Review E, vol. 90, no. 1

[5] TING, L., KLEIN, R. Viscous Vortical Flows. Berlin &Springer-Verlag, 1991.

[6] WU, J., MA, H., ZHOU, M. Vorticity and Vortex Dynamics. Berlin &Springer, 2016.

[7] MARSHALL, J., PLUMB, R. Atmospheric, Ocean and Climate Dynamics. Burlington:MA &Elsevier, 2008.

[8] HABCHI, C., ANTAR, G.Y. Direct numerical simulation of electromagnetically forced flows using OpenFOAM. Computers & Fluids, 2015,vol. 116, p. 1-9

Acknowledgements The research described in the paper was done during

the final-year project of Abi Raad’s Bacherlor’s in mechanical engineering. It was supervised by Dr. Marwan Darwish from the FEA at AUB, Lebanon and Dr. Ghassan Antar from the FAS, AUB, Lebanon, as well as with the help of Dr. Charbel Habchi. It was also done in collaboration with Christopher Nahed, Saleem El Jazaerly and Mohammad Bassel Kazzaz.

About Authors... Elie ABI RAAD was born in Beirut, Lebanon. After completing his Bachelor’s degree in Mechanical Engineering at the American University of Beirut (from which this project was the Final-Year Project), he completed his Master’s degree in Acoustical Engineering at ENSTA, France. He is currently pursuing his doctorate’s degree at the Institute for Technical Acoustics at RWTH, Aachen, Germany.