Patterns in Rotating Rayleigh-Bénard Convection at High Rotation Rates Presented by: P. L. Mutyaba pmutyaba@clunet.edu P. L. Mutyaba, Terri Kimmel, Janet D. Scheel California Lutheran University

Rayleigh-Bénard Convection (RBC) Rotation,Ω Ra Side View Ra –Temperature difference http://www.chemistrydaily.com/chemistry/upload/1/12/Convection_cells.png

Square Patterns in RBC Bulk Periphery Square Traveling wave Horizontal slice at mid plane. Overhead View

Previous Research Experiments Rotation rates 170 Cylindrical cells Aspect ratio 5 and 3 (radius to depth ratio) This was taken at an epsilon value of 0.09 Bajaj et al.(1998)

Sánchez-Álvarez et al.(2005) Previous Research Numerical Simulations Aspect Ratio 5 and 3 Ω =274 Aspect Ratio 3 Ω =180 Observations Traveling wave affects bulk Sánchez-Álvarez et al.(2005)

Current Research Goals Accurately simulate experiments Investigate interaction between the traveling wave and bulk Study effect of centrifugal forces on square pattern formation

Methods Boussinesq Equations Code written by Paul Fischer (Argonne) Experimentally realistic boundary conditions No slip for the velocity

Periodic Cell Random initial conditions Parameters KL Instability Aspect Ratio is 5, Ω = 274, ε=0.02 Oscillating Rolls KL Instability 90 ° The KL state does come in after the oscillating squares

Periodic Cell Non-random initial condition Super-imposed rolls, fade in and out Not a transient state Traveling wave is not necessary. Blank out the time and include the angles No conclusive evidence of stabilization of the bulk by tw

Results Aspect Ratio = 5, Ω=170, ε=0.09 Convert to epsilon –all slides, all rayleigh numbers Coriolis and centrifugal forces

Results Aspect Ratio = 5, Ω=170 , ε =0.09 Include Coriolis force only

Discussion The inclusion of the centrifugal and Coriolis forces provides better agreement with experiment. (Aspect Ratio = 5, Ω=170, ε=0.09) Include parameters( rotation rate and Ra) Coriolis and centrifugal forces Bajaj et al.(1998) Coriolis force

Sánchez-Álvarez et al.(2005) Discussion The inclusion of the Coriolis force only provides better agreement with other numerical simulations. (Aspect Ratio = 5,Ω=274,ε =0.004, ε=0.02 ) Include rotation rate and Ra and also remember to talk about the epsilon and how it gives a measure of how above threshold we are runnning the simulations at . Sánchez-Álvarez et al.(2005) Coriolis and centrifugal forces Coriolis force

Conclusion The oscillating rolls may be Küppers-Lortz Instability with a switching angle of 90 °. The centrifugal force should be included in order to numerically model the RBC experiments.

Future Work The effects of the fictitious forces on the growth rates of the modes are necessary to understand pattern formation. The cause of the square patterns The oscillation of the square bulk

Acknowledgements Dr. Janet Scheel Terri Kimmel Sam Walton Katelyn White Dr. Michael Cross The Swenson Family