# Flow periodicity and convection modes in rotating Rayleigh–Bénard convection at low Rayleigh numbers

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## Abstract

A three-dimensional numerical study on rotating Rayleigh–Bénard convection of water in a cylindrical container with a specific aspect ratio is performed in the present work. The simulations are carried out at four different Rayleigh numbers (\(3\times 10^4, 5\times 10^4,7\times 10^4\) and \(10^5\)) and a fixed Prandtl number (\(Pr=7\)) for a range of rotation rates. Flow structures and their evolution with the addition of rotation to the system are studied in detail. Emphasis is given on the analysis of wall mode and bulk mode convection that appear at different rotation rates. The changes in heat transfer and stability of the system are also investigated. Heat transfer rate is measured by calculating the average Nusselt number at the hot wall. The results show that rotation primarily has an inhibiting effect on heat transfer. For \(Ra\le 7\times 10^4\) the decrease in heat transfer is negligible at lower rotation rates, while it declines steeply for higher rotation rates. At \(Ra=10^5\) a small increase in Nusselt number is obtained at low rotation rates before it drops at higher rotation rates. Numerous probes placed at different points within the flow domain are used to investigate the flow regimes and convection modes. The flow initially remains steady at low rotation rates and transforms to a periodic stage with bulk-mode-dominated convection at moderate rotation rates. Further increase in rotation gives a wall mode convection accompanied by a drastic drop in heat transfer rate before finally approaching a static conductive stage. The dual role of rotation on the stability of Rayleigh–Bénard convection is clearly identified in the present study. At moderate rotation rates, the rotation force destabilizes the system to reach a periodic flow whereas extremely large rotation rate stabilizes it.

## Keywords

Rayleigh–Bénard convection wall modes bulk modes convection rolls periodic flows rotation## Nomenclature

- \(\alpha \)
thermal diffusivity

- \(\beta \)
thermal expansion coefficient

- \(\nu \)
kinematic viscosity

- \(\Gamma \)
aspect ratio

- \(\Omega \)
rotation rate

- \(\phi \)
azimuthal angle

- \(\theta \)
non-dimensional temperature

- \(\theta _{con}\)
conduction temperature profile

- \(\theta ^{'}\)
deviation from conduction profile

- \(\Delta T\)
temperature difference

*D*cylinder diameter

*f*frequency

*g*acceleration due to gravity

*H*cylinder height

*Nu*Nusselt number

- \(\langle Nu \rangle \)
average Nusselt number

*P*pressure

*Pr*Prandtl number

*R*cylinder radius

*Ra*Rayleigh number

*Ro*Rossby number

- \(S_k\)
skewness

*Ta*Taylor number

- \(T_{\circ }\)
mean temperature

- \(T_c\)
temperature of cold plate

- \(T_h\)
temperature of hot plate

- \(T_{p}\)
flow time period

- \({\varvec{U}}_{{\varvec{R}}}\)
velocity vector

*V*free-fall velocity

*w*vertical velocity component

- \(w_{rms}\)
vertical velocity (r.m.s.)

## Notes

### Acknowledgements

All simulations have been carried out in the newly set up ‘Param-Ishan’ computing facility at the institute.

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