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Natural convection and thermal drift
- Arman Abtahi, J. M. Floryan
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- Journal:
- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
- Published online by Cambridge University Press:
- 08 August 2017, pp. 553-582
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- Article
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An analysis of natural convection in a horizontal, geometrically non-uniform slot exposed to spatially non-uniform heating has been carried out. The upper plate is smooth and isothermal, and the lower plate has sinusoidal corrugations with a sinusoidal temperature distribution. The distributions of the non-uniformities are characterized in terms of the wavenumber
$\unicode[STIX]{x1D6FC}$ and their relative position is expressed in terms of the phase difference
$\unicode[STIX]{x1D6FA}_{TL}$. The analysis is limited to heating conditions which do not give rise to secondary motions in the absence of the non-uniformities. The heating creates horizontal temperature gradients which lead to the formation of vertical and horizontal pressure gradients which drive the motion regardless of the intensity of the heating. When the hot spots (points of maximum temperature) overlap either with the corrugation tips or with the corrugation bottoms, convection assumes the form of pairs of counter-rotating rolls whose size is dictated by the heating/corrugation wavelengths. The formation of a net horizontal flow, referred to as thermal drift, is observed for all other relative positions of the hot spots and corrugation tips. Both periodic heating as well as periodic corrugations are required for the formation of this drift, which can be directed in the positive as well as in the negative horizontal directions depending on the phase difference between the heating and corrugation patterns. The most intense convection and the largest drift occur for wavelengths comparable to the slot height, and their intensities increase proportionally to the heating intensity as well as proportionally to the corrugation amplitude, with the drift being a very strong function of the phase difference. Convection creates forces at the plates which would cause horizontal displacement of the corrugated plate and deform the corrugations if such effects were allowed. Tangential forces generated by the uniform heating always contribute to the corrugation buildup while similar forces generated by the periodic heating contribute to the buildup only when the hot spots overlap with the upper part of the corrugation. The processes described above are qualitatively similar for all Prandtl numbers
$Pr$, with the intensity of convection and the magnitude of the drift increasing with a reduction in
$Pr$.
Natural convection in a corrugated slot
- Arman Abtahi, J. M. Floryan
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- Journal:
- Journal of Fluid Mechanics / Volume 815 / 25 March 2017
- Published online by Cambridge University Press:
- 23 February 2017, pp. 537-569
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Analysis of natural convection in a horizontal slot formed by two corrugated isothermal plates has been carried out. The analysis is limited to subcritical Rayleigh numbers
$Ra$ where no secondary motion takes place in the absence of corrugations. The corrugations have a sinusoidal form characterized by the wavenumber, the upper and lower amplitudes and the phase difference. The most intense convection occurs for corrugation wavelengths comparable to the slot height; it increases proportionally to
$Ra$ and proportionally to the corrugation height. Placement of corrugations on both plates may either significantly increase or decrease the convection depending on the phase difference between the upper and lower corrugations, with the strongest convection found for corrugations being in phase, i.e. a ‘wavy’ slot, and the weakest for corrugations being out of phase, i.e. a ‘converging–diverging’ slot. It is shown that the shear forces would always contribute to the corrugation build-up if erosion was allowed, while the role of pressure forces depends on the location of the corrugations as well as on the corrugation height and wavenumber, and the Rayleigh number. Placing corrugations on both plates results in the formation of a moment which attempts to change the relative position of the plates. There are two limiting positions, i.e. the ‘wavy’ slot and the ‘converging–diverging’ slot, with the latter being unstable. The system would end up in the ‘wavy’ slot configuration if relative movement of the two plates was allowed. The presence of corrugations affects the conductive heat flow and creates a convective heat flow. The conductive heat flow increases with the corrugation height as well as with the corrugation wavenumber; it is largest for short-wavelength corrugations. The convective heat flow is relevant only for wavenumbers of
$O(1)$, it increases proportionally to
$Ra^{3}$ and proportionally to the second power of the corrugation height. Convection is qualitatively similar for all Prandtl numbers
$Pr$, with its intensity increasing for smaller
$Pr$ and with the heat transfer augmentation increasing for larger
$Pr$.
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