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Effect of Rayleigh Number with Rotation on Natural Convection in Differentially Heated Rotating Enclosure

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Effect of Rayleigh Number with Rotation on Natural Convection in Differentially Heated Rotating Enclosure Journal of Applied Fluid Mechanics, Vol. 10, No. 4, pp. 1125-1138, 2017. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.73.241.26514 Effect of Rayleigh Number with Rotation on Natural Convection in Differentially Heated Rotating Enclosure M. Narendra Kumar1, G. Pundarika2, K. R. Narasimha3 and K. N. Seetharamu4 1 Centre for Incubation, Innovation, Research and Consultancy (CIIRC), Jyothy Institute of Technology, Bengaluru, Karnataka, 560082, India 2 Government Engineering College, Ramanagara, Karnataka, 571511, India 3 KS Institute of Technology, Bengaluru, Karnataka, 560062, India 4 PES University, Bengaluru, Karnataka, 560085, India †Corresponding Author Email:[email protected] (Received May 7, 2016; accepted March 15, 2017) ABSTRACT A Numerical study is carried out to investigate the effect of Rayleigh number with rotation on the flow and heat transfer characteristics in a differentially heated enclosure rotating about the horizontal axis. A Fortran Code developed based on FVM is used to discretize governing equations. Upwind difference scheme for convective terms and fully implicit scheme for transient terms are used. The SIMPLE algorithm is employed to couple pressure and velocities on staggered grid arrangement. The results were obtained for a Taylor number(10 ≤ ≤ 10 ), rotation (10 ≤ Ω ≤ 25 ), and Rotational Rayleigh number (10 ≤ ≤10 ) for two different Rayleigh number (1.3 × 10 & 1.1 × 10) with fixed Prandtl number ( = 0.71). The results showed that the Coriolis force first tends to decrease heat transfer to a minimum and then starts to increase it with increase Rayleigh number and rotation. Minimum depends on Rayleigh number and corresponds to the balanced effects of interacting forces at the point of transition. At rotations, below minimum in average heat transfer, the circulations are counter clockwise. The direction of coriolis force is from core region, so both flow and heat transfer is reduced. When coriolis force is much larger than thermal buoyancy, motion is clockwise, and transition is prevented. coriolis force now tends to promote flow circulation and therefore increases the heat transfer. The frequency content of flow pattern reveals the structural changes in the flow and temperature fields with increasing Rayleigh number and rotation. The existence of different flow regimes dominated by these body forces complicates the time average heat transfer characteristics with a different behaviour in each of the regimes. Key words: Rayleigh number; Taylor number; Rotating enclosure; Natural convection; Coriolis force. NOMENCLATURE Dimensionless Centrifugal Force temperature of hot wall Dimensionless Coriolis Force temperature of cold wall gravity vector temperature thermal conductivity dimensionlessThermal Buoyancy length of enclosure Taylor Number local Nusselt number , dimensionless velocity component space average Nusselt number , dimensional velocity components time average Nusselt number dimensionless resultant velocity revolutions V velocity vector dimensionaless pressure , dimensionaless cartesion coordinates Prandtl Number , dimensional cartesion coordinates motion pressure Rotational Rayleigh Number thermal diffusity Rayleigh Number voefficient of thermal expansion time dimensionless temperature intial temperature dynamic viscosity M. Narendra Kumar et al. / JAFM, Vol. 10, No. 4, pp. 1125-1138, 2017. density Ω rotation dimensionless time angular velocity angular position of enclosure dimensionless frequency 1. INTRODUCTION circular pipe rotating around an axis perpendicular to its own axis. Their analysis revealed that the flow Natural convection in rotating enclosure is very resistance coefficient and Nusselt number (Nu) different from stationary enclosure. When the increased remarkably due to a secondary flow driven enclosure is rotated, the flow in it is simultaneously by the Coriolis force for laminar case; whereas the affected by the coriolis and centrifugal forces as well influence of the secondary flow and increase in Nu is as the thermal buoyancy. Natural convection flows less in case of turbulent flow as compared to laminar in rotating enclosure are important in many flow. Thermal convection in a vertical rotating engineering applications such as nuclear power cylinder heated from above has been studied by plants, cooling of electronic systems, atmospheric re- Homsy and Hudson (1971) and Abell and Hudsen entry of space vehicles, gas turbines, spin-stabilized (1975). Difference between rotating and non rotating missiles, and cooling of conventional rotating convection were presented in terms of the centrifugal machinery such as electrical motors, turbines, guided acceleration, which is a strong function of the radial missiles, manufacturing of single crystal wafer and position, and the Coriolis acceleration, which space-based manufacturing processes etc. Natural contributes significantly to the heat transfer as a convection in a rotating system also finds consequence of the induced secondary flow. applications in petroleum engineering for observing Pfotenhauer et al. (1987) reported experimental the movement of oil and gas through a reservoir. results for the effects of the cylinder geometry and When natural convection takes place in a rotating rotation on the onset of convection for low enclosure, transport processes, involving the temperature liquid helium. Both the Rayleigh coupling of fluid flow and heat transfer, become number associated with the convective onset and the more challenging in view of the complexities arising convection heat transfer were found to depend on the out of rotation and increased number of parameters rotation rate and aspect ratio of the cell. Numerical emanating from pseudo forces. These forces arising investigation of Xin et al. (1997) have provided due to rotation makes the flow and heat transfer detailed information regarding flow structure and characteristics more complex and still need to be heat transfer for a range of Rayleigh and Prandtl explored. numbers for the case of temperature extremes along the horizontal diameter. Their investigation revealed Most of the early studies initially focused only on the that the flow is essentially of the boundary layer type heat transfer in rotating cylindrical enclosure. Both with motion confined to a thin layer close to the theoretical and experimental studies, exists in the cylinder wall and the core being nearly isothermal open literature related to flow and heat transfer in and stagnant. They also carried out a linear stability circular cylinder. Ostrach (1950) considered the analysis to determine the critical Rayleigh number steady laminar flow generated in a horizontal for the loss of stability of steady convection. Ker et cylinder by an imposed cosine temperature al. (1998) experimentally studied the flow distribution on the cylinder wall. The temperature stabilization by rotation in convection of air in a extremes were along the horizontal diameter. vertical circular cylinder heated from below. Their Weinbaum (1964) investigated the convection investigation relived that for a given ∆ two phenomenon in a horizontal cylinder for different different ranges of rotation exist for the flow to be locations of temperature extremes on wall of the stabilised. Outside these ranges the flow oscillates cylinder. He also carried out a linear stability periodically or quasi-periodically in time. At given analysis for the bottom heating case. Brooks and ∆ and rotation, the temperature oscillation at Ostrach (1970) experimentally investigated natural various locations exhibits some change in amplitude convection in a horizontal cylinder by varying the with negligible change in frequency. However, location of the temperature extremes on the change of the oscillation frequency with rotation is periphery of the cylinder. Numerical study by non-monotonic and is rather significant at high Veronis (1968) revealed the significant effects of rotation rates. But this frequency changes when the Prandtl number on the flow and thermal structures. imposed ∆ is small. Yin et al. (2001) carried out For the limit of an infinite Prandtl number Kupper experimental studies to investigate the fluctuating and Lortz (1969) showed that no stable steady state characteristics and stabilization of the thermal convective flow exists beyond a certain critical value buoyancy driven water flow in a vertical axis rotating of Taylor number. Rossby (1969) experimentally cylinder heated from below. They found that the flow observed the subcritical instability in a water layer suppression by cylinder rotation causes the time for > 10 and in an air layer for < 10. In average temperature distribution along the cylinder addition, for water at > 10, the Nusselt number axis for high rotation to become linear. Besides, a was found to increase with the Taylor number. The finite range of the rotation exits for the thermal opposite trend is observed for air. Besides, at a large buoyancy driven flow to be stabilized. Significant Taylor number oscillatory convection is preferred in dependence of the oscillations amplitude was mercury. Mori et al. (1971) studied the laminar as revealed. Moreover, a non-monotonic variation of well as turbulent heat convection with fully the oscillation frequency with rotation rate was developed velocity and temperature fields in a observed. Hasan and Sanghi. (2004, 2007) 1126 M. Narendra Kumar et al. / JAFM, Vol. 10, No. 4, pp. 1125-1138,
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