• Journal of computational fluids engineering
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• v.11 no.2 s.33
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• pp.49-55
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• 2006

This study investigates the oscillatory thermal convection of a fluid with Pr=0.02 in a wide-gap horizontal annulus with constant heat flux inner wall. When Pr=0.02, dual steady-state flows are not found. After the first Hopf bifurcation from a steady to a time-periodic flow, five successive period-doubling bifurcations are recorded before chaos. The power spectrum shows the $period-2^4\;and\;2^5$ flows clearly, and a window of period $3{\times}2^3$ flow is found in the chaotic regime. The approximate value of the Feigenbaum number for the last three period-doubling bifurcations is 4.76. The transition route to chaos of the present simulations is consistent with the period-doubling route of Feigenbaum.

• Nuclear Engineering and Technology
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• v.44 no.3
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• pp.283-296
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• 2012

The natural convection in a horizontal enclosure heated from the bottom wall, cooled at the top wall, and having a square adiabatic body in the center is studied. Three different Prandtl numbers (0.01, 0.7 and 7) are considered for the investigation of the effect of the Prandtl number on natural convection. Adiabatic boundary conditions are employed for the side walls. A two-dimensional solution for unsteady natural convection is obtained, using an accurate and efficient Chebyshev spectral methodology for different Rayleigh numbers varying over the range of $10_3$ to $10_6$. It had been experimentally reported that the heat transfer mode becomes oscillatory when Pr is out of a specific Pr band beyond the critical Ra. In this study, we reproduced this phenomenon numerically. It was found that when Ra=$10_6$, only the case for intermediate Pr (=0.7) reached a non-changing steady state and the low and high Pr number cases (Pr=0.01 and 7) showed a periodically oscillatory fashion hydrodynamically and thermally. The variation of time- and surface-averaged Nusselt numbers on the hot and cold walls for different Rayleigh numbers and Prandtl numbers are presented to show the overall heat transfer characteristics in the system. Further, the isotherms and streamline distributions are presented in detail to compare the physics related to their thermal behavior.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.24 no.9
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• pp.1210-1218
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• 2000

Thermal convection in a horizontal annulus is considered, and the bifurcation phenomena of flows from time-periodic to chaotic convection are numerically investigated. The unsteady two-dimensional streamfunction-vorticity equation is solved with finite difference method. As Rayleigh number is increased, the steady flow bifurcates to a time-periodic flow with a fundamental frequency, and afterwards a period-tripling bifurcation occurs with further increase of the Rayleigh number. Chaotic convection is established after a period-doubling bifurcation. A periodic convection with period 4 appears after the first chaotic convection. At still higher Rayleigh numbers, chaotic flows reappear.

• Transactions of the Korean Society of Mechanical Engineers
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• v.17 no.9
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• pp.2304-2314
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• 1993

Natural convective flow and heat transfer in a two-dimensional square enclosure fitted with a horizontal partition are investigated numerically. The enclosure was composed of the lower hot and the upper cold horizontal walls and the adiabatic vertical walls, and a partition was situated perpendicularly at the one vertical insulated wall. The governing equations are solved by using the finite element method with Galerkin method. The computations were carried out with the variations of length, position and thermal conductivity of the partition, and Rayleigh number based on the temperature difference between two horizontal walls and the enclosure height with water(Pr=4.95). As the results, an oscillatory motion of natural convection is resulted in a sudden rise of overall heat transfer, but the increase of length of partition is significantly restrained the increase of Nusselt number. The maximum heat transfer was shown just before the transition of the direction of oscillating flow. An oscillatory motion of flow was perfectly shown the stability with the decrease of the length of partition and Rayleigh number. Also, the heat transfer was raised with the increase of the thermal conductivity in proportion to the increase of the length of partition. The stability and oscillation of flow are affected by the position of partition.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.10
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• pp.1963-1970
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• 1992

An oscillatory motion of the natural convection in a two dimensional, partially divided square enclosure heated from below, and fitted with a partition is investigated numerically. The enclosure was composed of the lower hot and the upper cold horizontal walls and the adiabatic vertical walls, and a partition was situated perpendicularly at the mid-height of the one vertical insulated wall. The governing equations are solved by using the finite element method with Galerkin method. The computations were performed with the variation of the length and the thermal conductivity of the partition, and Rayleigh number based on the temperature difference between horizontal walls and the enclosure height with water(Pr=4.95). also, the effect of the inclination angles was studied for the transition to the oscillating flow. As the results, it was found that the intensity and frequency of oscillatory motion were affected significantly by the Rayleigh number and the length of partition. The effect of oscillatory motion was weaken with the increase of the thermal conductivity of partition. The inclination angle for the transition was raised with the increase of Rayleigh number and the length of partition.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.13 no.2
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• pp.88-95
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• 2001

Transition to chaotic convection is investigated for natural convection of a fluid with Pr=0.1 in a wide-gap horizontal annuls. The unsteady two-dimensional stream-function-vorticity equation is solved with finite difference method. As the Rayleigh number is increased, the steady 'downward flow' bifurcates to a time-periodic flow with a fundamental frequency, and afterwards a period-doubling bifurcation occurs. As the Rayleigh number is increased further, the chaotic flow regime is reached after a sequence of successive Hopf bifurcation to quasi-periodic and chaotic flow regimes. The route to chaos shows the Ruelle-Takens-Newhouse scenario. The flow of chaotic regime displays complex coalescence and separation of eddies in the side and lower region of the annulus.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.25 no.3
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• pp.433-441
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• 2001

Natural convection of a fluid with intermediate Prand시 number of Pr=0.2 in a horizontal annulus is considered, and the bifurcation phenomena and chaotic flows are numerically investigated. The unsteady two-dimensional streamfunction-vorticity equation is solved with finite difference method. The steady downward flow with two counter-rotating eddies bifurcates to a simple periodic flow with a fundamental frequency. And afterwards, second Hopf bifurcation occurs, and a quasi-periodic flow with two incommensurable frequencies appears. However, a new time-periodic flow is established after experiencing quasi-periodic states. As Rayleigh number is increased further, the chaotic flow regime is reached after a sequence of successive Hopf bifurcation to quasi-periodic and chaotic flow regimes. A scenario similar to the Ruelle-Takens-Newhouse scenario of the onset of chaos is observed.

• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.2
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• pp.456-466
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• 1994

Experimental investigation was made to study the mechanism of fluid and thermal oscillation phenomena of surface-tension driven flow in a cylindrical liquid column heated from above which is the low-gravity floating zone simulated on earth. Hexadecane, octadecane, silicon oil (10cs), FC-40 and water are used as the test liquids. The onset of the oscillatory thermocapillary convection appears when Marangoni number exceeds its criteria value and is found to be due to the coupling among velocity and temperature field with the free surface deformation. The frequency of temperature oscillation decreases with increasing aspect ratio for a given diameter and Marangoni number and the oscillation level increases with Marangoni number. The flow pattern in the liquid column appears either as symmetric or asymmetric 3-D flow due to the oscillatory flow in the azimuthal direction. The free surface deformation also occurs either as symmetric or asymmetric mode and its frequency is consistent with those of flow and temperature oscillations. The amplitude of surface deformation also increases with Marangoni number.

• Journal of computational fluids engineering
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• v.16 no.3
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• pp.29-36
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• 2011

The natural convection in a horizontal enclosure heated from the bottom wall, cooled at the top wall, and having a square adiabatic body at its centered area was studied. Three different Prandtl numbers (0.01, 0.7 and 7) were considered for an effect of the Prandtl number on natural convection. A two-dimensional solution for unsteady natural convection was obtained, using Chebyshev spectral methodology for different Rayleigh numbers varying over the range of $10^4$ to $10^6$. It had been experimentally and numerically reported [1,2] that the heat transfer mode becomes oscillatory when Pr is out of a specific Pr band beyond the critical Ra. In this study, we reproduced this phenomenon numerically. The variation of time- and surface-averaged Nusselt numbers on the hot and cold walls for different Rayleigh numbers and Prandtl numbers was presented to show the overall heat transfer characteristics in the system. And also, the isotherms and streamline distributions were presented in detail to compare the physics related to their thermal behavior.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.20 no.9
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• pp.2953-2964
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• 1996

Heat Transfer with periodic fluctuation of fluid temperature caused by oscillatory flow or compression expansion can be out of phase with balk fluid-wall temperature difference. Newton's law of convection is inadequate to describe this phenomenon. In order to solve this problem the concept of the complex Nusselt number has been introduced by severla researchers. The complex Nusselt number expresses out of phase excellently while the first harmonic is dominant in the variations of both fluid-wall temperature difference and heat flux. However, in the case of oscillatory flow with non-linear wall temperature distribution, the complex Nusselt number is not appropriate to predict the heat transfer phenomena since the higher order harmonic components appear in periodic temperature variation. Analytic solutions to the heat transfer with an sinusoidal well temperature distribution were obtained to investagate the effect of non-linear wall temperature distribution. A new formula considering the thermal boundary layer was suggested based on the solutions. A comparison was also made with the complex Nusselt number. It was verified that the new formula describes well the heat transfer of oscillating flow even if the first harmonic component is not dominant in the fluid-wall temperature difference.