Toshio Tagawa

Magnetizing force modeled and numerically solved for natural convection of air in a cubic enclosure : effect of the direction of the magnetic field

Abstract Magnetizing force is modeled by considering magnetic susceptibility as a function of temperature and is included in a momentum balance equation as an external force term in addition to the buoyant force term. Under ideal gas behavior, the magnetizing force term can be represented by a new and simple non-dimensional parameter group γ, which represents the ratio of magnetizing force to the gravitational force. This magnetizing force acts on material of high magnetic susceptibility, like oxygen gas in a temperature gradient field, and affects the convection in addition to gravity. Sample computation was carried out for air in a cubic box that is heated from one vertical wall and cooled from the opposing wall, and has the other four walls thermally insulated. With an increase in the magnetic strength, the upward and downward natural convection in the gravity field becomes horizontal circulation under a cusp-shaped magnetic field.

DISCUSSION ON MOMENTUM INTERPOLATION METHOD FOR COLLOCATED GRIDS OF INCOMPRESSIBLE FLOW

Discussions are given of the different momentum interpolation methods to evaluate the interface velocity in the collocated grid system. It is pointed out that the interface velocity is used in three cases in the overall numerical procedure of the solution of Navier-Stokes equations by utilizing a collocated grid: in the continuity equation; in the interface flow rate computation for the determination of the coefficients in discretization equation; and in the mass residual in the coefficient Ap. Analysis shows that it is better to adopt the momentum interpolation method in the three cases. Two new momentum interpolation methods, called MMIM1 and MMIM2, are proposed. Analysis shows that the two new methods can achieve numerical solutions that are independent of both the underrelaxation factor and the time step size. Taking lid-driven cavity flow as an example, numerical computations are conducted for several Reynolds numbers and different mesh sizes using the SIMPLE algorithm, and the results are compared with benchmark solutions. Numerical tests demonstrate that both MMIM1 and MMIM2 can give unique solutions for different underrelaxation factors and time step sizes, solutions from MMIM1 are slightly better than that of the momentum interpolation of Majumdar, and solutions from MMIM2 have an appreciably better accuracy when the mesh is not fine.

Convection induced by a cusp-shaped magnetic field for air in a cube heated from above and cooled from below

Magnetizing force, which acts in a magnetic field of steep gradient, was applied to air in a cube heated from above and cooled from below, and with the four vertical walls thermally insulated. A four-poles magnet was installed to apply the cusp-shaped magnetic field to air in the cubic enclosure. A simple model equation was derived for magnetizing force and numerically computed for the system. Without a magnetic field, the conduction was stable, but under the magnetizing force a strong downward flow occurred from the center of the top heated plate and the average Nusselt number attained Nu = 1.17 at Ra = 10 5 and γ=0.5, which is equivalent to a temperature difference of 4 [°C] between the top and bottom walls under a maximum magnetic induction of 0.9 [T] inside a cube of (0.064) 3 [m 3 ] heated from above. The flow visualization experiment with hot incense smoke proved the downward flow from the top hot plate

EFFECT OF PRANDTL NUMBER AND COMPUTATIONAL SCHEMES ON THE OSCILLATORY NATURAL CONVECTION IN AN ENCLOSURE

Abstract Numerical calculations were carried out for natural convection of low-Prandtl-number fluid. These calculations include the inertial terms that were approximated by six kinds of schemes, i.e., upwind scheme, hybrid scheme, second-order central difference method, Kawamura-Kuwahara scheme, Utopia scheme, and fourth-order central difference method. The average Nusselt number depended significantly on the schemes. The occurrence of oscillatory flow also depended on the schemes for inertial terms. Higher order up-winding approximations for inertial terms appear to be required to calculate natural convection of low-Prandtl-number fluids like liquid metal, even if the Rayleigh number is not large enough.

Numerical computation for Rayleigh–Benard convection of water in a magnetic field

Abstract The derivation process for the model equation is shown for the natural convection of water (diamagnetic) under both gravity and magnetizing force fields and numerically solved for the Rayleigh–Benard convection in a shallow cylinder heated from below and cooled from above. The cylindrical enclosure was located at two levels in the bore of a super-conducting magnet, where the radial component of the magnetizing force is minimal and its axial component prevails. The cylindrical enclosure was assumed to be located coaxially with the bore of the magnet, and a two-dimensional model equation was presumed. Sample computations were carried out without or with a gravity force for various strengths of Rayleigh number and magnetic induction. When the enclosure was placed above the coil center, where the magnetizing force is opposed to the gravitational force, the average Nusselt number decreased with increasing strength of the magnetic field. When the enclosure was placed below the coil center, where the magnetizing force is parallel to gravity, the average Nusselt number increased above unity even at Ra =1000 and 1500. All of the data agreed favorably with the classical experimental data of Silveston when plotted against the magnetic Rayleigh number proposed by Braithwaite et al.

NUMERICAL COMPUTATION FOR NATURAL CONVECTION OF AIR IN A CUBIC ENCLOSURE UNDER COMBINATION OF MAGNETIZING AND GRAVITATIONAL FORCES

Numerical computations were carried out for natural convection of air in a cubic enclosure under both magnetizing and gravitational force fields. The air in the cubic enclosure is heated from one vertical wall and cooled from an opposing cold wall. Two electric wires to produce a magnetic field are located outside the vertical side walls, perpendicular to the hot and cold walls. Computation for a nongravity field revealed that the magnetizing force attracts the cold air and repels the hot air. As a result, convection roll cells can be seen from the top plate, although usual natural-convection roll cells can be seen through a vertical side wall. Computations were carried for the combined force field of gravity and magnetism for the ranges of parameters Pr=0.71, Ra=10 4 , 10 5 , 10 6 , and 10 7 , and n =0, 0.1, 1, and 10, where n represents the strength of magnetic field. As n increases, the effect of the magnetizing force prevails. This can be understood from the model equation and the flow modes computed. As the Rayleigh number increases, the magnetizing force is enhanced also.

Enhanced convection or quasi-conduction states measured in a super-conducting magnet for air in a vertical cylindrical enclosure heated from below and cooled from above in a gravity field

Magnetizing force was applied for natural convection of air in a shallow cylindrical enclosure heated from below and cooled from above. The cylinder measured 45 mm in diameter and 14.8 mm in height. The convection enclosure was located 66 mm above or below the coil center in the bore of a super-conducting magnet. The average Nusselt numbers were enhanced about twice at the location +66 mm above the coil center under 3.40 Tesla and decreased to Nu=1.12∼1.28 at the location -66 mm below the coil center for the Rayleigh number from 3520 to 6980. These two locations were selected as the most effective positions for application of the magnetizing force in this super-conducting magnet. A model equation for magnetizing force was derived and numerically computed for Pr =0.7 and Ra =2100 and 7000. One turn coil was presumed as a model of thousand turns real superconductor. The magnetic strength is represented by a new parameter y and varied from 2345 to 9124. By adjusting the location of the enclosure in the bore of the super-conducting magnet, the average Nusselt number of 1.14 at Ra =2100 varied from 1.8 to 1.0001 depending on the magnetic strength, and that of 2.02 at Ra=7000 varied from 2.6 to 1.0003. These data are plotted versus magnetic Rayleigh number Ra m =Ra(γ∂B 2 z /∂Z+1) R=0,Z = 0.5 at the center of the enclosure and agreed well with Silveston s data for a classical nonmagnetic field.

Enhanced Heat Transfer Rate Measured for Natural Convection in Liquid Gallium in a Cubical Enclosure Under a Static Magnetic Field

The heat transfer rate of natural convection in liquid gallium in a cubical enclosure was measured experimentally under an external magnetic field applied horizontally and parallel to the vertical heated wall and the opposing cooled wall of the enclosure. One vertical wall was heated with an electric heater and the opposing wall was cooled isothermally with running water. Experiments were conducted in the range of modified Rayleigh number from 1.85 x 10 6 to 4.76 x 10 6 and of Hartmann number from 0 to 573

The natural convection of liquid metal in a cubical enclosure with various electro-conductivities of the wall under the magnetic field

Abstract The natural convection of liquid metal in a cubical enclosure was numerically studied for various electro-conductivities of the wall from zero to infinity under a static magnetic field. The cubical enclosure was heated from one vertical wall and cooled from an opposing vertical wall both isothermally and four other walls were thermally insulated. The direction of the static magnetic field was perpendicular to the heated and cooled walls ( X -direction) or parallel to the heated and cooled walls ( Y -direction) for the present work. All calculations were carried out for the Rayleigh number 10 5 , the Prandtl number 0.025, and the Hartmann number 100. Under the X -directional magnetic field, the average heat transfer rate was effectively suppressed with an increase in the electro-conductivity of the wall. Under the Y -directional magnetic field, the average heat transfer rate was only slightly suppressed for the electrically insulated wall, but drastically decreased with an increase in the electro-conductivity of the wall.

Transient characteristics of convection and diffusion of oxygen gas in an open vertical cylinder under magnetizing and gravitational forces

Abstract A simplified model was derived for magnetizing force convection of oxygen gas and solved for transient characteristics of oxygen gas in a vertical open pipe. Oxygen gas in the cylindrical pipe initially flows out downward since oxygen gas is heavier than air. However, the magnetizing force works to attract the oxygen gas in the lower half of the vertical pipe, and the oxygen gas rises back to the central part of the pipe where the magnetic field is strongest. After a long time, all the oxygen gas in the pipe is replaced with air due to diffusion. This model represented moderately well the transient concentration of oxygen gas measured experimentally in a similar system.