A numerical study of thermal convection in a rotating spherical annulus with axial gravitational field by using parametric spline function approximation
01 Sep 1998-International Journal of Numerical Methods for Heat & Fluid Flow (MCB UP Ltd)-Vol. 8, Iss: 6, pp 673-688
TL;DR: In this paper, a parametric spline function approximation is used to study the steady combined convection of a Boussinesq fluid contained between two concentric rotating spheres, and the resulting flow patterns, temperature distributions, total heat flux and torque characteristics are presented for various cases considered.
Abstract: A parametric spline function approximation is used to study the steady combined convection of a Boussinesq fluid contained between two concentric rotating spheres. A spline function which depends on a parameter p > 0 is used for approximation. This approximation gives second order accuracy. The spheres which form the flow region are maintained at uniform but different temperatures and are allowed to rotate about a common vertical axis with different angular velocities. A uniform gravitational field acts in the direction parallel to that of the axis of rotation. The resulting flow patterns, temperature distributions, total heat flux and torque characteristics are presented for various cases considered and it is found that these results are in good agreement with the results of Dallman and Douglass in 1980.
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21 citations
TL;DR: In this paper, an implicit finite difference scheme in combination with the quasilinearization technique is applied to obtain the steady state non-similar solutions of the governing boundary layer equations for flow and temperature fields, and numerical results indicate that the heat transfer rate at the wall decreases significantly with an increasing thermal stratification parameter, but its effect on the skin friction coefficients is rather minimum.
Abstract: Purpose – The purpose of this paper is to study the influence of thermally stratified medium on a free convection flow from a sphere, which is rotating about the vertical axis, immersed in a stably thermally stratified medium.Design/methodology/approach – An implicit finite‐difference scheme in combination with the quasi‐linearization technique is applied to obtain the steady state non‐similar solutions of the governing boundary layer equations for flow and temperature fields.Findings – The numerical results indicate that the heat transfer rate at the wall decreases significantly with an increasing thermal stratification parameter, but its effect on the skin friction coefficients is rather minimum. In fact, the presence of thermal stratification of the medium influences the heat transfer at wall to be in opposite direction, that is, from fluids to the wall above a certain height. The heat transfer rate increases but the skin frictions decrease with the increase of Prandtl number. In particular, the effect...
3 citations
01 Jan 2007
TL;DR: In this article, the fundamental problem of spherical Couette flow (SCF) in the presence of heat and under the action of a gravitational field was studied. And the authors focused their attention to such aspect ratios (i.e., σ > 0.24) for which Taylor vortices formation was observed.
Abstract: We study the fundamental problem of spherical Couette flow (SCF) in the presence of heat and under the action of a gravitational field. The flow is considered as unsteady, axisymmetric and the fluid as viscous, incompressible and Newtonian. The numerical technique, which we use, is convergent and consistent and can be applied to 3D or 2D flow motions. We focus our attention to such aspect ratios () oi i σ =r -r r for which have the presence of the important physical phenomenon of Taylor vortices formation and especially for large annular gaps, σ>0.24. Results concerning the stream function, the temperature field, the function of vorticity, the skin friction and the rate of heat transfer for large annular gap are presented. Finally, we calculate the drag acting on the inner and outer sphere and the thermal energy which is convected from the inner sphere to the fluid and from the fluid to the outer sphere.
1 citations
Cites background from "A numerical study of thermal convec..."
...[9] S.H. Sohrab, Modified Form of the Helmholtz Vorticity Equation and its Solution for Spherical Flow Within a Droplet in Uniform or Counterflow Streams, IASME Transactions, Vol. 1, 2004, pp. 634-640....
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...In a related problem Sohrab [9] discussed the flow within a droplet either located in a uniform stream or at stagnation point of axisymmetric counterflow and presented interesting results referring to the formation of Hill ’s vortices within the droplet as well as to the form of these vortices as product solutions....
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...In addition Pearlman and Sohrab [10] studied the formation of ring vortices near the equatorial plane in the case of rotating spheres....
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...[10] H.G. Pearlman and S.H. Sohrab, Effects of Rotation on Cellular Premixed Flames Stabilized on Rotating Porous Spheres, Combustion and Flame, Vol. 92, 1993, pp. 469-474....
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TL;DR: In this paper, a general formulation to treat mixed boundary conditions using the spline approximation has been presented, where numerical solutions have been obtained by solving the Navier-Stokes and energy equations.
Abstract: The present work is devoted to the numerical study of laminar natural convection flow from a heated horizontal cylinder under diverse surface boundary conditions using the spline fractional step method. A general formulation to treat mixed boundary conditions using the spline approximation has been presented. Numerical solutions have been obtained by solving the Navier-Stokes and energy equations. The results for the isothermal boundary condition as well as for the uniform heat flux are in good agreement with published experimental data and with other solutions presently available in the literature. Some new computations at very high Rayleigh numbers indicate the existence of attached separation vortices in the downstream plume region, the appearance of these vortices being dependent on the values of the Biot number. All results were computed on a personal computer using unequally spaced grids that provided good results with a minimum number of computational points. The numerical scheme presented here app...
103 citations
TL;DR: In this article, a linear theory is presented for the steady, axially symmetric motion of a stratified fluid in a narrow, rotating spherical annulus with a spherically symmetric gravitational field.
Abstract: A linear theory is presented for the steady, axially symmetric motion of a stratified fluid in a narrow, rotating spherical annulus with a spherically symmetric gravitational field.The fluid is driven by a combination of differential rotation of the two shells and differential heating applied at the surfaces of the spheres.It is shown that the effect of stratification becomes increasingly important at lower latitudes with the Ekman layers on the spheres’ surfaces fading in strength as the geostrophic interior velocities themselves tend toward the shell speeds at lower latitudes.The Singularities In The Geostrophic Solutions At The Equator Are Removed By A Boundary Layer Whose Detailed Structure Depends On The Ratio Of Horizontal To Vertical Mixing Coefficients Of Momentum And Heat.
25 citations
TL;DR: In this paper, a spline function which depends on a parameter p > 0 has been constructed, which is of trigonometric and polynomial order one and reduces to ordinary cubic spline.
23 citations
TL;DR: A numerical method involving cubic splines is used to solve the classical problem of natural convection in cavities as mentioned in this paper, which allows the reduction of the usual block-tridiagonal matrix to three tridiagonal ones.
15 citations
TL;DR: In this paper, steady axisymmetric flow engendered by coaxially rotating concentric spheres is considered, with each sphere maintained at a uniform temperature in the presence of a uniform gravitational field parallel to the axis of rotation.
Abstract: With each sphere maintained at a uniform temperature in the presence of a uniform gravitational field parallel to the axis of rotation, consideration is given to steady axisymmetric flow engendered by coaxially rotating concentric spheres. All fluid properties except specific weight are treated as constant, and viscous dissipation is negligible. The mathematical formulation allows treatment of both positive and negative angular speed ratios as well as choice of which sphere is to be the hotter. Velocity components are expressed in terms of steam and swirl functions; these functions as well as temperature are expanded in power series of the square of Reynolds number, with two terms in each series being obtained. Flow regimes are defined based upon the nature of the zeroth-order secondary flow in the absence of thermal effects. As the strength of thermal influence (never dominant) is varied, the resulting flow patterns are discussed and illustrated for each flow regime.
9 citations