The Adiabatic Conditions on Inner Boundary for Natural Convection With Porous Medium in a Narrow Horizontal Cylindrical Annulus

Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow Horizontal Cylindrical annulus filled with porous medium due to a cosine temperature variation on the outer and adiabatic conditions at the inner boundaries The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two and Rayleigh number and the series extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but Pade approximation leads our result to be good even for much higher value of the similarity parameter.

Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow Horizontal Cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and adiabatic conditions at the inner boundaries with constant volumetric heat flux The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) Horizontal Cylindrical annulus filled with viscous fluid and periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of four, and Rayleigh number and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow horizontal cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and adiabatic conditions at the inner boundaries with constant volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow (micro) horizontal cylindrical annulus filled with porous medium due with linear volumetric heat generation. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but pade approximation leads our results to be good even for much higher value of the similarity parameter.


2005 ◽  
Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow Horizontal Cylindrical annulus filled with viscous fluid due to a sine temperature variation on the outer and adiabatic conditions at the inner boundaries. The solution is expanded in powers of a single combined similarity parameter introduced by [1], which is the product of the Gap ratio to the power of four, and Rayleigh number and the series extended by means of symbolic calculation up to 16 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow horizontal cylindrical annulus filled with porous medium due to a constant temperature variation on the outer and at the inner boundaries with periodic volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the Gap ratio to the power of two, and Rayleigh number. The series is extended by means of symbolic calculation up to 28 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small, but Pade approximation leads our results to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a narrow (Micro size) horizontal cylindrical annulus filled with viscous fluid and constant volumetric heat flux. The solution is expanded in powers of a single combined similarity parameter, which is the product of the gap ratio to the power of four, and Grashof number and the series extended by means of symbolic calculation up to 14 terms. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is almost zero but Pade approximation lead our result to be good even for much higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a circular cavity with periodic heat generation filled with viscous fluid subject to cosine temperature variation on the boundary. The solution is expanded for low Rayleigh number and extended to 16 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but pade approximation leads our result to be good even for higher value of the similarity parameter.


Author(s):  
Kamyar Mansour

We consider the two-dimensional problem of steady natural convection in a circular cavity filled with porous material due to a cosine temperature variation on the boundary. We use Darcy’s law for this cavity filled with porous material. The solution is governed by dimensionless parameter Darcy-Rayleigh number. The solution is expanded for low Darcy-Rayleigh number and extended to 18 terms by computer. Analysis of these expansions allows the exact computation for arbitrarily accuracy up to 50000 figures. Although the range of the radius of convergence is small but Pade approximation leads our result to be good even for higher value of the similarity parameter.


2020 ◽  
Author(s):  
Márk Szijártó ◽  
Attila Galsa

<p>Onset of thermal and haline convection was studied separately by Lapwood (1948) and Wooding (1956) in theoretical models using analytical methods. They established that the buoyancy force caused by difference in temperature (Δ<em>T</em>) or concentration (Δ<em>c</em>) can induce natural convection depending on the model properties (e.g. geometry, permeability, etc.). In the course of further numerical simulations, the thermal (<em>Ra<sub>T</sub></em>) and the haline Rayleigh number (<em>Ra<sub>H</sub></em>) proved itself useful to characterise the type, the intensity and the form of the natural convection (e.g. Diersch and Kolditz, 2002). The main purpose of our study was to examine numerically the combined effect of temperature- and salinity-driven natural convection in a two-dimensional homogeneous porous medium.</p><p>Two-dimensional finite element base model was set up in agreement with the Elder problem (Wooding, 1956) in order to verify the numerical calculation. First, it was established that (1) the critical Rayleigh numbers are mathematically equivalent in the two separated cases (<em>Ra</em><sub><em>Tcr</em></sub>=<em>Ra<sub>Hcr</sub></em>=4π<sup>2</sup>), and (2) time-dependent thermal or haline natural convections evolve, when the Rayleigh number lies within the range of 300–600. Numerical simulations were accomplished to investigate the interaction of the temperature- and salinity-driven natural convection. Non-dimensional thermal expansion and haline concentration were increased from αΔ<em>T</em>=0.01 to 1 and from βΔ<em>c</em>=10<sup>-5</sup> to 10<sup>-3</sup>, respectively, while the variation of the Darcy flux, the temperature, the concentration, the Nusselt and the Sherwood numbers was computed. The main points of this study were that (1) how the onset of the thermohaline convection is facilitated by the positive interaction of the thermal and haline effects (<em>Ra<sub>THcr</sub></em>); (2) under what conditions time-dependent flow evolves in the theoretical models; (3) whether a new non-dimensional number can be defined instead of the two separated Rayleigh numbers in order to characterise the behaviour of the thermonaline convection. These simulations draw attention to the importance of understanding the physical background of thermohaline convection, for instance, at the margin of confined and unconfined carbonate systems (e.g. Buda Thermal Karst), or in the case of groundwater flow induced by water pumping/injection of deep geothermal power plants.</p><p>The project was supported by the ÚNKP-19-3 and ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology, the Hungarian Research Fund (K 129279) and the János Bolyai Scholarship of the Hungarian Academy of Science. This research is a part of a project that has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 810980.</p><p>References:</p><p>Diersch, H.-J.G., Kolditz, O. (2002), Variable-density flow and transport in porous media: approaches and challenges. Advances in Water Resources, 25, 899-944.</p><p>Lapwood, E.R. (1948), Convection of a fluid in a porous medium. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 508-521.</p><p>Wooding, R.A. (1956), Steady state free thermal convection of liquid in a saturated permeable medium. Journal of Fluid Mechanics, 2, 273-285.</p>


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