scholarly journals Pressure-Velocity Coupling Schemes for Bouyancy Driven Flow in a Differentially Heated Cavity Using F.V.M and Matlab

2021 ◽  
pp. 19-39
Author(s):  
Purity Mberia ◽  
Stephen Karanja ◽  
Mark Kimathi
Keyword(s):  
Fluid Flow ◽  
Rayleigh Number ◽  
Bench Mark ◽  
Three Dimension ◽  
Two Dimensional ◽  
Central Difference ◽  
Square Enclosure ◽  
Simpler Algorithm ◽  
Simplec Algorithm ◽  
Rayleigh Numbers

Numerical analysis of fluid flow is anchored on the laws of conservation. A challenge in solving the momentum equation arises due to the unavailability of an explicit pressure equation. To avoid solving the pressure term most researchers have eliminated it by cross differentiating the x and the y two dimensional momentum equations and subtracting them. This method introduces more variables to be solved in comparison to the primitive variables and is  restricted to two-dimensional flows as streamlines do not exist in three-dimension. This method thus presents a serious limitation in analysis of fluid flow. In this study an equation for computing pressure has been developed using pressure - velocity coupling and used in solving the governing equations. The performance of three pressure velocity schemes namely; the Semi Implicit Method for Pressure linked Equation (SIMPLE), SIMPLE Revised (SIMPLER) and SIMPLE Consistent (SIMPLEC) for laminar buoyancy driven flow has been tested in order to establish the scheme that gives results consistent with bench mark data. The equations governing the flow are solved iteratively using finite volume method together with the central difference interpolating scheme. The solutions are presented for Rayleigh numbers of 103, 104, and 105. This resulted in the velocity profiles for the SIMPLE, SIMPLER, and SIMPLEC algorithm for a Rayleigh number of 104 and 105 converging to the same path. At a Rayleigh number of 103 however, SIMPLER algorithm undergoes a degradation in convergence with grid refinement at the baffle region. Results predicted by using the SIMPLEC algorithm are thus able to effectively compute the velocity of fluid flow in a differentially heated square enclosure with baffles for both low and higher Rayleigh numbers irrespective of the grid size.

Characteristics of high Rayleigh number two-dimensional convection in an open-top porous layer heated from below

1999 ◽  
Vol 394 ◽  
pp. 241-260 ◽  
Author(s):  
ABDELLAH S. M. CHERKAOUI ◽  
WILLIAM S. D. WILCOCK
Keyword(s):  
Rayleigh Number ◽  
Control Volume ◽  
Flow Patterns ◽  
Convective System ◽  
Two Dimensional ◽  
Simpler Algorithm ◽  
Convective Pattern ◽  
Volume Method ◽  
Rayleigh Numbers ◽  
Type Of Transition

Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.

scholarly journals Classification of Flow Modes for Natural Convection in a Square Enclosure with an Eccentric Circular Cylinder

Energies ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2788
Author(s):  
Hyun-Sik Yoon ◽  
Yoo-Jeong Shim
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Circular Cylinder ◽  
Flow Structures ◽  
Two Dimensional ◽  
Square Enclosure ◽  
Wide Range ◽  
Secondary Vortices ◽  
Rayleigh Numbers

The present study investigated the natural convection for a hot circular cylinder embedded in a cold square enclosure. The numerical simulations are performed to solve a two-dimensional steady natural convection for three Rayleigh numbers of 103, 104 and 105 at a fixed Prandtl number of 0.7. This study considered the wide range of the inner cylinder positions to identify the eccentric effect of the cylinder on flow and thermal structures. The present study classifies the flow structures according to the cylinder position. Finally, the present study provides the map for the flow structures at each Rayleigh number (Ra). The Ra = 103 and 104 form the four modes of the flow structures. These modes are classified by mainly the large circulation and inner vortices. When Ra = 105, one mode that existed at Ra = 103 and 104, disappears in the map of the flow structures. The new three modes appear, resulting in total six modes of flow structures at Ra = 105. New modes at Ra = 105 are characterized by the top side secondary vortices. The corresponding isotherms are presented to explain the bifurcation of the flow structure.

scholarly journals Effect of Rayleigh Number on Internal Eccentricity in a Heated Horizontal Elliptical Cylinder to its Coaxial Square Enclosure

2020 ◽  
Vol 25 (3) ◽  
pp. 17-29
Author(s):  
Abdelkrim Bouras ◽  
Djedid Taloub ◽  
Zied Driss
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Outer Cylinder ◽  
Boussinesq Approximation ◽  
Square Enclosure ◽  
Nusselt Numbers ◽  
Flow And Heat Transfer ◽  
Commercial Code ◽  
Volume Method ◽  
Rayleigh Numbers

AbstractThis paper deals with numerical investigation of a natural convective flow in a horizontal annular space between a heated square inner cylinder and a cold elliptical outer cylinder with a Newtonian fluid. Uniform temperatures are imposed along walls of the enclosure. The governing equations of the problem were solved numerically by the commercial code Fluent, based on the finite volume method and the Boussinesq approximation. The effects of Geometry Ratio GR and Rayleigh numbers on fluid flow and heat transfer performance are investigated. The Rayleigh number is varied from 103 to 106. Throughout the study the relevant results are presented in terms of isotherms, and streamlines. From the results, we found that the increase in the Geometry Ratio B leads to an increase of the heat transfer coefficient. The heat transfer rate in the annulus is translated in terms of the average Nusselt numbers along the enclosure’s sides. Tecplot 7 program was used to plot the curves which cleared these relations and isotherms and streamlines which illustrate the behavior of air through the channel and its variation with other parameters. The results for the streamlines, isotherms, local and average Nusselt numbers average Nusselt numbers are compared with previous works and show good agreement.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

scholarly journals Numerical Investigation of Fluid Flow and Heat Transfer Inside a 2D Enclosure with Three Hot Obstacles on the Ramp under the Influence of a Magnetic Field

2017 ◽  
Vol 7 (3) ◽  
pp. 1647-1657
Author(s):  
M. M. Keshtkar ◽  
M. Ghazanfari
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Fluid Flow ◽  
Relaxation Times ◽  
Appreciable Effect ◽  
Two Dimensional ◽  
Square Enclosure ◽  
Flow And Heat Transfer ◽  
Contour Lines ◽  
Considerable Impact

This paper focuses on solving the fluid flow and heat transfer equations inside a two-dimensional square enclosure containing three hot obstacles affected by gravity and magnetic force placed on a ramp using Boltzmann method (LBM) applying multiple relaxation times (MRT). Although, the Lattice Boltzmann with MRT is a complex technique, it is a relatively new, stable, fast and high-accurate one. The main objective of this research was to numerically model the fluid flow and ultimately obtaining the velocity field, flow and temperature contour lines inside a two-dimensional enclosure. The results and their comparisons for different types of heat transfer revealed that free or forced heat transfer has a considerable impact on the heat transfer and stream lines. This can be controlled by modifying the Richardson number. It is revealed that changing the intensity of the magnetic field (Hartman number) has an appreciable effect on the heat transfer.

scholarly journals Effect of Various Tilted Positions of a Thin Fin on Natural Convection of Laminar Viscous Flow in a Square Cavity

2021 ◽  
Vol 39 (5) ◽  
pp. 1634-1642
Author(s):  
Syed Fazuruddin ◽  
Seelam Sreekanth ◽  
G Sankara Sekhar Raju
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Fluid Flow ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Finite Difference ◽  
Average Nusselt Number ◽  
Vertical Position ◽  
Inclination Angles ◽  
Rayleigh Numbers

An exhaustive numerical investigation is carried out to analyze the role of an isothermal heated thin fin on fluid flow and temperature distribution visualization in an enclosure. Natural convection within square enclosures finds remarkable pragmatic applications. In the present study, a finite difference approach is performed on two-dimensional laminar flow inside an enclosure with cold side walls and adiabatic horizontal walls. The fluid flow equations are reconstructed into vorticity - stream function formulation and these equations are employed utilizing the finite-difference strategy with incremental time steps. The parametric study includes a wide scope of Rayleigh number, Ra, and inclination angle ϴ of the thin fin. The effect of different Rayleigh numbers ranging Ra = 104-106 with Pr=0.71 for all the inclination angles from 0°-360° with uniform rotational length of angle 450 of an inclined heated fin on fluid flow and heat transfer have been investigated. The heat transfer rate within the enclosure is measured by means of local and average Nusselt numbers. Regardless of inclination angles of the thin fin, a slight enhancement in the average Nusselt number is observed when Rayleigh number increased for both the cases of the horizontal and vertical position of the thin fin. When the fin has inclined no change in average Nusselt number is noticed for distinct Rayleigh numbers.

scholarly journals Free convection of a nanofluid in a square cavity with a heat source on the bottom wall and partially cooled from sides

2014 ◽  
Vol 18 (suppl.2) ◽  
pp. 283-300 ◽  
Author(s):  
Mostafa Mahmoodi ◽  
Arani Abbasian ◽  
Sebdani Mazrouei ◽  
Saeed Nazari ◽  
Mohammad Akbari
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Fluid Flow ◽  
Rayleigh Number ◽  
Average Nusselt Number ◽  
Volume Fraction ◽  
Heat Sinks ◽  
Square Cavity ◽  
Flow And Heat Transfer ◽  
Rayleigh Numbers

The problem of free convection fluid flow and heat transfer in a square cavity with a flush mounted heat source on its bottom wall and two heat sinks on its vertical side walls has been investigated numerically. Via changing the location of the heat sinks, six different arrangements have been generated. The cavity was filled with Cu-water nanofluid. The governing equations were discretized using the finite volume method and SIMPLER algorithm. Using the developed code, a parametric study was undertaken, and effects of Rayleigh number, arrangements of the heat sinks and volume fraction of the nanoparticles on fluid flow and heat transfer inside the cavity were investigated. Also for the middle-middle heat sinks arrangement, capability of five different water based nanofluids on enhancement of the rate of heat transfer was examined and compared. From the obtained results it was found that the average Nusselt number, for all six different arrangements of the heat sinks, was an increasing function of the Rayleigh number and the volume fraction of the nanoparticles. Also it was found that at high Rayleigh numbers, maximum and minimum average Nusselt number occurred for middle-middle and top-bottom arrangement, respectively. Moreover it was found that for the middle-middle arrangement, at high Rayleigh numbers, maximum and minimum rate of heat transfer was obtained by Cu-water and TiO2-water nanofluids respectively.

Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders

1976 ◽  
Vol 76 (2) ◽  
pp. 337-362 ◽  
Author(s):  
Jean-Paul Caltagirone
Keyword(s):  
Porous Medium ◽  
Finite Elements ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Different Types ◽  
Rayleigh Numbers ◽  
Steady Convection

The study of natural convection in a saturated porous medium bounded by two concentric, horizontal, isothermal cylinders reveals different types of evolution according to the experimental conditions and the geometrical configuration of the model. At small Rayleigh numbers the state of the system corresponds to a regime of pseudo-conduction. The isotherms are coaxial with the cylinders. At larger Rayleigh numbers a regime of steady two-dimensional convection sets in between the two cylinders. Finally, for Rayleigh numbers above the critical Rayleigh number Ra*c the phenomena become three-dimensional and fluctuating. The appearance of these different regimes depends, moreover, on the geometry considered and, in particular, on two numbers: R, the ratio of the radii of the cylinders, and A, the ratio of the length of the cylinders to the radius of the inner one. In order to approach these experimental observations and to obtain realistic theoretical models, several methods of solving the equations have been used.The perturbation method yields information about the thermal field and the heat transfer between the cylinders under conditions close to the equilibrium state.A numerical two-dimensional model enables us to extend the range of investigation and to represent properly the phenomena when steady convection appreciably modifies the temperature distribution and the velocities within the porous layer.Neither of these models allows account to be taken of the instabilities observed experimentally above a critical Rayleigh number Ra*c. For this reason, a study of stability has been carried out using a Galerkin method based on equations corresponding to an initial state of steady convection. The results obtained show the importance of three-dimensional effects for the onset of fluctuating convection. The critical transition Rayleigh number Ra*c is thus determined in terms of the ratio of the radii R by solving an eigenvalue problem.A numerical three-dimensional model based on the method of finite elements has thus been developed in order to point out the different types of evolution with time. Steady two-dimensional convection and fluctuating three-dimensional convection have been actually found by calculation. The solution of the system of equations by the method of finite elements is briefly described.The experimental and theoretical results are then compared and a general physical interpretation is given.

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