Effect of Wall Thermal Boundary Conditions on the Development of Three-Dimensional, Unsteady Natural Convective Flow in a Horizontal Enclosure With a Heated Strip on the Lower Surface

2004 ◽  
Author(s):  
Patrick H. Oosthuizen ◽  
Jane T. Paul
Keyword(s):  
Boundary Condition ◽  
Rayleigh Number ◽  
Low Temperature ◽  
Cross Section ◽  
Three Dimensional ◽  
Lower Surface ◽  
Vertical Side ◽  
Thermal Boundary Conditions ◽  
Side Walls ◽  
Rayleigh Numbers

Flow in a rectangular enclosure with a square vertical cross-section normal to the longitudinal coordinate direction and having a strip on the lower horizontal surface which is heated to a uniform high temperature has been numerically studied. Two wall thermal boundary conditions have been considered. In one, the longitudinal vertical side walls are cooled to a uniform low temperature and the horizontal top surface is adiabatic while in the other the longitudinal vertical side walls and the horizontal top surface are cooled to a uniform low temperature. In both cases, the square vertical end walls of the enclosure are adiabatic. It has been assumed that the flow is laminar and that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces. The unsteady, three-dimensional governing equations, expressed in dimensionless form, have been solved using a finite-difference procedure. The solution was started with no flow in the enclosure. The solution, in general, has the following parameters: the Rayleigh Number, Ra, the Prandtl number, Pr, the dimensionless longitudinal length of the enclosure relative to the size of the square cross-section, Ay, the dimensionless width of the heated strip on the lower surface relative to the size of the square cross-section, wH, and the thermal boundary condition on the upper surface. Results have only been obtained for a Prandtl number of 0.7 and only results for wH = 1/3 will be presented. Results have been obtained for values of Ay between 0.5 and 2 for Rayleigh numbers up to 5×105. In all cases, three-dimensional unsteady flow has been found to exist at the higher Rayleigh numbers. The conditions under which this unsteady flow develops and the effect of Ay on the variation of the mean Nusselt number with Rayleigh number and the effect of the wall surface boundary condition on these results has been investigated.

Free Convection in a Rectagular Enclosure With Three Heated Sections on the Lower Surface and Isothermal Side and Top Surfaces

2005 ◽  
Author(s):  
Patrick H. Oosthuizen ◽  
Jane T. Paul
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Unsteady Flow ◽  
Three Dimensional ◽  
Lower Surface ◽  
Vertical Side ◽  
Longitudinal Length ◽  
Vorticity Vector ◽  
Rayleigh Numbers ◽  
Vertical Height

The development of unsteady, three-dimensional free convective flow in a rectangular enclosure with multiple heated elements on the bottom horizontal surface has been numerically studied. The enclosure considered has rectangular horizontal lower and upper surfaces and rectangular vertical side surfaces. The horizontal width of enclosure is twice the vertical height of the enclosure while the longitudinal length of the enclosure is equal to the vertical height of the enclosure. There are three square symmetrically placed isothermal heated sections on the lower surface, the rest of this surface being adiabatic. The vertical side-walls and the horizontal rectangular upper surface of the enclosure are kept at a uniform low temperature. It has been assumed that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces, this having been treated by using the Boussinesq approach. The solution has been obtained by numerically solving the unsteady, three-dimensional governing equations written in dimensionless form, the solution being obtained in terms of the vorticity vector and vector potential functions. The solution has the following parameters: the Rayleigh number, Ra, the Prandtl number, Pr, the dimensionless size, wH, of the square heated sections and the dimensionless distance between the heated sections on the lower surface, wS. Results have only been obtained for a Prandtl number of 0.7. In a given geometrical situation it was found in all cases that a steady flow exists at low Rayleigh numbers, that an unsteady flow develops at higher Rayleigh numbers and that the flow then again becomes steady at still higher Rayleigh numbers. The conditions under which unsteady flow develops and ceases and the variation of mean Nusselt number with Rayleigh number have been explored.

Laminar Natural Convection in an Inclined Rectangular Box With the Lower Surface Half-Heated and Half-Insulated

1983 ◽  
Vol 105 (3) ◽  
pp. 425-432 ◽  
Author(s):  
P. K.-B. Chao ◽  
H. Ozoe ◽  
S. W. Churchill ◽  
N. Lior
Keyword(s):  
Heat Transfer ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Lower Surface ◽  
The Other ◽  
Uniform Temperature ◽  
Side Walls ◽  
Laminar Natural Convection ◽  
Rayleigh Numbers ◽  
Good Agreement

The pattern of circulation and the rate of heat transfer were determined experimentally and also by three-dimensional, finite-difference calculations for an inclined 2 × 1 × 1 rectangular enclosure with a 1 × 1 segment of the lower 2 × 1 surface at a uniform temperature, the other 1 × 1 segment and four side walls insulated, and the upper surface at a lower uniform temperature. As contrasted with an enclosure heated and cooled on the horizontal surfaces, a fluid motion occurs and the rate of heat transfer exceeds that for pure conduction for all temperature differences and orientations. The effects of elevation of the heated and insulated segments were investigated, as well as of inclination about the longer dimension. Despite differences in the Prandtl and Rayleigh numbers, the observed and predicted patterns of circulation are in good agreement, and the measured and predicted rates of heat are in qualitative agreement.

Numerical Investigation on Mixed Convection in a Horizontal Channel Heated From Below

2007 ◽  
Author(s):  
Giuseppe Foglia ◽  
Nicola Lanzaro ◽  
Oronzio Manca ◽  
Sergio Nardini
Keyword(s):  
Mixed Convection ◽  
Fluid Dynamic ◽  
Three Dimensional ◽  
Horizontal Channel ◽  
Uniform Heat Flux ◽  
Vertical Side ◽  
Parallel Wall ◽  
Horizontal Wall ◽  
Side Walls ◽  
Rayleigh Numbers

In this work mixed convection in a horizontal channel with the lower wall heated at uniform heat flux is studied numerically. A three dimensional problem is modeled and solved by means of the FLUENT code. The domain is made of a principal channel and two channels with adiabatic walls, one upstream the principal channel and the other downstream. The principal channel is formed by a uniformly heated horizontal wall, a parallel wall located above and two adiabatic vertical side walls. The aim of this paper is to investigate the effect of Reynolds and Rayleigh numbers on thermal and fluid dynamic behavior in mixed convection in a horizontal channel heated from below. The analysis is carried out in transient regime in order to evaluate the thermal and fluid dynamic parameters as functions of the time. The Reynolds and Rayleigh numbers investigated are between 5 and 500 and 1.37×106 and 2.75×106 respectively. The corresponding Richardson number, Ri = Gr/Re2, holds values in the range 7.76 – 1.55 × 105. Wall temperature distributions and profiles along longitudinal and transversal coordinates are reported for different time values. Air velocity and temperature in the principal channel are presented along the longitudinal and transversal sections for some time values.

scholarly journals Effects of Anisotropy on Natural Convection in a Vertical Porous Cavity Filled with a Heat Generation

2020 ◽  
pp. 12-27
Author(s):  
Degan Gerard ◽  
Sokpoli Amavi Ernest ◽  
Akowanou Djidjoho Christian ◽  
Vodounnou Edmond Claude
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Temperature Fields ◽  
Poisson Equations ◽  
Gravitational Vector ◽  
Side Walls ◽  
Rayleigh Numbers ◽  
Vertical Cavity

This research was devoted to the analytical study of heat transfer by natural convection in a vertical cavity, confining a porous medium, and containing a heat source. The porous medium is hydrodynamically anisotropic in permeability whose axes of permeability tensor are obliquely oriented relative to the gravitational vector and saturated with a Newtonian fluid. The side walls are cooled to the temperature  and the horizontal walls are kept adiabatic. An analytical solution to this problem is found for low Rayleigh numbers by writing the solutions of mathematical model in polynomial form of degree n of the Rayleigh number. Poisson equations obtained are solved by the modified Galerkin method. The results are presented in term of streamlines and isotherms. The distribution of the streamlines and the temperature fields are greatly influenced by the permeability anisotropy parameters and the thermal conductivity. The heat transfer decreases considerably when the Rayleigh number increases.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Numerical Calculation of Three-Dimensional Turbulent Natural Convection in a Cubical Enclosure Using a Two-Equation Model for Turbulence

1986 ◽  
Vol 108 (4) ◽  
pp. 806-813 ◽  
Author(s):  
H. Ozoe ◽  
A. Mouri ◽  
M. Hiramitsu ◽  
S. W. Churchill ◽  
N. Lior
Keyword(s):  
Natural Convection ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Vertical Wall ◽  
Three Dimensional ◽  
Spiral Flow ◽  
Vertical Side ◽  
Turbulent Natural Convection ◽  
Cubical Enclosure ◽  
Side Walls

This paper presents a model and numerical results for turbulent natural convection in a cubical enclosure heated from below, cooled on a portion of one vertical side wall and insulated on all other surfaces. Three-dimensional balances were derived for material, energy, and the three components of momentum, as well as for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy ε. The constants used in the model were the same as those used by Fraikin et al. for two-dimensional convection in a channel. Illustrative transient calculations were carried out for Ra = 106 and 107 and Pr = 0.7. Both the dominant component of the vector potential and the Nusselt number were found to converge to a steady state. Isothermal lines and velocity vectors for vertical cross sections normal to the cooled wall indicated three-dimensional effects near the side walls. A top view of the velocity vectors revealed a downward spiral flow near the side walls along the cooled vertical wall. A weak spiral flow was also found along the side walls near the wall opposing the partially cooled one. The highest values of the eddy diffusivity were 2.6 and 5.8 times the molecular kinematic viscosity for Ra = 106 and 107, respectively. A coaxial double spiral movement, similar to that previously reported for laminar natural convection, was found for the time-averaged flow field. This computing scheme is expected to be applicable to other thermal boundary conditions.

Three-Dimensional Laminar Natural Convection in an Inclined Air Slot With Hexagonal Honeycomb Core

1991 ◽  
Vol 113 (4) ◽  
pp. 906-911 ◽  
Author(s):  
Y. Asako ◽  
H. Nakamura ◽  
Z. Chen ◽  
M. Faghri
Keyword(s):  
Natural Convection ◽  
Numerical Solutions ◽  
Transfer Problem ◽  
Three Dimensional ◽  
Convection Heat Transfer ◽  
Temperature Fields ◽  
Honeycomb Core ◽  
Inclination Angles ◽  
Side Walls ◽  
Rayleigh Numbers

Numerical solutions are obtained for a three-dimensional natural convection heat transfer problem in an inclined air slot with a hexagonal honeycomb core. The air slot is assumed to be long and wide such that the velocity and temperature fields repeat themselves in successive enclosures. The numerical methodology is based on an algebraic coordinate transformation technique, which maps the complex cross section onto a rectangle, coupled with a calculation procedure for fully elliptic three-dimensional flows. The calculations are performed for Rayleigh numbers in the range of 103 to 105, inclination angles in the range of −90 to 80 deg, Prandtl number of 0.7, and for five values of the aspect ratio. Three types of thermal boundary condition for the honeycomb side walls are considered. The average Nusselt number results are compared with those for a rectangular two-dimensional enclosure.

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 simulation of convective heat transport within the nanofluid filled vertical tube of plain and uneven side walls

2019 ◽  
Author(s):  
M. J. Uddin ◽  
A. K. M. Fazlul Hoque ◽  
M. M. Rahman ◽  
K. Vajravelu
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Numerical Data ◽  
Vertical Tube ◽  
Vertical Side ◽  
Left Wall ◽  
Special Cases ◽  
Side Walls ◽  
Horizontal Side ◽  
Thermal Rayleigh Number

Two-dimensional transient natural convective flow in a vertical tube of plain and uneven side-walls containing cobalt-kerosene nanofluids is analyzed using a nonhomogeneous dynamic model. The vertical right wall of the enclosure is maintained at a constant low temperature and the left wall is heated by a uniform thermal condition whereas the horizontal side-walls are insulated. The Brownian motion and thermophoretic phenomena of the nanoparticles are considered in the model. The governing nonlinear momentum, energy, and concentration equations are solved numerically using a Galerkin weighted residual finite element method. The thermal, flow and concentration fields are obtained to understand the flow dynamics of cobalt-kerosene nanofluid in two types of enclosures. The local and average Nusselt numbers are analyzed for plain and uneven side walls of the tube for different parameters of the problem. The simulated results are compared with the experimental as well as with the numerical data available in the literature for some special cases. The outcomes show that the tube of having uneven vertical side-walls give higher heat transfer for lower values of the thermal Rayleigh number; whereas for the higher values of the thermal Rayleigh number, the tube of plain vertical side-walls exhibit significantly higher heat transfer rate.

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