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.

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.

scholarly journals A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus

1999 ◽  
Vol 381 ◽  
pp. 27-61 ◽  
Author(s):  
MARK P. DYKO ◽  
KAMBIZ VAFAI ◽  
A. KADER MOJTABI
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Three Dimensional ◽  
Flow Stability ◽  
Secondary Flows ◽  
Flow Structures ◽  
Wide Range ◽  
First Time ◽  
Rayleigh Numbers

A numerical and experimental study of buoyancy-driven flow in the annulus between two horizontal coaxial cylinders at Rayleigh numbers approaching and exceeding the critical values is presented. The stability of the flow is investigated using linear theory and the energy method. Theoretical predictions of the critical Rayleigh number for onset of secondary flows are obtained for a wide range of radius ratio R and are verified by comparison with results of previous experimental studies. A subcritical Rayleigh number which provides a necessary condition for global flow stability is also determined. The three-dimensional transient equations of fluid flow and heat transfer are solved to study the manifestation of instabilities within annuli having impermeable endwalls, which are encountered in various applications. For the first time, a thorough examination of the development of spiral vortex secondary flow within a moderate gap annulus and its interaction with the primary flow is performed for air. Simulations are conducted to investigate factors influencing the size and number of post-transitional vortex cells. The evolution of stable three-dimensional flow and temperature fields with increasing Rayleigh number in a large gap annulus is also studied. The distinct flow structures which coexist in the large gap annulus at high Rayleigh numbers preceding transition to oscillatory flow, including transverse vortices at the end walls which have not been previously identified, are established numerically and experimentally. The solutions for the large-gap annulus are compared to those for the moderate-gap case to clarify fundamental differences in behaviour. Heat transfer results in the form of local Nusselt number distributions are presented for both the moderate- and large-gap cases. Results from a series of experiments performed with air to obtain data for validation of the numerical scheme and further information on the flow stability are presented. Additionally, the change from a crescent-shaped flow pattern to a unicellular pattern with centre of rotation at the top of the annulus is investigated numerically and experimentally for a Prandtl number of 100. Excellent agreement between the numerical and experimental results is shown for both Prandtl numbers studied. The present work provides, for the first time, quantitative three-dimensional descriptions of spiral convection within a moderate-gap annulus containing air, flow structures preceding oscillation in a large-gap annulus for air, and unicellular flow development in a large-gap annulus for large Prandtl number fluids.

A Numerical Study of Three-Dimensional Natural Convection in a Low Aspect Ratio Horizontal Enclosure With a Uniform Heat Flux on the Lower Surface

1999 ◽  
Author(s):  
Patrick H. Oosthuizen
Keyword(s):  
Heat Flux ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Numerical Study ◽  
Lower Surface ◽  
Horizontal Surface ◽  
Uniform Heat Flux ◽  
Cross Sectional ◽  
Vertical Side ◽  
Uniform Heat

Abstract Flow in a rectangular enclosure with a square horizontal cross-section and with a uniform heat flux applied across the lower horizontal surface and with the upper horizontal surface cooled to a uniform low temperature has been numerically studied. The vertical side-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 form of the governing equations, written in terms of the vector potential and vorticity vector functions and expressed in dimensionless form, have been solved using a finite-difference procedure based on the equations. The solution was started with no flow in the enclosure. The solution, in general, has the following parameters: the heat flux Rayleigh number Raq, the Prandtl number and the size A of the square cross-sectional shape compared to the height of the enclosure. Results have only been obtained for a Prandtl number of 0.7. Results for values of A between 0.5 and 3 for various relatively low and moderate values of Rayleigh number (up to 40000) have been obtained. The results have been used to determine the effect of A on the value of Raq below which there is no fluid motion in the enclosure and to examine the various flow patterns that arise as the value of Raq is increased with various values of A. The effect of A on the variation of the mean dimensionless lower surface temperature with Ra has also been examined.

Finite amplitude cellular convection

1958 ◽  
Vol 4 (3) ◽  
pp. 225-260 ◽  
Author(s):  
W. V. R. Malkus ◽  
G. Veronis
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Heat Transport ◽  
Stability Problem ◽  
Dominant Role ◽  
Linear Equations ◽  
Finite Amplitude ◽  
Set Up ◽  
Rayleigh Numbers ◽  
Linear Stability Problem

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

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 ).

A Numerical Study of Laminar and Turbulent Natural Convective Heat Transfer From an Isothermal Vertical Plate With a Wavy Surface

2010 ◽  
Author(s):  
Patrick H. Oosthuizen
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Turbulent Flow ◽  
Convective Heat Transfer ◽  
Transfer Rate ◽  
Surface Waviness ◽  
Dimensionless Amplitude ◽  
Buoyancy Forces ◽  
Rayleigh Numbers

Natural convective heat transfer from a wide isothermal plate which has a wavy surface, i.e., has a surface which periodically rises and falls, has been numerically studied. The main purpose of the study was to examine the effect of the surface waviness on the conditions under which transition from laminar to turbulent flow occurred and to study the effect of the surface waviness on the heat transfer rate. The surface waves, which have a saw-tooth cross-sectional shape, are normal to the direction of flow over the surface and have a relatively small amplitude. The range of Rayleigh numbers considered in the present study extends from values that for a non-wavy plate would be associated with laminar flow to values that would be associated with fully turbulent flow. The flow has been assumed to be steady and fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. A standard k-epsilon turbulence model with full account being taken of the effects of the buoyancy forces has been used in obtaining the solution. The solution has been obtained using the commercial CFD solver FLUENT. The solution has the following parameters: the Rayleigh number based on the plate height, the Prandtl number, the dimensionless amplitude of the surface waviness, and the dimensionless pitch of the surface waviness. Results have been obtained for a Prandtl number of 0.7 and for a single dimensionless pitch value for Rayleigh numbers between approximately 106 and 1012. The effects of Rayleigh number and dimensionless amplitude on the mean heat transfer rate have been studied. It is convenient in presenting the results to introduce two mean heat transfer rates, one based on the total surface area and the other based on the projected frontal area of the surface.

scholarly journals The global characteristics of the three-dimensional thermal convection inside a spherical shell

1997 ◽  
Vol 4 (1) ◽  
pp. 19-27 ◽  
Author(s):  
J. Arkani-Hamed
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Spherical Shell ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Boundary Layer Theory ◽  
Physical Parameters ◽  
Rayleigh Numbers

Abstract. The Rayleigh number-Nusselt number, and the Rayleigh number-thermal boundary layer thickness relationships are determined for the three-dimensional convection in a spherical shell of constant physical parameters. Several models are considered with Rayleigh numbers ranging from 1.1 x 102 to 2.1 x 105 times the critical Rayleigh number. At lower Rayleigh numbers the Nusselt number of the three-dimensional convection is greater than that predicted from the boundary layer theory of a horizontal layer but agrees well with the results of an axisymmetric convection in a spherical shell. At high Rayleigh numbers of about 105 times the critical value, which are the characteristics of the mantle convection in terrestrial planets, the Nusselt number of the three-dimensional convection is in good agreement with that of the boundary layer theory. At even higher Rayleigh numbers, the Nusselt number of the three-dimensional convection becomes less than those obtained from the boundary layer theory. The thicknesses of the thermal boundary layers of the spherical shell are not identical, unlike those of the horizontal layer. The inner thermal boundary is thinner than the outer one, by about 30- 40%. Also, the temperature drop across the inner boundary layer is greater than that across the outer boundary layer.

Free convection in an open thermosyphon, with special reference to turbulent flow

1955 ◽  
Vol 230 (1183) ◽  
pp. 502-530 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Boundary Layer Flow ◽  
Radius Ratio ◽  
Tube Length ◽  
Prandtl Numbers ◽  
Layer Flow ◽  
Rayleigh Numbers

This paper describes an experimental investigation of heat transfer by free convection of a fluid in a heated vertical tube, sealed at its lower end. Heated fluid adjacent to the wall is discharged from the open end into a suitably cooled large reservoir, while a central core of cool fluid is continuously drawn into the tube by way of replacement. The system constitutes an unusual case of natural convection because the two streams of fluid, moving in opposite directions, are compelled to create their own internal boundary. Such an arrangement forms a static simulation of the Schmidt system (1951) for cooling high-temperature gas turbine blades, where sealed radial passages in the blades communicate with a reservoir in the rotor drum, and large centrifugal accelerations replace that due to gravity in the static system. The use of a scaled-up static tube in large measure compensates for the relatively small gravitational acceleration, when determining the working range of Rayleigh numbers, in this case from 10 7 to 10 13 . These are based on tube length, the fluid property values being referred to tube-wall temperature. Separate assessments are made of the effect of fluid Prandtl number (covering values from 7600 to 0·69) and tube length radius ratio (ranging from 7·5 to 47·5). In laminar flow the former is not found to be significant, but the quotient of the Rayleigh number (based on radius) and tube length-radius ratio determines the ranges of three laminar flow régimes. High values of the quotient correspond to 'boundary-layer flow’ and greatest heat transfer. This is followed first by ‘impeded non-similarity flow’ and then by ‘impeded similarity flow’ as the quotient becomes smaller, where the two streams of fluid mingle. These findings are in close agreement with theoretical prediction (Lighthill 1953). Turbulence arises in two ways. For Prandtl numbers near unity, transition occurs during the laminar impeded-flow régimes, resulting in a mixing effect and reduced heat transfer. This is predicted by Lighthill, but his discussion of turbulent flow is restricted to a Prandtl number of unity. For larger Prandtl numbers, transition takes place during laminar boundary-layer flow, yielding a conventional turbulent boundary-layer régime with increased heat transfer. The mean transitional Grashof numbers (based on radius) are in the range 10 4.4 to 10 4.6 ; they compare favourably with a pre­dicted range of from 10 4.0 to 10 4.3 . The tendency for the cool entering fluid to become turbulent renders turbulent boundary-layer flow potentially unstable. Both modes of transition eventually lead to a stable ‘fully mixed' régime where the two turbulent streams mix. This causes reduced circulation and heat transfer, the extent of the reduction varying directly with length-radius ratio and inversely with Prandtl number. The régime was predicted by Lighthill, but there are considerable dis­crepancies between estimated and experimental heat-transfer rates, and in the duration of the régime. In practice it appears to persist indefinitely, whereas Lighthill forecasts its replace­ment at high Rayleigh numbers by a stable boundary-layer flow. Empirical correlations show that fully mixed flow yields optimum heat transfer at a length-radius ratio, which is determined by the Rayleigh number. The suitability of the Schmidt system for blade cooling is briefly discussed in the light of the investigation.

Natural Convective Heat Transfer Through a Liquid Between a Heated Horizontal Surface and a Vertical Fin Attached to a Cooled Surface

1999 ◽  
Author(s):  
Patrick H. Oosthuizen
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Prandtl Number ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Lower Surface ◽  
Horizontal Surface ◽  
Finite Element Procedure ◽  
Liquid Depth ◽  
Rayleigh Numbers

Abstract Natural convective heat transfer from a heated horizontal surface directly exposed to a liquid into which vertical fins, attached to a cooled horizontal surface, project vertically downwards has been numerically studied. It has been assumed that the flow is steady, laminar and two-dimensional. The governing equations have been written in terms of dimensionless variables and have been solved using the finite element procedure. The solution has the following parameters: the Rayleigh number, the Prandtl number, the dimensionless half-width i.e. the ratio of the half-distance between the fins to the liquid depth, the dimensionless gap between the bottom of the fin and the lower surface, and the ratio of the thermal conductivity of the fin material to that of the liquid. Because of the application being considered, the Prandtl number has been taken as 5. Solutions have been obtained for Rayleigh numbers of between 10 and 1,000,000 for dimensionless half-widths of between 1 and 0.2 for thermal conductivity ratios of 20, 10 and 4 for a range of dimensionless fin gaps and widths.

Relation between Nusselt number and Rayleigh number in turbulent thermal convection

1976 ◽  
Vol 73 (3) ◽  
pp. 445-451 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Similarity Theory ◽  
Unknown Constant ◽  
Velocity Defect ◽  
Fluid Layers ◽  
Rayleigh Numbers ◽  
Existing Data

A theory is developed for the dependence of the Nusselt number on the Rayleigh number in turbulent thermal convection in horizontal fluid layers. The theory is based on a number of assumptions regarding the behaviour in the molecular boundary layers and on the assumption of a buoyancy-defect law in the interior analogous to the velocity-defect law in flow in pipes and channels. The theory involves an unknown constant exponentsand two unknown functions of the Prandtl number. For eithers= ½ ors= 1/3, corresponding to two different theories of thermal convection, and for a given Prandtl number, constants can be chosen to give excellent agreement with existing data over nearly the whole explored range of Rayleigh numbers in the turbulent case. Unfortunately, comparisons with experiment do not permit a definite choice ofs, but consistency with the chosen form of the buoyancy-defect law seems to suggests= 1/3, corresponding to similarity theory.

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