A Numerical Investigation of Thermal Convection in a Heat-Generating Fluid Layer

1980 ◽  
Vol 102 (3) ◽  
pp. 531-537 ◽  
Author(s):  
A. A. Emara ◽  
F. A. Kulacki
Keyword(s):  
Thermal Convection ◽  
Large Scale ◽  
Fluid Layer ◽  
Lower Surface ◽  
Rectangular Domain ◽  
Temperature Fields ◽  
Free Boundaries ◽  
Low Velocity ◽  
Nusselt Numbers ◽  
Rayleigh Numbers

Finite difference solutions of the equations governing thermal convection driven by uniform volumetric energy sources are presented for two-dimensional flows in a rectangular domain. The boundary conditions are a rigid, (i.e., zero slip), zero heat-flux lower surface, rigid adiabatic sides, and either a rigid or free (i.e., zero shear) isothermal upper surface. Computations are carried out for Prandtl numbers from 0.05 to 20 and Rayleigh numbers from 5 × 104 to 5 × 108. Nusselt numbers and average temperature profiles within the layer are in good agreement with experimental data for rigid-rigid boundaries. For rigid-free boundaries, Nusselt numbers are larger than in the former case. The structure of the flow and temperature fields in both cases is dominated by rolls, except at larger Rayleigh numbers where large-scale eddy transport occurs. Generally, low velocity upflows over broad regions of the layer are balanced by higher velocity downflows when the flow exhibits a cellular structure. The hydrodynamic constraint at the upper surface and the Prandtl number are found to influence only the detailed nature of flow and temperature fields. No truly steady velocity and temperature fields are found despite the fact that average Nusselt numbers reach steady values.

scholarly journals Natural convection air flow in vertical upright-angled triangular cavities under realistic thermal boundary conditions

2016 ◽  
Vol 20 (5) ◽  
pp. 1407-1420 ◽  
Author(s):  
Jaime Sieres ◽  
Antonio Campo ◽  
José Martínez-Súarez
Keyword(s):  
Natural Convection ◽  
Vertical Wall ◽  
Aperture Angle ◽  
Temperature Fields ◽  
Uniform Heat Flux ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Horizontal Wall ◽  
Rayleigh Numbers

This paper presents an analytical and numerical computation of laminar natural convection in a collection of vertical upright-angled triangular cavities filled with air. The vertical wall is heated with a uniform heat flux; the inclined wall is cooled with a uniform temperature; while the upper horizontal wall is assumed thermally insulated. The defining aperture angle ? is located at the lower vertex between the vertical and inclined walls. The finite element method is implemented to perform the computational analysis of the conservation equations for three aperture angles ? (= 15?, 30? and 45?) and height-based modified Rayleigh numbers ranging from a low Ra = 0 (pure conduction) to a high 109. Numerical results are reported for the velocity and temperature fields as well as the Nusselt numbers at the heated vertical wall. The numerical computations are also focused on the determination of the value of the maximum or critical temperature along the hot vertical wall and its dependence with the modified Rayleigh number and the aperture angle.

An Analytical Study of Natural Convective Heat Transfer within a Trapezoidal Enclosure

1980 ◽  
Vol 102 (4) ◽  
pp. 640-647 ◽  
Author(s):  
L. Iyican ◽  
Y. Bayazitogˇlu ◽  
L. C. Witte
Keyword(s):  
Heat Transfer ◽  
Analytical Study ◽  
Temperature Fields ◽  
Nusselt Numbers ◽  
Method Of Solution ◽  
Different Temperatures ◽  
Nonlinear Form ◽  
Trapezoidal Enclosure ◽  
Energy Equations ◽  
Rayleigh Numbers

The natural convection motion and the heat transfer within a trapezoidal enclosure with parallel cylindrical top and bottom walls at different temperatures and plane adiabatic sidewalls are studied. Two-dimensional natural convective fields for a range of Rayleigh numbers, up to 2.7 × 106, and enclosure tilt angles, 0 to 180 deg measured from vertical, are investigated. The Galerkin’s method of solution is applied to nonlinear form of the momentum and energy equations to determine the velocity and temperature fields. The average and local Nusselt numbers are also presented.

Correlation Equations for Turbulent Thermal Convection in a Horizontal Fluid Layer Heated Internally and from Below

1978 ◽  
Vol 100 (3) ◽  
pp. 416-422 ◽  
Author(s):  
F. B. Cheung
Keyword(s):  
Boundary Layer ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Lower Surface ◽  
Rate Variation ◽  
Correlation Equations ◽  
Volumetric Heating ◽  
Local Boundary ◽  
Simple Boundary ◽  
Horizontal Fluid Layer

High Rayleigh number thermal convection in a horizontal fluid layer with uniform volumetric energy sources and a constant rate of bottom heating is studied analytically by a simple boundary layer approach. Heat transfer characteristics of the layer are defined in terms of local boundary-layer variables. Correlation equations are derived for the upper and the lower surface Nusselt numbers as functions of two independent Rayleigh numbers, based respectively on the surface to surface temperature difference and the volumetric heating rate. Variation of the turbulent core temperature, which so far has not been determined successfully by existing analytical methods, is obtained. This is found to depend on a single dimensionless parameter which measures the relative rates of internal and external heating. Results of this study are presented with available experimental data.

On the occurrence of cellular motion in Bénard convection

1967 ◽  
Vol 30 (4) ◽  
pp. 651-661 ◽  
Author(s):  
E. Palm ◽  
T. Ellingsen ◽  
B. Gjevik
Keyword(s):  
Kinematic Viscosity ◽  
Silicone Oil ◽  
Fluid Layer ◽  
Critical Value ◽  
Free Boundaries ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Numbers ◽  
Effect Of Surface ◽  
Cellular Motion

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.

scholarly journals Plume generation in natural thermal convection at high Rayleigh and Prandtl numbers

2001 ◽  
Vol 434 ◽  
pp. 1-21 ◽  
Author(s):  
C. LITHGOW-BERTELLONI ◽  
M. A. RICHARDS ◽  
C. P. CONRAD ◽  
R. W. GRIFFITHS
Keyword(s):  
Boundary Layer ◽  
Thermal Convection ◽  
Large Scale ◽  
Lower Boundary ◽  
Secular Change ◽  
Scaling Analysis ◽  
Temperature Dependent Viscosity ◽  
Earth’S Mantle ◽  
Dependent Viscosity ◽  
Rayleigh Numbers

We study natural thermal convection of a fluid (corn syrup) with a large Prandtl number (103–107) and temperature-dependent viscosity. The experimental tank (1 × 1 × 0.3m) is heated from below with insulating top and side boundaries, so that the fluid experiences secular heating as experiments proceed. This setup allows a focused study of thermal plumes from the bottom boundary layer over a range of Rayleigh numbers relevant to convective plumes in the deep interior of the Earth's mantle. The effective value of Ra, based on the viscosity of the fluid at the interior temperature, varies from 105 at the beginning to almost 108 toward the end of the experiments. Thermals (plumes) from the lower boundary layer are trailed by continuous conduits with long residence times. Plumes dominate flow in the tank, although there is a weaker large-scale circulation induced by material cooling at the imperfectly insulating top and sidewalls. At large Ra convection is extremely time-dependent and exhibits episodic bursts of plumes, separated by periods of quiescence. This bursting behaviour probably results from the inability of the structure of the thermal boundary layer and its instabilities to keep pace with the rate of secular change in the value of Ra. The frequency of plumes increases and their size decreases with increasing Ra, and we characterize these changes via in situ thermocouple measurements, shadowgraph videos, and videos of liquid crystal films recorded during several experiments. A scaling analysis predicts observed changes in plume head and tail radii with increasing Ra. Since inertial effects are largely absent no transition to ‘hard’ thermal turbulence is observed, in contrast to a previous conclusion from numerical calculations at similar Rayleigh numbers. We suggest that bursting behaviour similar to that observed may occur in the Earth's mantle as it undergoes secular cooling on the billion-year time scale.

scholarly journals Numerical simulation of turbulent convection over wavy terrain

1992 ◽  
Vol 237 ◽  
pp. 261-299 ◽  
Author(s):  
Kilian Krettenauer ◽  
Ulrich Schumann
Keyword(s):  
Numerical Simulation ◽  
Large Scale ◽  
Fluid Layer ◽  
Domain Size ◽  
Lower Surface ◽  
Constant Heat Flux ◽  
Computational Domain ◽  
Turbulent Structures ◽  
Convective Boundary ◽  
Scale Motion

Thermal convection of a Boussinesq fluid in a layer confined between two infinite horizontal walls is investigated by direct numerical simulation (DNS) and by large-eddy simulation (LES) for zero horizontal mean motion. The lower-surface height varies sinusoidally in one horizontal direction while remaining constant in the other. Several cases are considered with amplitude δ up to 0.15H and wavelength λ of H to 8H (inclination up to 43°), where H is the mean fluid-layer height. Constant heat flux is prescribed at the lower surface of the initially at rest and isothermal fluid layer. In the LES, the surface is treated as rough surface (z0 = 10−4H) using the Monin-Oboukhov relationships. At the flat top an adiabatic frictionless boundary condition is applied which approximates a strong capping inversion of an atmospheric convective boundary layer. In both horizontal directions, the model domain extends over the same length (either 4H or 8H) with periodic lateral boundary conditions.We compare DNS of moderate turbulence (Reynolds number based on H and on the convective velocity is 100, Prandtl number is 0.7) with LES of the fully developed turbulent state in terms of turbulence statistics and Characteristic large-scale-motion structures. The LES results for a flat surface generally agree well with the measurements of Adrian et al. (1986). The gross features of the flow statistics, such as profiles of turbulence variances and fluxes, are found to be not very sensitive to the variations of wavelength, amplitude, domain size and resolution and even the model type (DNS or LES), whereas details of the flow structure are changed considerably. The LES shows more turbulent structures and larger horizontal scales than the DNS. To a weak degree, the orography enforces rolls with axes both perpendicular and parallel to the wave crests and with horizontal wavelengths of about 2H to 4H. The orography has the largest effect for λ = 4H in the LES and for λ = 2H in the DNS. The results change little when the size of the computational domain is doubled in both horizontal directions. Most of the motion energy is contained in the large-scale structures and these structures are persistent in time over periods of several convective time units. The motion structure persists considerably longer over wavy terrain than over flat surfaces.

Effect Of Dust Particles On Thermal Convection In A Ferromagnetic Fluid

2005 ◽  
Vol 60 (7) ◽  
pp. 494-502 ◽  
Author(s):  
◽  
Anupama Sharma ◽  
Divya Sharma ◽  
R. C. Sharma
Keyword(s):  
Rayleigh Number ◽  
Thermal Convection ◽  
Uniform Magnetic Field ◽  
Fluid Layer ◽  
Dust Particles ◽  
Free Boundaries ◽  
Stationary Convection ◽  
Ferromagnetic Fluid ◽  
Critical Wavenumber ◽  
Thermal Rayleigh Number

This paper deals with the theoretical investigation of the effect of dust particles on the thermal convection in a ferromagnetic fluid subjected to a transverse uniform magnetic field. For a flat ferromagnetic fluid layer contained between two free boundaries, the exact solution is obtained, using a linear stability analysis. For the case of stationary convection, dust particles and non-buoyancy magnetization have always a destabilizing effect. The critical wavenumber and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of the buoyancy magnetization parameter M1. The results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because the heat capacity of the clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles. The oscillatory modes are introduced by the dust particles. A sufficient condition for the non-existence of overstability is also obtained.

A Numerical Study of Local and Average Natural Convection Nusselt Numbers for Simultaneous Convection Above and Below a Uniformly Heated Horizontal Thin Plate

1997 ◽  
Vol 119 (1) ◽  
pp. 102-108 ◽  
Author(s):  
B. Chambers ◽  
Tien-Yu T. Lee
Keyword(s):  
Natural Convection ◽  
Power Law ◽  
Numerical Study ◽  
Lower Surface ◽  
Plate Edge ◽  
Nusselt Numbers ◽  
Plate Center ◽  
Heated Plates ◽  
Rayleigh Numbers ◽  
Plate Width

Numerical simulations were conducted to determine local and average natural convection Nusselt numbers for uniformly heated horizontal plates with convection occurring simultaneously from upper and lower surfaces. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 86 to 1.9 × 108. Upper surface Nusselt numbers were found to be smaller than corresponding lower surface Nusselt numbers. The local Nusselt number was largest at the plate edge and decreased towards the plate center for both surfaces. This variation followed approximately a minus 1/3-power law variation with the non-dimensionalized x coordinate on the upper surface for modified Rayleigh numbers greater than 104, and a minus 1/9-power law variation on the lower surface for all modified Rayleigh numbers. Comparative simulations were also performed for upward and downward facing uniformly heated plates (single sided convection). For these cases, Nusselt numbers on the upward facing plates were larger than for downward facing plates.

Three-dimensional thermal convection in a spherical shell

1989 ◽  
Vol 206 ◽  
pp. 75-104 ◽  
Author(s):  
D. Bercovici ◽  
G. Schubert ◽  
G. A. Glatzmaier ◽  
A. Zebib
Keyword(s):  
Heat Flow ◽  
Spherical Shell ◽  
Thermal Convection ◽  
Flow Direction ◽  
Three Dimensional ◽  
Maximum Velocity ◽  
Temperature Fields ◽  
Free Boundaries ◽  
Onset Of Convection ◽  
Axisymmetric Solution

Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra, the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra.

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