Showing papers on "Rayleigh number published in 1976"

An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders

Abstract: An experimental and theoretical-numerical investigation has been carried out to extend existing knowledge of velocity and temperature distributions and local heat-transfer coefficients for naturel convection within a horizontal annulus. A Mach—Zehnder interferometer was used to determine temperature distributions and local heat-transfer coefficients experimentally. Results were obtained using water and air at atmospheric pressure with a ratio of gap width to inner-cylinder diameter of 0·8. The Rayleigh number based on the gap width varied from 2·11 × 104to 9·76 × 105. A finite-difference method was used to solve the governing constant-property equations numerically. The Rayleigh number was changed from 102 to 105 with the influence of Prandtl number and diameter ratio obtained near a Rayleigh number of 104. Comparisons between the present experimental and numerical results under similar conditions show good agreement.

Thermal convection with strongly temperature-dependent viscosity

Abstract: This paper experimentally investigates the heat transport and structure of convection in a high Prandtl number fluid layer whose viscosity varies by up to a factor of 300 between the boundary temperatures. An appropriate definition of the Rayleigh number R uses the viscosity at the average of the top and bottom boundary temperatures. With rigid boundaries and heating from below, the Nusselt number N normalized with the Nusselt number N0 of a constant-viscosity fluid decreases slightly as the viscosity ratio increases. The drop is 12% at a variation of 300. A slight dependence of N/N0 on R is consistent with a decrease in the exponent in the relation N ∝ Rβ from its constant-viscosity value of 0·281 to 0·25 for R [lsim ] 5 × 104. This may be correlated with a transition from three- to two-dimensional flow. At R ∼ 105 and viscosity variation of 150, the cell structure is still dominated by the horizontal wavelength of the marginally stable state. This is true with both free and rigid upper boundaries. The flow is strongly three-dimensional with a free upper boundary, while it is nearly two-dimensional with a rigid upper boundary.

Nonlinear double-diffusive convection

Abstract: The two-dimensional motion of a fluid confined between two long horizontal planes, heated and salted from below, is examined. By a combination of perturbation analysis and direct numerical solution of the governing equations, the possible forms of large-amplitude motion are traced out as a function of the four non-dimensional parameters which specify the problem: the thermal Rayleigh number RT, the saline Rayleigh number ES, the Prandtl number σ and the ratio of the diffusivities τ. A branch of time-dependent asymptotic solutions is found which bifurcates from the linear oscillatory instability point. In general, for fixed σ, τ and RS, as RT increases three further abrupt transitions in the form of motion are found to take place independent of the initial conditions. At the first transition, a rather simple oscillatory motion changes into a more complicated one with different structure, at the second, the motion becomes aperiodic and, at the third, the only asymptotic solutions are time independent. Disordered motions are thus suppressed by increasing RT. The time-independent solutions exist on a branch which, it is conjectured, bifurcates from the time-independent linear instability point. They can occur for values of RT less than that at which the third transition point occurs. Hence for some parameter ranges two different solutions exist and a hysteresis effect occurs if solutions obtained by increasing RT and then decreasing RT are followed. The minimum value of RT for which time-independent motion can occur is calculated for fourteen different values of σ, τ and RS. This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of σ and RS and the smaller values of τ, is less than the critical value of time-dependent linear theory.

Natural Convection in Enclosed Spaces—A Review of Application to Solar Energy Collection

Abstract: A useful solar-thermal converter requires effective control of heat losses from the hot absorber to the cooler surroundings. Based upon the theory and some experimental measurements it is shown that the spacing between the tilted hot solar absorber and successive glass covers should be in the range 4 to 8 cm to assure minimum gap conductance. Poor choice of spacing can significantly affect thermal conversion efficiency, particularly when the efficiency is low or when selective black absorbers are used. Recommended data for gap Nusselt number are presented as a function of the Rayleigh number for the high aspect ratios of interest in solar collector designs. It is also shown that a rectangular cell structure placed over a solar absorber is an effective device to suppress natural convection, if designed with the proper cell spacing d, height to spacing ratio L/d and width to spacing ratio W/d needed to give a cell Rayleigh number less than the critical value.

Studies of finite amplitude non‐Newtonian thermal convection with application to convection in the Earth's mantle

Abstract: Studies of non-Newtonian thermal convection have been made to determine the importance of non-Newtonian viscosity on flow in the earth's mantle. Finite difference solutions have been obtained with a viscosity law representing the sum of deformation rates due to diffusion and dislocation creep. Non-Newtonian flow structures differ only slightly from corresponding Newtonian flows. The principal effect of the non-Newtonian viscosity is to increase the effective Rayleigh number. An effective Rayleigh number based on a strain rate squared averaged viscosity provides a good correlation between Newtonian and non-Newtonian flows over a wide range of Rayleigh numbers.

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

Abstract: 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.

Onset of Convection in a Porous Medium With Internal Heat Generation

Abstract: The conditions leading to the onset of thermal convection in a horizontal porous layer are determined analytically using the method of linear stability of small disturbances. The lower boundary is treated as a rigid surface and the upper boundary as a free surface. The critical internal and external Rayleigh numbers are determined for both stabilizing and destabilizing boundary temperatures. The predicted critical external Rayleigh number in the limit of no heat generation is in agreement with the critical number predicted for a porous medium heated from below. (16 refs.)

Transient combined laminar free convection and radiation in a rectangular enclosure

Abstract: The mass, momentum and energy-transfer equations are solved to determine the response of a rectangular enclosure to a fire or other high-temperature heat source. The effects of non-participating radiation, wall heat conduction, and laminar natural convection are examined. The results indicate that radiation dominates the heat transfer in the enclosure and alters the convective flow patterns significantly. At a dimensionless time of 5·0 the surface of the wall opposite a vertical heated wall has achieved over 99% of the hot-wall temperature when radiation is included but has yet to change from the initial temperature for pure convection in the enclosure. At the same time the air at the centre of the enclosure achieves 33% and 13% of the hot-wall temperature with and without radiation, respectively. For a hot upper wall the convection velocities are not only opposite in direction but an order of magnitude larger when radiation transfer between the walls is included.

Numerical models for hydrothermal circulation in the oceanic crust

Abstract: Two-dimensional numerical calculations of hydrothermal circulation in permeable oceanic crust have been carried out. The effects of Rayleigh number, impermeable and permeable upper boundaries, and permeability variations with depth have been investigated. Flow and temperature fields as well as surface heat flux distributions are presented. Spatial distributions of surface heat flux are compared with observations at the Galapagos spreading center. The hydrothermal circulation alters the spatial distribution of surface heat flow but not its mean value. It is concluded that the oceanic crust has a permeability of about 4.5×10−12 cm2. With a permeable upper boundary the mass of seawater circulating through the crust equals the mass of the crust in about 2.1 m.y. A corresponding typical circulation time is estimated to be about 4 years.

Axisymmetric convection in a cylinder

Abstract: In three-dimensional BBnard convection regions of rising and sinking fluid are dissimilar. This geometrical effect is studied for axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries. Near the critical Rayleigh number R, the solution is obtained from a perturbation expansion, valid only if both the Reynolds number and the PBclet number are small. For values of the Nusselt number N 1 there is a viscous regime with N M 2(R/Rc)i; when R/R, 2 pb, N increases more rapidly, approximately as R0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.

Convective heat transfer in a liquid saturated porous layer

Abstract: The convective heat transfer in fluid-saturated porous beds either heated from below or heated by distributed sources is investigated for several bed thicknesses and permeabilities. For the case of heating from below, Rayleigh numbers range from about 10 to 10,000. For distributed heat sources, Rayleigh numbers range from about 10 to 1,000. Critical Rayleigh numbers for the onset of convection are estimated as 38 for heating from below and 31.8 for distributed heat sources. Heat transfer results for convection induced by heating from below are in good agreement with analytical upper bound estimates obtained by Gupta and Joseph. (12 refs.)

An experimental study of turbulent convection in air

Abstract: An experiment was performed in a 3·5 by 3·5 m variable-height, closed convection box, with conditions ranging from a Rayleigh number of 4 × 104 up to 7 × 109, using air as the working fluid. Heat-flux measurements made at Rayleigh numbers up to 7 × 109 yielded a Nusselt number Nu = 0·13Ra0·30. Velocities and temperatures were measured up to Ra = 1·7 × 107, and Fourier spectra calculated to find the predominant horizontal scales of the motion midway between the boundaries. The predominant scale at Ra ∼ 105 was approximately four times the distance between plates, changing to six as Ra increased to 106. With side walls introduced so that the transverse aspect ratio was equal to five, Fourier spectra indicated considerable smaller scale motions, approximately equal to the layer depth. These motions decreased in size as Ra was increased. The results are discussed in relation to previous experimental and theoretical work.

Numerical simulation of three-dimensional Benard convection in air

Abstract: A numerical model is developed to simulate three-dimensional Benard convection. This model is used to investigate thermal convection in air for Rayleigh numbers between 4000 and 25000. According to experiments, this range of Rayleigh numbers in air covers three regimes of thermal convection: (i) steady two-dimensional convection, (ii) time-periodic convection and (iii) aperiodic convection. Numerical solutions are obtained for each of these regimes and the results are compared with the available experimental data and theoretical predictions.At the Rayleigh number Ra = 4000 the present model is able to produce experimentally realistic wavelengths for the two-dimensional convection. The small amplitude wave disturbances at Ra = 6500 have period τ = 0·24. When they become finite amplitude travelling waves, the period is τ = 0·27. These values are in good agreement with theoretical and experimental results. A detailed study of the form of these waves and of their energetics is given in appendix A. As the Rayleigh number is increased to Ra = 9000 and 25 000, the convection manifests progressively more complex spatial and temporal variations.The vertical heat transport and other mean properties of the convection are calculated for the range of Ra considered and compared with experimental and theoretical data. A detailed comparison is also made between the mean properties of two- and three-dimensional convection at the larger values of Ra. It is found that the heat flux Nu is nearly independent of the dimensionality of the convection.

Relation between Nusselt number and Rayleigh number in turbulent thermal convection

Abstract: 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 exponent s and two unknown functions of the Prandtl number. For either s = ½ or s = 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 of s, but consistency with the chosen form of the buoyancy-defect law seems to suggest s = 1/3, corresponding to similarity theory.