Showing papers on "Natural convection published in 1968"

Effect of a stabilizing gradient of solute on thermal convection

Abstract: A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Benard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

Freezing and shattering of supercooled water drops

Abstract: Supercooled water drops of about 0·05 cm radius were frozen at temperatures between −4°C and −25°C under various conditions. Explosive shattering and ejected ice splinters were observed when supercooled drops were frozen under conditions of free convection in hydrogen and helium at 1·0 atm pressure, in air below 0·13 atm pressure and in carbon dioxide below 0·05 atm pressure. Drops did not fracture when ventilated at terminal velocity unless they were rotated about an axis normal to the airstream during freezing. Fracture was also observed when either the drops were not allowed sufficient time to attain thermal and solution equilibrium with the environment before nucleation, or when drops were frozen in carbon dioxide at a pressure above 0·3 atm. A condition for drop fracture is the establishment of a strong shell of ice around the drop, which in turn depends on the distribution of heat transfer to the environment during freezing and on the effective thermal conductivity of the environment. The volume of gas dissolved in the drop is a secondary factor, except in the case of a very soluble gas such as carbon dioxide where splinters are produced by vigorous effervescence during freezing. In the atmosphere the shattering of an individual drop appears possible only if the drop rotates during freezing. This appears unlikely since any asymmetry in shape would tend to orient the drop as it fell.

Transient natural convection in a vertical cylinder

Abstract: An experimental and analytical study is reported of transient natural convection in a vertical cylinder. For the experiments a cylinder was partially filled with liquid and subjected to a uniform heat flux at the walls. Thermocouples were used to measure the unsteady temperature field within the liquid; dye tracers were used to study flow patterns. Parameters that were varied included the test liquid (water-glycerin mixtures), the liquid depth, and the wall heat flux. A range of Prandtl number from 2 to 8,000, L/D ratio from 1 to 3, and Grashof number from 103 to 1011 were studied, encompassing both laminar and turbulent flow regimes. An analytical model was developed by dividing the system into three regions: a thin boundary layer rising along the heated walls, a mixing region at the top where the boundary layer discharges and mixes with the upper core fluid, and a main core region which slowly falls in plug flow. The temperature of the core fluid was assumed to vary in the vertical direction but not in the horizontal direction. Natural convection boundary-layer equations were modified to allow for a temperature variation at the outer edge of the boundary layer. The model may be used with a step-by-step computational procedure to predict the temperature distribution in the fluid as a function of time for an arbitrary set of conditions. Results computed by using the model were in good agreement with the experimental data.

The stability of the laminar natural convection boundary layer

Abstract: This paper concerns the stability characteristics of laminar natural convection in external flows. Until recently, very little was known about such stability because of the inherent complexity of temperature-coupled flows and because of the complicated mechanisms of disturbance propagation. In this work the stability of the laminar natural convection boundary layer is examined more closely in an attempt to predict the experimental results recently obtained. In particular, it is shown that an important thermal capacity coupling exists between the fluid and the wall which generates the flow. This thermal capacity coupling is shown to have a first-order effect for particular Grashof-number wave-number products. Solutions are obtained for a Prandtl number of 0·733 and several values of relative wall thermal capacity. These solutions indicate the important role of this wall coupling. In particular, the results predict the experimental data previously obtained.In addition, solutions with ‘zero wall storage’ are obtained for a range of Prandtl numbers from 0·733 to 6·9. The relative disturbance u-velocity and temperature amplitudes and their phases are shown for Pr = 0·733 and several wall-storage parameters, and for Pr = 6·9 with zero wall storage. A comparison between the disturbance temperature distribution and the data obtained from a recent experimental investigation shows close agreement when the thermal capacity of the wall is taken into account.In the appendix, it is shown that for temperature-coupled flows and wall-coupled boundary conditions the flow is unstable at a lower Grashof number for two-dimensional disturbances than it is for three-dimensional disturbances. This result has been supported by the recent experimental observations.

The Influence of the Boundary Conditions and Rotation on Convection in the Earth's Mantle

Abstract: The external nonhydrostatic gravity field must be maintained by flow within the mantle. An analysis of axisymmetric poloidal circulation in a rotating compressible sphere shows it to be more important than was previously believed. However, such flow is probably unable to produce the observed nonhydrostatic equatorial bulge. The basic equations are nonlinear, but can be solved if the viscosity is sufficiently large. The conditions which must be satisfied before this approximation is valid are examined in some detail because of their relevance to convection in the mantle. The horizontal temperature differences between oceans and continents must also drive convection currents, though such flow can only be of local importance at present. Two-dimensional convection driven by horizontal temperature variations can successfully account for most of the phenomena which are believed to be the surface expressions of flow in the mantle. The only exception is the gravity anomaly over the rising current, which is positive in the model. Though this linear theory is not self consistent, it does show that convection of heat is the only important nonlinear term. Further understanding of the problem will require numerical analysis.

Laminar natural convection about an isothermally heated sphere at small Grashof number

Abstract: The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.

Haline convection induced by the freezing of sea water

Abstract: The onset of haline convection induced by the freezing of sea water is investigated. The linear analysis appropriate to the onset of thermal convection is applied to haline convection by replacing the temperature variables in the equations with salinity variables. Expressions for the time necessary for the onset of manifest convective behavior and the horizontal spacing of the convection cells are derived for various manners of increasing the salinity at the top surfaces. For molecular coefficients of viscosity and diffusion and cooling conditions that may be appropriate for the freezing season in the Antarctic it is found that the initial horizontal spacing of convection cells is about 0.3 cm.

Free convection from a flat plate

Abstract: A numerical solution is presented for the development of free convection from a semi-infinite vertical flat plate which is uniformly heated up to a length l from the base and insulated for the rest of its length. At great heights above the heated part of the plate, the velocity and temperature distributions behave as if the heat were put in as a line source of heat at the base of the plate. Matching of the solutions for the heated and the insulated parts of the plate, by keeping the fluxes of heat and momentum continuous, determines the position of the effective origin of the similarity solution for the insulated plate in terms of the length, l, of the heated part of the plate. Graphs of the dimensionless velocity, temperature, heat flux and axial length parameters are given for different values of the Prandtl number.

Laminar Free Convection From a Downward-Projecting Fin

Abstract: A theoretical analysis of conduction through and free convection from a tapered, downward-projecting fin immersed in an isothermal quiescent fluid is presented. The problem is solved by assuming quasi-one-dimensional heat conduction in the fin and matching the solution to that of the convection system, which is treated as a boundary layer problem. For an infinite Prandtl number, solutions are derived which take the form of a power law temperature distribution along the fin. The effect of this power (n) on heat transfer, drag, and the corresponding boundary layer profiles is discussed. It is shown that n is independent of the fin profile and dependent on a single nondimensional group χ. The theoretical results for infinite Prandtl number are compared with corresponding results derived from previous work using a Prandtl number of unity. The effect of Prandtl number on the determination of n and consequently the fin effectiveness is found to be extremely small. The results of an experimental program are also presented. These consist of temperature profiles and the n — χ relation for different fin geometries and surrounding fluids. Comparison with the theoretical predictions reveals good agreement.

A study of instability in free convection from an inclined plate

Abstract: The results of an experimental study of the stability of boundary layer free convection from an inclined plane surface are presented. In particular, the relative contributions of hydrodynamic and thermal effects are investigated. It is demonstrated that thermal effects, through surface inclinations, are increasingly significant as the surface departs further from the vertical position and that positive and negative inclinations do not produce equal and opposite effects.