Convection In Horizontal Layer Research Articles

Convection in horizontal layers with internal heat generation. Theory

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable.The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R, exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal (x, y) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Bénard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R, a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P, and which for water is about 3Rc), and provided the wave-number of the pattern lies within certain limits dependent on R and P.

Thermal Convection in Horizontal Fluid Layers

The problem of heat transfer by convection in a horizontal layer of fluid across which the imposed temperature gradient gives rise to an unstable density gradient is classical in the theory of fluid mechanics. The central question, of the instability to small perturbations of a layer of incompressible, non-radiating fluid contained between rigid horizontal boundaries, has been completely answered, and has provided the key to a number of investigations of the patterns of finite amplitude convection that do indeed occur in unstable conditions. In recent years an increased awareness of the importance of convective processes in many situations of geophysical and astrophysical interest has stimulated a considerable amount of theoretical work on convective instability and heat transport under conditions rather less restrictive than those of the central problem mentioned. For example, the effects of radiative heat transfer and compressibility have been taken into account, as has the effect of conditional instability in meteorology, and the penetration of convective motions into possibly stable bounding layers when the conditions of horizontal rigid boundaries are relaxed. The purpose of the present paper is to review and criticize these recent developments, and, it is hoped, to focus attention on the areas of greatest theoretical inadequacy. It is not intended that the paper should be merely a bibliography, nor does it claim to be complete in this respect, but rather an exposition of the present (early 1966) state of the theory. Consequently, certain papers, not necessarily the first or best in their field but those which in the writer's opinion expose important features of the problem, are widely quoted, whilst other similar papers may be only briefly acknowledged.

Natural Convection in Horizontal Liquid Layers

Natural convection in horizontal layers is examined. The eigenvalue problem yielding the characteristic Rayleigh numbers for the first ten modes is solved using linear theory. The contribution to heat transfer is determined and compared with experiment.

Experiments on the onset of thermal convection in horizontal layers of gases

Precise data are presented on the Rayleigh-Jeffreys instability in air, argon and carbon dioxide at pressures between 0·6 and 6·0 atm in layers of depths 1/8, 1/4, and 3/4in. A new method yields better resolution and repeatability than the methods employed in the past. Here the gas layer is brought through the state of marginal stability by increasing the pressure while both the temperature and the temperature difference are held constant. The onset of convection is detected by means of a type of heat flow meter ascribed to Prof. L. M. K. Boelter. The equipment and procedure are described and analysed in detail.The experimentally-determined value of the critical Rayleigh number is 1793, repeatable within a probable deviation of 1% due to random errors. Considering in addition the systematic errors of the instrumentation but not the uncertainties of property values, we place the absolute value within 1793 ± 80. The corresponding theoretical value is 1708.Finite rates of pressure rise inhibit the development of convection. The results achieved at the least rates of pressure rise are in agreement with steady-state determinations.The approach employed in this investigation shows great promise as a method for measuring the transport properties of gaseous mixtures or of gases at combined high temperature and high pressure.

Experiments on the inhibition of thermal convection by a magnetic field

Experiments on the magnetic inhibition of thermal convection in horizontal layers of mercury heated from below are described. A large 36½ in. cyclotron magnet reconditioned for hydromagnetic studies was used in these experiments. By using layers of mercury of depth 3 to 6 cm and magnetic fields of strength 500 to 8000 gauss, it has been possible to determine the dependence of the critical Rayleigh number for the onset of instability on the parameter Q 1 ( = σH 2 d 2 / π 2 ρν , where H denotes the strength of the field, σ the electrical conductivity, ν the coefficient of kinematic viscosity, ρ the density and d the depth of the layer) for Q 1 varying between 40 and 1·6 × 10 6 . The experiments fully confirm the theoretical relation derived by Chandrasekhar.