Showing papers on "Rayleigh number published in 1968"

Onset of Thermohaline Convection in a Porous Medium

Abstract: The problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis. It is shown that oscillatory instability may be possible when a strongly stabilizing solute gradient is opposed by a destabilizing thermal gradient, but attention is concentrated on monotonic instability. The eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, is obtained, by a Fourier series method, for a general set of boundary conditions. Numerical solutions are found for some special limiting cases, extending existing results for the thermal problem. When the thermal and solute boundary conditions are formally identical, the net destabilizing effect is expressed by the sum of R and S.

On the thermal instability of a rotating-fluid sphere containing heat sources

Abstract: The theory of marginal convection in a uniformly rotating, self-gravitating, fluid sphere, of uniform density and containing a uniform distribution of heat sources, is developed to embrace modes of convection which are asymmetric with respect to the axis of rotation It is shown that these modes are the most unstable, except for the smallest Taylor numbers, T (a measure of the rotation rate); ie for any T and o) (Prandtl number), the lowest Rayleigh number (a measure of the temperature gradients in the sphere) is associated with an asymmetric motion This is demonstrated both by an expansion method suitable for small T, and by asymptotic theory for T oo For large T, the eigenmode most easily excited is small in amplitude everywhere except near a cylindrical surface, of radius about half that of the sphere, and coaxial with the diameter parallel to the angular velocity vector

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.

Finite amplitude convection with changing mean temperature. Part 1. Theory

Abstract: When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude e. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.

Large-amplitude Bénard convection in a rotating fluid

Abstract: Linear stability theory of Benard convection in a rotating fluid (Chandrasekhar 1961) has shown that fluids with large ([ges ] 1) Prandtl number, σ, exhibit behaviour markedly different from that of fluids with σ [les ] 1. This difference in behaviour extends also into the finite-amplitude range (Veronis 1959, 1966I). In this paper we report a numerical study of two-dimensional Be´nard convection in a rotating fluid confined between free boundaries, with σ = 6·8 and σ = 0·2 for the range of Taylor number 0 [les ] [Fscr ]2 [Lt ] 105 and for Rayleigh numbers, R, extending an order of magnitude from the critical value of linear stability theory. The behaviour of water (σ = 6·8) is dominated by the rotational constraint even for relatively moderate values (∼ 103) of [Fscr ]2. A study of the resultant velocity and temperature fields shows how rotation controls the system, with the principal behaviour reflected by the thermal wind balance; i.e. the horizontal temperature gradient is largely balanced by the vertical shear of the velocity component normal to the temperature gradient. A fluid with a small Prandtl number (σ = 0·2) becomes unstable to finite-amplitude disturbances at values of the Rayleigh number significantly below the critical value of linear stability theory. The subsequent steady vorticity and temperature fields exhibit a structure which is quite different from that of fluids with large σ. The rotational constraint is balanced primarily by non-linear processes in a limited range of Taylor number ([Fscr ]2 [les ] 103·6). For larger values of [Fscr ]2 the system first becomes unstable to infinitesimal oscillatory disturbances but a steady, finite-amplitude flow is established at supercritical values of R which are none the less smaller than the values that one would expect from linear theory. The ranges of Taylor number in which the above phenomena occur are different from those which were estimated on the basis of an earlier study (Veronis 1966 I) which made use of a minimal representation of the finite-amplitude velocity and temperature fields. No subcritical, finite-amplitude oscillatory motions were found in the present study. Comparison with some of the experimental features observed and reported by Rossby (1966) is also discussed and it is pointed out that some of the differences between theory and experiment may be traced to the restrictive conditions (two-dimensionality and free boundaries) of the present study.

Finite amplitude convection with changing mean temperature. Part 2. An experimental test of the theory

Abstract: It has been found in part 1 (Krishnamurti 1968) that when the mean temperature of a fluid layer is changing at a constant rate η, hexagonal flows are stable in a range of Rayleigh numbers near the critical. The direction of flow depends upon the sign of η. The static state is unstable to finite amplitude disturbances at Rayleigh numbers below the critical point predicted by linear theory. The validity of this theory is tested in an experiment in which the heat flux is measured as a function of η and Rayleigh number. The horizontal plan form is determined from the side by continuously exposing a photographic film moving in a vertical direction as tracers in different regions of the fluid are illuminated. Finite amplitude instability and hexagonal cells are indeed observed.

An asymptotic solution for large Prandtl number free convection

Abstract: The set of ordinary differential equations governing free convection boundary layer flow past an isothermal semi-infinite vertical flat plate is solved for large Prandtl numbers by means of the method of matched asymptotic expansions. The analysis leads to an expression for heat transfer which contains the Prandtl number explicitly and which is very accurate for sufficiently large values of the Prandtl number. On the other hand the analysis also has qualitative assets. Before choosing the mathematical method of solution, the physical aspects of the large Prandtl number free convection boundary layer are investigated. The mathematical solution serves to enlarge our understanding of the physical implications of a free convection boundary layer in a large Prandtl number fluid.

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 Rayleigh-Jeffreys problem with boundary slab of finite conductivity

Abstract: Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of infinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).

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.

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.

Stability of a Fluid Layer with Time‐Dependent Density Gradients

Abstract: The stability of a fluid layer with adverse time‐dependent density gradients is examined by two computational techniques. The most direct solution, that of integrating the time‐dependent equations, is compared with the results obtained by invoking the often used quasistationary state approximation. It is seen that this latter approximation is not valid if the instability stems from a sudden change in the concentration or temperature at one surface of the fluid layer. The particular problem studied here is that of a liquid film resting on an impermeable wall with gas absorption into the liquid. In this case the system evolves from one stable state to another with the essential problem being to assess the stability of the system during its evolution between these stable states. It is shown that a single critical Rayleigh number does not exist as in the usual case with linear time‐independent density gradients, but that the critical Rayleigh number depends, to some extent, on intangible factors such as the initial perturbation magnitude and the mode of observation.

Onset of Convection in a Layer of Water Formed by Melting Ice from Below

Abstract: The onset of convection, or the critical Rayleigh number in a layer of water formed continuously by melting ice from below, has been determined experimentally. Homogeneous, bubble‐free ice was prepared, and used in all the experiments. The critical Rayleigh number Rac for a fluid undergoing phase change and density inversion is not a single value but may be correlated empirically as a function of warm plate temperature T8 by Rac = 14 200 exp (−6.64 × 10−2T8). This relation is valid for T8 varying from 6.72‐25.50°C. The initial ice sample temperature T0 was varied from −4.8 to −22.00°C. The effect of T0 was found to be insignificant.

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.

Thermal free convection from an ice sphere in water

Abstract: Thermal free convection in water is studied by melting ice spheres in water, the uniform temperature of which is varied between 0 and 10°C. Flow patterns as well as local heat transfer are examined. In particular, the effect of anomalous thermal expansion is investigated. From our observations a more accurate picture of the flow phenomena can be obtained that agrees not only with our experimental heat transfer data but also with theoretical results from existing literature.

Classical cellular convection with a spatial heat source

Abstract: This paper considers the problem of the stability of an infinite horizontal layer of a viscous fluid which loses heat throughout its volume at a constant rate. The variation of the critical Rayleigh number, R t , and the cell aspect ratio, a , with the rate of heat loss, is calculated with two sets of boundary conditions corresponding to two free and two rigid boundaries. In both cases we find that, as the rate of heat loss increases, R t decreases, showing that the layer becomes more unstable, and a increases, showing that the cells become narrower. We also consider the possibility that a double layer of cells is formed for large values of the rate of heat loss, by the stable layer at the top, and find that this does not occur while the temperature of the upper surface of the layer is less than that of the lower.