Showing papers on "Rayleigh number published in 1989"

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

Abstract: An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to . A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Transition to chaos in a differentially heated vertical cavity

Abstract: We investigate numerically the transition from laminar to chaotic flow of a Boussinesq fluid with Pr = 0.71 in two-dimensional closed, differentially heated, vertical cavities having aspect ratios near unity. The cavities have rigid conducting sidewalls, and rigid insulating top and bottom walls. The physical nature of the resulting flow is a function of the aspect ratio and Rayleigh number.It is shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, and chaotic flow exist for the flow regimes investigated. We find that for aspect ratios of approximately three or larger the the first transition from steady-state is due to instability of the sidewall boundary layers, while for small aspect ratios, but larger than ½, it is due to internal waves near the departing corners. For both instabilities we obtain the critical Rayleigh number as a function of aspect ratio and write expressions relating the fundamental frequencies of the oscillatory flow to the Rayleigh number and aspect ratio. When Ra is increased significantly above the first critical value, the flow becomes complex since both types of instabilities can be present. With a further increase in Rayleigh number the flow becomes chaotic and eventually turbulent. The above results are illustrated for different Rayleigh numbers and aspect ratios using time histories, spectral analysis, and streamlines at different values of time.

Turbulence in helium-gas free convection.

Abstract: Results on a Rayleigh-B\'enard experiment in helium gas at 5 K in a cylindrical cell of aspect ratio 1 are presented. The Rayleigh number spans a range from ${10}^{5}$ to ${10}^{12}$. A large-scale coherent flow is observed via the correlation of two adjacent temperature probes. This flow-velocity measurement shows clear transitions between different turbulent states. In hard turbulence, the dimensionless velocity [V/(\ensuremath{\kappa}/L)] scales with the Rayleigh number, with an exponent close to 1/2. The horizontal temperature difference across the cell is another measure of the different turbulent states. The temperature signals in the side-wall region (the large mean vertical velocity region) give clear pictures of various turbulent states. The measured velocity has been compared with the calculated free-fall velocity and also the heat transfer rate with the one calculated from the flow advection. The coherent frequency ${\ensuremath{\omega}}_{p}$ is found to be associated with the large-scale flow. In the side-wall region the power spectrum of the local temperature signal has a power-law dependence for Rayleigh numbers between ${10}^{8}$ and ${10}^{11}$. Both the exponent and the range of the power law change with the Rayleigh number. For Rayleigh numbers above ${10}^{11}$, a power law independent of Rayleigh number (exponent 1.4) develops at low frequency.

Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth

Abstract: The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant micro-organisms is examined for a layer of finite depth. In the steady basic state there is no bulk fluid motion, and the upwards swimming of the cells is balanced by diffusion resulting from randomness in their shape, orientation and swimming behaviour. This leads to a bulk density stratification with denser fluid on top. The theory is based on the continuum model of Pedley, Hill & Kessler (1988), and employs both asymptotic and numerical analysis. The suspension is characterized by five dimensionless parameters: a Rayleigh number, a Schmidt number, a layer-depth parameter, a gyrotaxis number G, and a geometrical parameter measuring the ellipticity of the micro-organisms. For small values of G, the most unstable mode has a vanishing wavenumber, but for sufficiently large values of G, the predicted initial wavelength is finite, in agreement with experiments. The suspension becomes less stable as the layer depth is increased. Indeed, if the layer is sufficiently deep an initially homogeneous suspension is unstable, and the equilibrium state does not form. The theory of Pedley, Hill & Kessler (1988) for infinite depth is shown to be appropriate in that case. An unusual feature of the model is the existence of overstable or oscillatory modes which are driven by the gyrotactic response of the micro-organisms to the shear at the rigid boundaries of the layer. These modes occur at parameter values which could be realized in experiments.

High Rayleigh Number Laminar Natural Convection in an Asymmetrically Heated Vertical Channel

Abstract: Experiments have been performed to determine local heat transfer data for the natural convective flow of air between vertical parallel plates heated asymmetrically. A uniform heat flux was imposed along one heated wall, with the opposing wall of the channel being thermally insulated. Local temperature data along both walls were collected for a wide range of heating rates and channel wall spacings corresponding to the high modified Rayleigh number natural convection regime. Laminar flow prevailed in all experiments. Correlations are presented for the local Nusselt number as a function of local Grashof number along the channel. The dependence of both average Nusselt number and the maximum heated wall temperature on the modified Rayleigh number is also explored. Results are compared to previous analytical and experimental work with good agreement.

A reassessment of the heat transport by variable viscosity convection with plates and lids

Abstract: The heat transport by a viscous fluid with temperature dependent viscosity has been studied numerically. As opposed to previous models, the top surface of the fluid clearly defines a tectonic plate with horizontally uniform velocity and subduction. Past studies failed to incorporate plates resulting in inefficient heat transport. With tectonic plates, the heat transport is as efficient as Rayleigh-Benard convection with constant viscosity; there is a strong buffering between internal temperature and heat loss. Past studies of parameterized convection which incorporated parameters indicative of a strong buffering between internal temperature and total heat output still provide the most physically plausible representation of the Earth's thermal evolution.

Compositional convection in viscous melts

Abstract: DURING solidification of multi-component melts, gradients in temperature and composition develop on different scales because of the large difference between their respective molecular diffusivities. Two consequences are the development of double-diffusive convection1 and the creation of mushy zones in which solid and liquid intimately coexist with a complex small-scale geometry2,3. Theoretical analysis requires simplifying assumptions that must be verified by laboratory experiments. Hitherto, experiments have been carried out with aqueous solutions which do not accurately represent the dynamics of melts with high Prandtl numbers, such as magmas. Here we describe the characteristics of compositional convection using a new experimental technique which allows the viscosity of the solution to be varied independently of chemical composition and liquidus temperature. A supereutectic melt was cooled from below, causing the growth of a horizontal layer of crystals. Convective instability occurred when the local solutal Rayleigh number of the compositional boundary layer ahead of the advancing crystallization front attained a value of ∼3 on average. We observed a novel regime of convection in which the thermal boundary layer above the crystallization front was essentially unmodified by the motion of the plumes. The plumes carried a small heat flux and did not mix the fluid to a uniform temperature.

The onset of convective instability in a triply diffusive fluid layer

Abstract: The onset of instability is investigated in a triply diffusive fluid layer in which the density depends on three stratifying agencies having different diffusivities. It is found that, in some cases, three critical values of the Rayleigh number are required to specify the linear stability criteria. As in the case of another problem requiring three Rayleigh numbers for the specification of linear stability criteria (the rotating doubly diffusive case studied by Pearlstein 1981), the cause is traceable to the existence of disconnected oscillatory neutral curves. The multivalued nature of the stability boundaries is considerably more interesting and complicated than in the previous case, however, owing to the existence of heart-shaped oscillatory neutral curves. An interesting consequence of the heart shape is the possibility of ‘quasi-periodic bifurcation’ to convection from the motionless state when the twin maxima of the heart-shaped oscillatory neutral curve lie below the minimum of the stationary neutral curve. In this case, there are two distinct disturbances, with (generally) incommensurable values of the frequency and wavenumber, that simultaneously become unstable at the same Rayleigh number. This work complements the earlier efforts of Griffiths (1979a), who found none of the interesting results obtained herein.

Free thermohaline convection in sediments surrounding a salt column

Abstract: Complex groundwater convection patterns are possible near salt domes because groundwater is subject to both lateral heat and salinity gradients. In order to assess the mechanisms responsible for driving convection near salt domes we use dimensional analysis and numerical simulations to investigate coupled heat and salt transport in homogeneous sediments surrounding a cylindrical salt column. The dimensional analysis does not require the Boussinesq assumption. The coupled heat, solute, and groundwater transport equations are controlled by three dimensionless parameters: the Rayleigh number, the Lewis number, and the buoyancy ratio. The buoyancy ratio is the ratio of salinity to temperature effects on groundwater density, and it directly affects the groundwater flow equation. A finite difference numerical multigridding algorithm is used to iteratively solve the coupled transport equations. The multigridding technique was about 3 times faster than a point-wise successive overrelaxation solution. Boundary conditions for the numerical simulations were adjusted to represent different contrasts in the thermal gradient between the salt and the overlying sediments. The contrast in thermal gradient is parameterized by the thermal conductivity ratio and is responsible for isotherms being elevated near the salt. The analysis suggests that a wide range of convective flow patterns are possible, with flow occurring either up or down along the salt flank. The sense of convection is dependent mainly on the value of the buoyancy ratio and how sharply isotherms are pulled up near the salt. These factors in turn depend on the regional salinity variation, the time since diapirism, and the thermal conductivity of water saturated sediments. While this analysis provides useful insight into the mechanisms driving free convection near salt domes, the assumptions about medium and fluid properties may limit the applicability of dimensional analysis in simulating flow in specific geologic settings.

Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots

Abstract: The stability of the buoyancy-driven parallel shear flow of a variable-viscosity Newtonian fluid between vertical or inclined plates maintained at different temperatures is studied theoretically. The analysis is capable of dealing with arbitrary viscosity-temperature relations. Depending on the Prandtl number, angle of inclination, and form of the viscosity-temperature variation, the flow may become unstable with respect to two-dimensional longitudinal or transverse disturbances. Outstanding questions arising in previous investigations of the stability of parallel free-convection flows of constant-viscosity fluids in inclined slots and of variable-viscosity fluids in vertical slots are discussed. We find that, in a variable-viscosity fluid, non-monotonic dependence of the critical Rayleigh number on the inclination angle can occur at significantly higher Prandtl numbers than is possible in the constant-viscosity case. Results are also presented for the stability of the free-convection flow of several glycerol-water solutions in an inclined slot.

Transient flow in a side-heated cavity at high Rayleigh number: a numerical study

Abstract: A series of two- and three-dimensional numerical simulations of transient flow in a side-heated cavity has been conducted. The motivation for the work has been to resolve discrepancies between a flow description based on scaling arguments and one based on laboratory experiments, and to provide a more detailed description of the approach to steady state. All simulations were for a Rayleigh number of 2 × 109, and a water-filled cavity of aspect ratio 1. The simulations (beginning with an isothermal fluid at rest) generally agree with the results of the scaling arguments. In addition, the experimental observations are entirely accounted for by the position of the measurement instruments and the presence of an extremely weak, stabilizing temperature gradient in the vertical.

Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below

Abstract: Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm x 12 cm x 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60 and 90 percent glycerin-water solutions, and 100 percent glycerin. The depth ratio d, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger d in order to keep the temperature difference across the tank within reasonable limits. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed: (1) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (2) an eightfold decrease in the critical wavelength between d = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen and Chen, 1988).

Combined buoyant-thermocapillary flow in a cavity

Abstract: We treat the problem of combined buoyancy-thermocapillary convection in a cavity with a free surface heated differentially in the horizontal. Attention is focused on the structure and strength of the flow for large ΔT, i.e. large Marangoni and Rayleigh numbers. In the combined problem, the boundary-layer scalings for buoyant and thermocapillary convection suggest that in the limit of large ΔT, thermocapillarity will dominate the large-scale flow. Accurate numerical solutions are used to study this question at fixed cavity aspect ratio and Prandtl number, with G = Ra/Ma as a parameter. For G = 1, the flow evolves toward its boundary-layer limit in a fashion identical to that for G = 0, i.e. pure thermocapillary flow. For G = 10, the evolution is from a buoyancy-dominated structure, through a transition, to a thermocapillary-dominated structure. We infer that thermocapillarity will ultimately dominate all such flows at sufficiently large ΔT, for any fixed values of G, the aspect ratio, and the Prandtl number.

High-accuracy bench mark solutions to natural convection in a square cavity

Abstract: A fourth-order high-accuracy finite difference method is presented for the bouyancy-driven flow in a square cavity with differentially heated vertical walls. The two bench mark solutions against which other solutions can be compared were obtained. The present solution is seemed to be accurate up to fifth decimal. The proposed scheme is stable and convergent for high Rayleigh number, and will be applicable to general problems involving flow and heat transfer, especially in three dimensions.

Modal exchange mechanisms in Lapwood convection

Abstract: Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Benard problem: Lapwood convection in a two-dimensional saturated porous cavity heated from below. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.

Penetrative convective flows induced by internal heating and mantle compressibility

Abstract: Penetrative convective flows induced in a spherical shell by combined effects of internal heating and mantle compressibility are investigated using mathematical and numerical formulations for compressible spherical shell convection. Isothermal stress-free boundary conditions applied at the top and the bottom of the shell are solved using a time-dependent finite difference code in a temperature, vorticity, stream function formulation for Rayleigh numbers ranging from the critical Rc up to 2000 Rc. Results indicate that compressibility, together with internal heating, could be a mechanism capable of generating spontaneously layered convection and local melting in the mantle and that non-Boussinesq effects must be considered in interpretations of geophysical phenomena.

Numerical simulation of thermal convection in a two-dimensional finite box

Abstract: The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Benard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

Time‐dependent convection with non‐Newtonian viscosity

Abstract: A numerical model of bottom-heated, two-dimensional convection in boxes of aspect ratios 2.5 and 4.0 was used to study the differences in time-dependent convection between Newtonian and stress-dependent viscosity. The onset of time dependence due to boundary layer instability is found at approximately the same effective Rayleigh number for both rheologies. However, with increasing Rayleigh number, the temporal and spatial fluctuations in the flow field become much more pronounced with non-Newtonian rheology; also, the tendency for breakup of long cells into smaller ones is stronger. The presence of stress-dependent rheology can cause long-wavelength lateral viscosity variations up to an order of magnitude; this result could have strong implications for the interpretation of mantle viscosity from postglacial rebound.

Oscillatory convection in a viscoelastic fluid through a porous layer heated from below

Abstract: The stability of a viscoelastic fluid in a densely packed horizontal porous layer heated from below is considered using an Oldroyd model. Critical Rayleigh number, wave number, and frequency for overstability are determined by applying the linear stability theory. It is shown that the critical Rayleigh number is invariant under all relevant boundary combinations. Also, it is found that the effect of elasticity of the fluid is to destabilize the system and that of porosity is to stabilize the same. The limiting case of very high Prandtl number and the degenerate case corresponding to the Maxwell model are analyzed in some detail.