Showing papers on "Rayleigh number published in 1999"

Numerical modeling of the geodynamo: Mechanisms of field generation and equilibration

Abstract: Numerical calculations of fluid dynamos powered by thermal convection in a rotating, electrically conducting spherical shell are analyzed. We find two regimes of nonreversing, strong field dynamos at Ekman number 10 -4 and Rayleigh numbers up to 11 times critical. In the strongly columnar regime, convection occurs only in the fluid exterior to the inner core tangent cylinder, in the form of narrow columnar vortices elongated parallel to the spin axis. Columnar convection contains large amounts of negative helicity in the northern hemisphere and positive helicity in the southern hemisphere and results in dynamo action above a certain Rayleigh number, through a macroscopic α 2 mechanism. These dynamos equilibrate by generating concentrated magnetic flux bundles that limit the kinetic energy of the convection columns. The dipole-dominated external field is formed by superposition of several flux bundles at middle and high latitudes. At low latitudes a pattern of reversed flux patches propagates in the retrograde direction, resulting in an apparent westward drift of the field in the equatorial region. At higher Rayleigh number we find a fully developed regime with convection inside the tangent cylinder consisting of polar upwelling and azimuthal thermal wind flows. These motions modify the dynamo by expelling poloidal flux from the poles and generating intense toroidal fields in the polar regions near the inner core. Convective dynamos in the fully developed regime exhibit characteristics that can be compared with the geomagnetic field, including concentrated flux bundles on the core-mantle boundary, polar minima in field intensity, and episodes of westward drift.

Prandtl number effects in convective turbulence

Abstract: The effect of Prandtl number on the dynamics of a convective turbulent flow is studied by numerical experiments. In particular, three series of experiments have been performed; in two of them the Rayleigh number spanned about two decades while the Prandtl number was set equal to 0.022 (mercury) and 0.7 (air). In the third series, in contrast, we fixed the Rayleigh number at 6×105 and the Prandtl number was varied from 0.0022 up to 15. The results have shown that, depending on the Prandtl number, there are two distinct flow regimes; in the first (Pr[lsim ]0.35) the flow is dominated by the large-scale recirculation cell that is the most important ‘engine’ for heat transfer. In the second regime, on the other hand, the large-scale flow plays a negligible role in the heat transfer which is mainly transported by the thermal plumes.For the low-Pr regime a model for the heat transfer is derived and the predictions are in qualitative and quantitative agreement with the results of the numerical simulations and of the experiments. All the hypotheses and the consequences of the model are directly checked and all the findings are consistent with the predictions and with experimental observations performed under similar conditions. Finally, in order to stress the effects of the large-scale flow some counter examples are shown in which the large-scale motion is artificially suppressed.

Infinite prandtl number convection

Abstract: We prove an inequality of the type N≤CR 1/3(1+log+ R)2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh–Benard convection in the limit of infinite Prandtl number

Heat transport in stagnant lid convection with temperature‐ and pressure‐dependent Newtonian or non‐Newtonian rheology

Abstract: A numerical model of two-dimensional Rayleigh-Benard convection is used to study the relationship between the surface heat flow (or Nusselt number) and the viscosity at the base of the lithosphere. Newtonian or non-Newtonian, temperature- and pressure-dependent rheologies are considered. In the high Rayleigh number time-dependent regime, calculations yield Nu ∝ RaBL1/3beff−4/3 where beff is the effective dependence of viscosity with temperature at the base of the upper thermal boundary layer and RaBL is the Rayleigh number calculated with the viscosity νBL (or the effective viscosity) at the base of the upper thermal boundary layer. The heat flow is the same for Newtonian and non-Newtonian rheologies if the activation energy in the non-Newtonian case is twice the activation energy in the Newtonian case. In this chaotic regime the heat transfer appears to be controlled by secondary instabilities developing in thermal boundary layers. These thermals are advected along the large-scale flow. The above relationship is not valid at low heat flow where a stationary regime prevails and for simulations forced into steady state. In these cases the Nusselt number follows a trend Nu ∝ RaBL1/5beff−1 for a Newtonian rheology, as predicted by the boundary layer theory. We argue that the equilibrium lithospheric thickness beneath old oceans or continents is controlled by the development of thermals detaching from the thermal boundary layers. Assuming this, we can estimate the viscosity at the base of the stable oceanic lithosphere. If the contribution of secondary convection to the surface heat flux amounts to 40 to 50 mW m−2, the asthenospheric viscosity is predicted to be between 1018 and 2×l019 Pa s.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

Abstract: The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Mixing and compositional stratification produced by natural convection: 1. Experiments and their application to Earth's core and mantle

Abstract: An extensive series of laboratory experiments is used to quantify the circumstances under which fluids can be mixed by natural convection at high flux Rayleigh number. A compositionally buoyant fluid was injected at a fixed rate into an overlying layer of ambient fluid from a planar, horizontally uniform source. The nature of the resulting compositional convection was found to depend on two key dimensionless parameters: a Reynolds number Re and the ratio U of the ambient fluid viscosity to the input fluid viscosity. Increasing the Reynolds number corresponded to increasing the vigor of the convection, while the viscosity ratio was found to determine the spacing between plumes and whether buoyant fluid rose as sheets (U 1). From measurements of the final density profile in the fluid after the experiments we quantified the extent to which buoyant liquid was mixed in terms of a thermodynamic mixing efficiency E. The mixing efficiency was found to be high (E > 0.9) when either the Reynolds number was large (Re > 100) or the viscosity ratio was small (U 200. The amount of mixing was related to whether ascending plumes generated a large-scale circulation in the ambient fluid. When our results are applied to the differentiation of the Earth's core, we suggest that the convection resulting from the release of buoyant residual liquid into the liquid outer core due to crystallization at the boundary between the inner and the outer core will probably lead to nearly complete mixing. In the dynamically very different context of the mantle, mantle plumes are predicted to ascend through the mantle and pond beneath the lithosphere, whereas convection driven by the subduction of oceanic lithosphere is expected to produce moderate to extensive mixing of the mantle. When the competing plate and plume modes of mantle convection are considered together, we find that owing to a larger driving buoyancy flux, the plate-scale flow will destroy any stratification at the top of the mantle produced by mantle plumes. Applying our results to the “stagnant lid” style of thermal convection predicted to occur in the mantles of the Moon, Mercury, Mars, Venus, and pre-Archean Earth, we expect the respective flows to produce minor thermal stratification at the respective core-mantle boundaries. In part 2 of this study [Jellinek and Kerr, this issue] we apply our results to the differentiation of magma chambers and komatiite lava flows.

High Rayleigh Number Turbulent Convection in a Gas near the Gas-Liquid Critical Point

Abstract: ${\mathrm{SF}}_{6}$ in the vicinity of its critical point was used to study turbulent convection up to exceptionally high Rayleigh numbers, Ra (up to $5\ifmmode\times\else\texttimes\fi{}{10}^{14}$) and to verify for the first time the generalized scaling laws for the heat transport and the large scale circulation velocity as a function of Ra and the Prandtl number, Pr, in a very wide range of these parameters. Both scaling laws obtained are consistent with theoretical predictions by Shraiman and Siggia [Phys. Rev. A 42, 3650 (1990)].

Weak turbulence and chaos for low Prandtl number gravity driven convection in porous media

Abstract: Low Prandtl number convection in porous media is relevant to modern applications of transport phenomena in porous media such as the process of solidification of binary alloys. The transition from steady convection to chaos is analysed by using Adomian's decomposition method to obtain an analytical solution in terms of infinite power series. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The solution shows a transition from steady convection to chaos via a Hopf bifurcation producing a 'solitary limit cycle’ which may be associated with an homoclinic explosion. This occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. Periodic windows within the broad band of parameter regime where the chaotic solution persists are identified and analysed. It is evident that the further transition from chaos to a high Rayleigh number periodic convection occurs via a period halving sequence of bifurcations.

Random Roughness of Boundary Increases the Turbulent Convection Scaling Exponent

Abstract: The influence of the boundary layer properties on the heat transport in turbulent thermal convection is experimentally investigated in a cell with a rough bottom plate. It is shown that the standard 2/7 exponent of the convective heat flow dependence on the Rayleigh number, usually observed in a cell with smooth boundaries, increases if the roughness has power law distributed asperity heights and the thermal boundary layer thickness is smaller than the maximum asperity size. In contrast a periodic roughness does not influence the heat transport law exponent.

Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients

Abstract: The Darcy model with the Boussinesq approximations is used to study double-diffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, e, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.

Convective instability of a boundary layer with temperature-and strain-rate-dependent viscosity in terms of ‘available buoyancy’

Abstract: Summary Cold mantle lithosphere is gravitationally unstable with respect to the hotter buoyant asthenosphere beneath it, leading to the possibility that the lower part of the mantle lithosphere could sink into the mantle in convective downwelling. Such instabilities are driven by the negative thermal buoyancy of the cold lithosphere and retarded largely by viscous stress in the lithosphere. Because of the temperature dependence of viscosity, the coldest, and therefore densest, parts of the lithosphere are unavailable for driving the instability because of their strength. By comparing theory and the results of a finite element representation of a cooling lithosphere, we show that for a Newtonian fluid, the rate of exponential growth of an instability should be approximately proportional to the integral over the depth of the lithosphere of the ratio of thermal buoyancy to viscosity, both of which are functions of temperature, and thus depth. We term this quantity ‘available buoyancy’ because it quantifies the buoyancy of material sufficiently weak to flow, and therefore available for driving convective downwelling. For non-Newtonian viscosity with power law exponent n and temperature-dependent pre-exponential factor B, the instabilities grow superexponentially, as described by Houseman & Molnar (1997), and the appropriate timescale is given by the integral of the nth power of the ratio of the thermal buoyancy to B. The scaling by the ‘available buoyancy’ thus offers a method of determining the timescale for the growth of perturbations to an arbitrary temperature profile, and a given dependence of viscosity on both temperature and strain rate. This timescale can be compared to the one relevant for the smoothing of temperature perturbations by the diffusion of heat, allowing us to define a parameter, similar to a Rayleigh number, that describes a given temperature profile’s tendency toward convective instability. Like the Rayleigh number, this parameter depends on the cube of the thickness of a potentially unstable layer; therefore, mechanical thickening of a layer should substantially increase its degree of convective instability, and could cause stable lithosphere to become convectively unstable on short × cales. We estimate that convective erosion will, in 10 Myr, reduce a layer thickened by a factor of two to a thickness only 20 to 50 per cent greater than its pre-thickened value. Thickening followed by convective instability may lead to a net thinningof a layer if thickening also enhances the amplitude of perturbations to the layer’s lateral temperature structure. For the mantle lithosphere, the resulting influx of hot asthenosphere could result in rapid surface uplift and volcanism.

Fingering instability in combustion: an extended view.

Abstract: We detail the experimental situation concerning the fingering instability that occurs when a solid fuel is forced to burn against a horizontal oxidizing wind. The instability appears when the Rayleigh number for convection is below criticality. The focus is on the developed fingering state. We present direct measurements of the depletion of oxygen by the front as well as new results that connect heat losses to the characteristic scale of the instability. In addition, we detail the experimental system, elaborate (qualitatively and quantitatively) on the results that were previously presented, and discuss new observations. We also show that the same phenomenological model applies to electrochemical deposition.

Local and global transitions to chaos and hysteresis in a porous layer heated from below

Abstract: The routes to chaos in a fluid saturated porous layer heated from below are investigated by using the weak nonlinear theory as well as Adomian's decomposition method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation is equivalent to the familiar Lorenz equations with different coefficients which correspond to the porous media convection. While the weak nonlinear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one of the two steady state convective solutions. On the other hand, the Adomian's decomposition method provides an analytical solution to the problem in terms of infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task transform the otherwise analytical results into a computational solution achieved up to a finite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains previously obtained computational results for low Prandtl number convection in porous media suggesting a transition from steady convection to chaos via a Hopf bifurcation, represented by a solitary limit cycle at a sub-critical value of Rayleigh number. A simple explanation of the well known experimental phenomenon of Hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results.

Numerical analysis of partially heated vertical parallel plates in natural convective cooling

Abstract: This paper addresses the significance of adding insulated extensions to a parallel-plate channel in which the plates receive a uniform heat flux and a natural convection airflow is responsible for the cooling. The wall temperatures may decrease or increase, depending on whether the channel extensions are appended at the inlet or at the exit of the channel. The full elliptic conservation equations are solved numerically in an I-type composite computational domain. For the two cases treated, the pertinent results are reported in terms of wall temperature profiles, induced mass flow rates, and pressure profiles. The insulated extension placed downstream of the heated part implies a reduction of the maximum wall temperature. This effect is less relevant as the Rayleigh number increases. In addition, correlations have been obtained between the induced mass flow rate as well as the maximum wall temperatures and the Rayleigh number and the extension ratio in the investigated range of parameters.

Characteristics of high Rayleigh number two-dimensional convection in an open-top porous layer heated from below

Abstract: Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.

The potential for free and mixed convection in sedimentary basins

Abstract: Free thermal convection and mixed convection are considered as potential mechanisms for mass and heat transport in sedimentary basins. Mixed convection occurs when horizontal flows (forced convection) are superimposed on thermally driven flows. In cross section, mixed convection is characterized by convection cells that migrate laterally in the direction of forced convective flow. Two-dimensional finite-element simulations of variable-density groundwater flow and heat transport in a horizontal porous layer were performed to determine critical mean Rayleigh numbers for the onset of free convection, using both isothermal and semi-conductive boundaries. Additional simulations imposed a varying lateral fluid flux on the free-convection pattern. Results from these experiments indicate that forced convection becomes dominant, completely eliminating buoyancy-driven circulation, when the total forced-convection fluid flux exceeds the total flux possible due to free convection. Calculations of the thermal rock alteration index (RAI=q·∇T) delineate the patterns of potential diagenesis produced by fluid movement through temperature gradients. Free convection produces a distinct pattern of alternating positive and negative RAIs, whereas mixed convection produces a simpler layering of positive and negative values and in general less diagenetic alteration.

Buoyancy-driven convection with a uniform magnetic field. Part 2. Experimental investigation

Abstract: In this paper, an experimental study of laminar magnetohydrodynamic (MHD) buoyancy-driven flow in a cylindrical cell with axis horizontal is described. A steady uniform magnetic field is applied vertically to the mercury-filled cell, which is also subjected to a horizontal temperature gradient. The main features of this internal MHD thermogravitational flow are made experimentally evident from temperature and electric potential measurements. Whatever the level of convection, raising the Hartmann number Ha to a value of the order of 10 is sufficient to stabilize an initially turbulent flow. At much higher values of the Hartmann number (Ha∼100) the MHD effects cause a change of regime from boundary-layer driven to core driven. In this latter regime an inviscid inertialess MHD core flow is bounded by a Hartmann layer on the horizontal cylindrical wall and viscous layers on the endwalls. Since the Hartmann layer is found to stay electrically inactive along the cell, the relevant asymptotic (Ha[Gt ]1) laws for velocity and heat transfer are found from the balance between the curl of buoyancy and Lorentz forces in the core, together with the condition that the flow of electric current between core and Hartmann layer is negligible. A modified Rayleigh number RaG/Ha2, which is a measure of the ratio of thermal convection to diffusion when there is a balance between buoyancy and Lorentz forces, is the determining parameter for the flow.

Transient natural convection in a square enclosure with horizontal walls submitted to periodic temperatures

Abstract: Natural convection in enclosures is of considerable interest in several practical problems. Applications range from thermal design of buildings to solar collectors, electronic and computer equipment, etc. Here, natural convection in a square cavity with its horizontal walls submitted to different heating models has been investigated by a finite difference procedure. The hot temperature of the bottom surface varies sinusoidally with time, while that of the opposite surface (cold temperature) is maintained constant or varied sinusoidally. The vertical walls are considered adiabatic. Parameters of the problem are the amplitude of the variable temperature(s) (0 {le} a {le} 0.5), its (their) period (0.001 {le} {tau} {le} 1), the dephasing between hot and cold temperatures ({Phi} = 0 and {pi}), the Rayleigh number (5 x 10{sup 5} {le} Ra {le} 10{sup 6}) and the Prandtl number (Pr = 0.72). The effect of these parameters on heat transfer can be enhanced or reduced, with respect to the case of constant temperatures, by proper choice of the variable heating mode, the parameters related to the periodic temperature(s), and the Rayleigh number. By varying the two imposed temperatures, fundamental differences have been observed comparatively with the case of variable hot temperature and that of constantmore » thermal boundary conditions.« less