Axisymmetric Convection Research Articles

Turbulent Rayleigh–Bénard convection in low Prandtl–number fluids

An experimental investigation of Rayleigh–Bénard convection in liquid sodium has been performed in cylindrical test cells with aspect ratios between 20 and 4.5 for a range of Rayleigh numbers 10 4< Ra<5×10 6. The Prandtl number is between 6.0×10 −3±0.1×10 −3. For low Rayleigh number Ra<10 4 the dimensionless heat flux, Nu is mainly conduction controlled and close to 1; a correlation Nu=0.115 × Ra 0.25 describes the observations adequately for 2×10 4< Ra<5×10 6. It is shown that the Nusselt numbers are smaller for sodium than for other fluids with larger Prandtl numbers such as mercury and helium. This contrasts with the relation Nu∼ Ra 2/7 Pr 1/7 recently proposed by Siggia [High Rayleigh number convection. Ann. Rev. Fluid Mech. (1994)] but conforms with a relation Nu∼( Ra Pr) 1/4 for low Prandtl number fluids derived by Jones et al. [Axisymmetric convection in a cylinder. J. Fluid Mech. 73 (1976) 353–358] and with recent numerical simulations of low Prandtl number convection by Vercicco and Camussi [Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9 (5) (1997) 1287–1295]. The heat transfer as well as the statistical properties of the turbulent convection are significantly influenced by mode transitions in the test cell of small aspect ratio. The formation of thermal boundary layers occurs only at high Rayleigh numbers of the order Ra≳10 6. An instability in these thermal boundary layers triggers a new mode of large scale fluctuating motion. The analysis of the temperature time signals shows that the temperature field behaves essentially dissipative up to Rayleigh numbers Ra∼5×10 6.

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.

Deposition of waste water into deep mines

Transport phenomena in contaminated water from waste disposal sites deposited into deep mines are analysed in the paper. Critical transport path for pollutant migration by diffusion of contaminants and buoyancy-driven convection of contaminated water is assumed to be in a vertical cylindrical enclosure. In the first approximation, both transport phenomena are modeled and then numerically simulated separately. The models presented describe ion diffusion and axisymmetric turbulent convection in a vertical cylinder with the radius R=3·5 m heated from the sidewalls. One-dimensional diffusion equation is used for simulating the ion diffusion in an enclosure of height H=1000 m. Three low-Reynolds number two-equation turbulence models contained in the FIDAP 7·5 software package were tested in our computational simulations of the natural convection. Two typical case studies of buoyancy-driven convection represent the aspect ratios H/R=14·3 and H/R=286. The former corresponds to the initial state just after deposition of the surplus water, the latter to the whole height of the cylindrical enclosure filled by the underground water. The numerical results obtained show that ion diffusion is not dangerous as far as the contamination of subsurface water is concerned. On the other hand, it is concluded that the buoyancy-driven convection in the enclosures with the above aspect ratios can lead to the transport of contaminants into subsurface water and hence the protection of receiving waters from contamination must be designed. Copyright © 1999 John Wiley & Sons, Ltd.

Symmetries of volcanic distributions on Mars and Earth and their mantle plume dynamics

The symmetries in the distribution of major volcanic centers on Mars have been analyzed by employing a technique involving the angular self‐similarity of point fields. The distribution of all central vent volcanoes exhibits an axisymmetry along with a reflecting symmetry about a plane whose normal is the axis of axisymmetry. The latter axis is skewed by about 30° from the rotational axis and is suggested to represent the early axis of symmetry of the interior dynamics. The high level of symmetry of Mars' volcanic distribution may be the result of a focusing of seismic energy from a major impact in the Hellas area that formed the Hellas basin and may have triggered the onset of volcanic activity in the greater Tharsis area. If the highland paterae near the Hellas region are not taken into account in the analysis, an asymmetry with respect to the reflecting plane appears. This is probably representative of the Late Hesperian and Amazonian, when the once global volcanic activity had retreated to the Tharsis province. These angular symmetries suggest a simple pattern of long‐wavelength axisymmetric martian mantle convection involving a large stationary plume, as shown by our three‐dimensional (3‐D) spherical‐shell simulations. On the basis of these calculations, we venture to propose that the volcanic history of Mars reflects the transformation of a bipolar mantle plume structure into a single megaplume by the process of plume‐plume collision acting in concert with the deep Martian phase transitions. This transformation might have overcome the original pattern induced by a major impact forming the Hellas basin. We have repeated the procedure for the Earth's hotspot distribution, which revealed both axisymmetry and a certain level of reflecting symmetry. This similarity between the two planets is striking. The symmetries of the Earth's hotspot distribution are probably due to the interaction of the lower mantle megaplumes with the endothermic perovskite to γ‐spinel phase transition.

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T 1 of the inner and T 2 of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined.

Geoid anomalies and dynamic topography from time-dependent, spherical axisymmetric mantle convection

Geoid anomalies and dynamic topography are two important diagnostics of mantle convection. We present geoid and topography results for several time-dependent convection models in spherical axisymmetric geometry for Rayleigh numbers between 10 6 and 10 7 with depth-dependent viscosity and mixtures of bottom and internal heating. The models are strongly chaotic, with boundary layer instabilities erupting out of both thermal boundary layers. In some instances, instabilities from one boundary layer influence the development of instabilities in the other boundary layer. Such coupling between events at the top and bottom of the mantle has been suggested to play a role in a mid-Cretaceous episode of enhanced volcanism in the Pacific. These boundary layer instabilities produce large temporal variations in the geoid anomalies and dynamic topography associated with the convection. The amplitudes of these fluctuations depend on the detailed model parameters, but fluctuations of 30–50% relative to the time-averaged geoid and topography are common. The convective planform is strongly sensitive to the specific initial conditions. Convection cells with larger aspect ratio tend to have larger fractional fluctuations in their geoid and topography amplitudes, because boundary layer instabilities have more time to develop in long cells. In some instances, we observe low-amplitude topographic highs adjacent to the topographic lows produced by cold downwellings. We discuss applications of these results to several situations, including the temporal variability of hotspots such as Hawaii, the topography of subduction zone outer rises, and the topography of coronae on Venus.

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (=length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. Pr< 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Rec→ 1784, as Pr→ 0, the most dangerous mode being m = 2. While for Pr⩾ 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 ⩽Pr⩽ 0.8 and m = 2 for Pr⩾ 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent linear-theory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a nondeformable free surface.

Nonlinear front evolution of hydrodynamic chemical waves in vertical cylinders

The nonlinear stability of three-dimensional reaction-diffusion fronts in vertical cylinders is considered using the viscous hydrodynamic fluid equations in the limit of infinite thermal diffusivity. A nonlinear front evolution equation is presented and used to examine the transition from nonaxisymmetric to axisymmetric convection observed in experiments performed in cylinders. Comparisons with experiments show excellent agreement in both the shape and speed of the front. @S1063-651X~97!13409-2#

The global characteristics of the three-dimensional thermal convection inside a spherical shell

Abstract. The Rayleigh number-Nusselt number, and the Rayleigh number-thermal boundary layer thickness relationships are determined for the three-dimensional convection in a spherical shell of constant physical parameters. Several models are considered with Rayleigh numbers ranging from 1.1 x 102 to 2.1 x 105 times the critical Rayleigh number. At lower Rayleigh numbers the Nusselt number of the three-dimensional convection is greater than that predicted from the boundary layer theory of a horizontal layer but agrees well with the results of an axisymmetric convection in a spherical shell. At high Rayleigh numbers of about 105 times the critical value, which are the characteristics of the mantle convection in terrestrial planets, the Nusselt number of the three-dimensional convection is in good agreement with that of the boundary layer theory. At even higher Rayleigh numbers, the Nusselt number of the three-dimensional convection becomes less than those obtained from the boundary layer theory. The thicknesses of the thermal boundary layers of the spherical shell are not identical, unlike those of the horizontal layer. The inner thermal boundary is thinner than the outer one, by about 30- 40%. Also, the temperature drop across the inner boundary layer is greater than that across the outer boundary layer.

Magnetic damping of axisymmetric thermocapillary convection in float zones in microgravity

Magnetic damping of axisymmetric thermocapillary convection in float zones in microgravity

Modelling plume-related uplift, gravity and melting on Venus

Three roughly circular regions on Venus about 1000 km across are identified as potential Hawaii-scale plume sites on the basis of their gravity (70–90 mgal) and topography (1.6–2.0 km) anomalies, and signs of melt generation and rifting. Axisymmetric isoviscous convection models are used to reproduce gravity profiles across these plumes and the line-of-sight acceleration of the Magellan spacecraft as it passes over them. The best fitting models have a conductive lid thickness of less than 150 km, mantle viscosity of 8.9 × 10 19 to 9.6 × 10 20 Pa s and a basal heat flux of 15–25 m W m −2. The lid thickness is constrained by requiring a modest amount of melt generation and a potential temperature of about 1300°C. The high mantle viscosity relative to that of Earth is probably a consequence of the absence of water in the mantle, and may help to explain the current absence of plate tectonics on Venus. The low heat flux suggests that the thermal evolution of Venus has differed from that of the Earth.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Theoretical analysis for axisymmetric mixed convection between rotating coaxial disks

This paper deals with a similarity analysis of axisymmetric mixed convection between two horizontal infinite coaxial disks. Governing equations with respect to a rotating frame of reference are formulated, and Boussinesq approximation is invoked to explore centrifugal-buoyancy effects. The rotational conditions of two disks rotating at different rates are mainly concerned. Upon considering Reynolds number up to 2000 and the buoyancy parameter ( B = βΔT) of | B | ⩽ 0.1, the rotational effects on the flow and heat transfer characteristics are examined. Analytic features of the high-Re solution behaviors are also studied. By using the present similarity model, flow structure patterns of co-rotating, counter-rotating and rotor-stator systems under the influence of buoyancy effect are disclosed. Furthermore, the results also reveal the significance of centrifugal buoyancy in flow and heat transfer characteristics. For co-rotating disks, the centrifugal-buoyancy effects are reflected in the velocity field and skin friction. In the case of counter-rotating disks, remarkable buoyancy effects exert on the temperature field, heat transfer rate, and the radial skin friction. For rotor-stator systems, the buoyancy effects on boundary derivatives are insignificant, while those on core-flow in high-Re cases are noticeable.

Magnetic damping of buoyant convection during semiconductor crystal growth in microgravity. Continuous random g-jitters

This paper treats the buoyant convection of a liquid metal in a circular cylinder with a uniform, steady, axial magnetic field and with the random residual accelerations encountered on earth-orbiting vehicles. The objective is to model the magnetic damping of the melt motion during semiconductor crystal growth by the Bridgman process in space. For a typical process with a magnetic flux density of 0.2 T, the convective heat transfer and the nonlinear inertial effects are negligible, so that the governing equations are linear. Therefore, for residual accelerations or ‘‘g-jitters’’ whose directions are random functions of time, the buoyant convection is given by a superposition of the convections for two unidirectional accelerations: an axial acceleration which is parallel to the cylinder’s axis and a transverse acceleration which is perpendicular to this axis. Similarly, the response to accelerations whose amplitudes are random functions of time is given by a Fourier-transform superposition of the buoyant convections for sinusoidally periodic accelerations for all frequencies. Since the temperature gradient in the Bridgman process is primarily axial, the magnitude of the three-dimensional buoyant convection for the transverse acceleration is much larger than the magnitude of the axisymmetric convection for the axial acceleration. At a low frequency, only the magnetic damping limits the magnitude of the buoyant convection. As the frequency is increased, linear inertial effects augment the magnetic damping, so that the magnitude of the convection decreases, and its phase shifts to a quarter-period lag after the acceleration.

Nonaxisymmetric and Axisymmetric Convection in Propagating Reaction-Diffusion Fronts

Observations of steady nonaxisymmetric chemical wave fronts are reported for upward propagation in iodate- arsenous acid solutions within vertical capillary tubes. These observations confirm a recent prediction of hydrodynamic stability theory that the onset of convection in such fronts should be nonaxisymmetric. The nonaxisymmetric waveform reflects the presence of a single convective roll in the vicinity of the moving front.

Convection near threshold for Prandtl numbers near 1

Rayleigh-B\'enard convection in two cylindrical geometries with different sidewalls and with radius-to-height ratios \ensuremath{\Gamma}=41 and 43 was studied with shadowgraph imaging and heat-transport measurements. The working fluid was ${\mathrm{CO}}_{2}$ at 25.3 bar with a Prandtl number \ensuremath{\sigma}=0.93. For one of the cells (\ensuremath{\Gamma}=43), axisymmetric convection rolls were stable above onset up to \ensuremath{\epsilon}==\ensuremath{\Delta}T/\ensuremath{\Delta}${\mathit{T}}_{\mathit{c}}$-1=0.19, where \ensuremath{\Delta}${\mathit{T}}_{\mathit{c}}$ is the critical temperature difference. The amplitude of the center of the concentric patterns, the umbilicus, much larger near onset than that of the outlying region, grew as ${\mathrm{\ensuremath{\epsilon}}}_{1}^{\mathrm{\ensuremath{\beta}}}$, with ${\mathrm{\ensuremath{\beta}}}_{1}$=0.27\ifmmode\pm\else\textpm\fi{}0.04. Outside the umbilicus, the amplitude grew as ${\mathrm{\ensuremath{\epsilon}}}_{2}^{\mathrm{\ensuremath{\beta}}}$, with ${\mathrm{\ensuremath{\beta}}}_{2}$=0.48\ifmmode\pm\else\textpm\fi{}0.04. For \ensuremath{\Gamma}=41, axisymmetric convection rolls, coexisting with an annular cross-roll state confined near the sidewall, were the preferred patterns above onset and remained stable up to an \ensuremath{\epsilon} of 0.08. The umbilicus of the concentric convection pattern shifted slightly off center as \ensuremath{\epsilon} was increased, but the shift never exceeded 50% of the cell depth. For \ensuremath{\epsilon}g0.10, the patterns were unstable and evolved over many horizontal thermal diffusion times to straight-roll patterns with defects and grain boundaries. The process involved the umbilicus moving toward the sidewall, accompanied by radially traveling waves emitted by the umbilicus. While the umbilicus emitted, the crossrolls adjacent to the sidewall traveled parallel to the wall, and rolls were created and destroyed at a source and sink, respectively, presumably owing to large-scale flows generated by the off-centered position of the umbilicus.

Natural Convection Within Spherical Annuli by Symbolic Algebra

An asymptotic truncated series solution for steady, axisymmetric convection between concentric spheres is obtained through effective use of symbolic algebra. The approximate solution is given in powers of the square root of the Grashof number based on outer radius. The solution differs from that of Hardee (1966) and Singh and Elliot (1981) in that we give a method for determining the series solution to any order. The results obtained here were used as the basic flows in the hydrodynamic stability analysis of TeBeest. 14 refs., 1 fig., 2 tabs.

Distribution of magnetic energy in αΩ-dynamos, I: The method

Abstract In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B 2/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 1014cm2s−1, which is an order of magnitude larger than usually adopted in solar mean field models.

Linear and weakly nonlinear variable viscosity convection in spherical shells

A theoretical study of linear and weakly nonlinear thermal convection in a spherical shell is performed. The Boussinesq fluid is of infinite Prandtl number and its viscosity is temperature dependent. The linear stability eigenvalue problem is derived and solved by a shooting method assuming isothermal, stress-free boundaries, a self-gravitating fluid, and corresponding to two heating models. The first is heating from below, and the second is a model of combined heating from below and within, such that convection is described by a self-adjoint linear stability formulation. In addition, nonlinear, hemispherical, axisymmetric convection is computed by a finite volume technique for a shell with 0.5 aspect ratio. It is shown that 2-cell convection occurs as transcritical bifurcation for a viscosity constrast across the shell up to about 150. Motions with four cells are also possible. As expected, the subcritical range is found to increase with increasing viscosity contrast, even when the linear operator is self-adjoint.

Mantle phase transitions and layered convection

Numerical simulations with an anelastic, spherical, axisymmetric mantle convection model have been conducted to address the question of the radial mixing length in the general circulation of the mantle. Continuing debate centers on the question as to whether or not the 670 km seismic discontinuity (which we now understand to exist as a consequence of an endothermic phase change of mantle mineralogy from the spinel phase to a mixture of perovskite and periclase) in combination with the 400 km discontinuity (associated with the exothermic phase change from olivine to spinel) will impose a sufficient barrier to the circulation so as to induce layering. We argue herein that the mantle must currently be converting in a partially layered style but that the degree of layering is highly time dependent. Moreover, in the perhaps not too distant past the propensity to layering was greater, possibly to the extent that soon after planetary formation mantle mixing occurred in two distinct reservoirs. As the planet cooled and the Rayleigh number fell, we suggest that the circulation was transformed from the layered state to the partially layered state that obtains today.