Showing papers on "Rayleigh number published in 2004"

Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes

Abstract: Our unifying theory of turbulent thermal convection [Grossmann and Lohse, J. Fluid. Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001); Phys. Rev. E 66, 016305 (2002)] is revisited, considering the role of thermal plumes for the thermal dissipation rate and addressing the local distribution of the thermal dissipation rate, which had numerically been calculated by Verzicco and Camussi [J. Fluid Mech. 477, 19 (2003); Eur. Phys. J. B 35, 133 (2003)]. Predictions for the local heat flux and for the temperature and velocity fluctuations as functions of the Rayleigh and Prandtl numbers are offered. We conclude with a list of suggestions for measurements that seem suitable to verify or falsify our present understanding of heat transport and fluctuations in turbulent thermal convection.

Heat transport by turbulent Rayleigh-B'enard Convection in cylindrical cells with aspect ratio one and less

Abstract: We present high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for cylindrical samples of water (Prandtl number sigma = 4.4) with a diameter D of 49.7 cm and heights L = 116.3, 74.6, and 50.6 cm, as well as for D = 24.8 cm and L = 90.2 cm. For each aspect ratio Gamma = D/L = 0.28, 0.43, 0.67, and 0.98 the data cover a range of a little over a decade of R. The maximum R ~= 10^12 and Nusselt number N ~= 600 were reached for Gamma = 0.43 and D = 49.7. The data were corrected for the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid to obtain estimates of N_infty for plates with infinite conductivity. The results for N_infty and Gamma >= 0.43 are nearly independent of Gamma. For Gamma = 0.275 N_infty falls about 2.5 % below the other data. For R ~ 1/3.

Cartesian convection driven dynamos at low Ekman number.

Abstract: The Cartesian dynamo model of Childress and Soward [Phys. Rev. Lett. 29, 837 (1972)] is studied numerically in the regime of low viscosity. Dynamos with Ekman numbers $E$ in the range ${10}^{\ensuremath{-}4}\ensuremath{\geqslant}E\ensuremath{\geqslant}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ are discussed and compared with the corresponding nonmagnetic states and with results obtained for imposed magnetic fields. We find that in the range of investigated Ekman numbers, a transition occurs from a flow regime where the planform of convection is only weakly affected by the dynamo generated field to a regime where the typical length scales of the flow are largely controlled by the Lorentz forces. The magnetic field acts to facilitate convection and leads to an increase in both the heat transport and in the amplitude of the flow. We demonstrate that this convection promoting effect allows for dynamo action even for Rayleigh numbers below the critical Rayleigh number for the onset of nonmagnetic convection.

High-Rayleigh-number convection in a fluid-saturated porous layer

Abstract: The Darcy-Boussinesq equations at infinite Darcy-Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number Ra. High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu, from onset at Ra = 4π 2 up to Ra = 10 4 . Over an intermediate range of increasing Rayleigh numbers, the simulations display the 'classical' heat transport Nu ∼ Ra scaling. As the Rayleigh number is increased beyond Ra = 1255, we observe a sharp crossover to a form fitted by Nu 0.0174 x Ra 0.9 over nearly a decade up to the highest Ra. New rigorous upper bounds on the high-Rayleigh-number heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: Nu ≤ 0.0297 x Ra. The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of Ra, including hysteretic effects observed in the simulations as the Rayleigh number is decreased from 1255 back down to onset

Convection driven by differential heating at a horizontal boundary

Abstract: We report laboratory and numerical experiments with the convective circulation that develops in a long channel driven by heating and cooling through opposite halves of the horizontal base. The problem is similar to that posed by Stommel (Proc. Natl Acad. Sci. vol. 48, 1962, p. 766) and Rossby (Deep-Sea Res. vol. 12, 1965, p. 9; Tellus vol. 50, 1998, p. 242), where flow forced by a linear temperature variation along the ocean surface or the base of a tank presented a demonstration of the smallness of sinking regions in the meridional overturning circulation of the oceans. In contrast to the previous experiments, we use small aspect ratio, larger Rayleigh numbers, piecewise uniform boundary conditions and an imposed input heat flux. The flow is characterized by a vigorous overturning circulation cell filling the box length and depth. A stable thermocline forms above the cooled base and is advected over the heated part of the base, where it is eroded from below by small-scale three-dimensional convection, forming a ‘convective mixed layer’. At the endwall, the convective mixing is overshadowed by a narrow but turbulent plume rising through the full depth of the box. The return flow along the top of the box is turbulent with large slowly migrating eddies, and occupies approximately a third of the total depth. Theoretical scaling laws give temperature differences, thermocline thickness and velocities that are in good agreement with the experimental data and two-dimensional numerical solutions. The measured and computed density structure is largely similar to the thermocline and abyssal stratification in the oceans.

Effects of nonperfect thermal sources in turbulent thermal convection

Abstract: The effects of the plates thermal properties on the heat transfer in turbulent thermal convection are investigated by direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation. It has been found that the governing parameter is the ratio of the thermal resistances of the fluid layer Rf and the plates Rp; when this ratio is smaller than a threshold value (Rf/Rp≈300 arbitrarily defined by requiring that the actual heat transfer differs by less than 2% from its ideal value), the finite conductivity of the plates limits the heat transfer in the cell. In addition, since Rf decreases for increasing Rayleigh numbers, any experimental apparatus is characterized by a threshold Rayleigh number that cannot be exceeded if the heat transfer in the cell has not to be influenced by the thermal properties of the plates. It has been also shown that the plate effects cannot be totally corrected by subtracting the temperature drop occurring within the plates from the measured total tempera...

Structure and dynamics of sheared mantle plumes

Abstract: [1] An extensive series of laboratory experiments is used to investigate the behavior of sheared thermal plumes. The plumes are generated by heating a small circular plate on the base of a cylindrical tank filled with viscous fluid and then sheared by rotating a horizontal lid at the fluid surface. The motion of passive tracers in the plumes is visualized by the release of several dye streams on the hot plate. We systematically examine the dependence of the convective flow on four dimensionless numbers: a velocity ratio, a Rayleigh number, the viscosity ratio, and an aspect ratio. We identify and delineate two transitions in the convective behavior: from a regime where the plume can spread upstream against the shear to a regime where the entire plume is advected downstream, and from a regime of negligible cross-stream circulation to a regime with significant cross-stream circulation and thermal entrainment. Our analysis of the steady profiles of the plumes shows that they initially rise with a constant vertical rise velocity. This rise velocity depends on the buoyancy flux and ambient viscosity but is almost independent of the centerline plume viscosity, which suggests that most of the thermal plume has a viscosity that is much closer to the ambient viscosity than the centerline viscosity. As the plumes approach the lid, they decelerate as the viscous drag on them steadily increases. The lateral spreading of the plumes under the lid is found to be well described by similarity solutions derived for the spreading of compositional plumes on a rigid surface, if the effective viscosity of the thermal plumes is taken to be the ambient viscosity rather than the centerline viscosity. A similar theoretical model is found to roughly predict the upstream spreading of thermal plumes at low shear, but it breaks down at moderate to high shear, where the entire plumes are advected downstream. When our results are applied to the Earth, we find that mantle plumes are mostly divided into only two flow regimes in the upper mantle: plumes under slow moving plates experience upstream flow and negligible cross-stream circulation, while plumes under faster moving plates (including all Pacific plumes) experience significant cross-stream circulation and are advected downstream. We also demonstrate that geochemical heterogeneities in a plume's source region will result in an azimuthally zoned plume and in an asymmetric geographical distribution of geochemical heterogeneities in the erupted hot spot basalts, as is seen in the Hawaiian, Galapagos, Marquesas, and Tahiti/Society island chains. For individual mantle plumes, we determine their diameter and vertical rise velocity as well as the extent of upstream spreading and the rate of lateral spreading under the lithosphere.

On the validity of two-dimensional numerical approaches to time-dependent thermal convection

Abstract: High-resolution computer simulations of two-dimensional (2D) convection are often used to investigate turbulent flows. In this paper we compare numerical simulations in 2D with three-dimensional (3D) simulations. We investigate flows at a fixed Rayleigh number of Ra = 106. The velocity boundary conditions are rigid for the upper and lower boundary and stress-free for the side walls. For high values of the Prandtl number the flow structure and global quantities such as the Nusselt number (Nu) and the Reynolds number (Re) show similar behaviour in 3D and 2D simulations. For values of the Prandtl number smaller than unity, however, the 2D results diverge strongly form the 3D findings. This is true not only for global properties (Nu, Re) but also for local properties such as the structure of the boundary layer and the shapes of the up- and down-wellings. We relate this behaviour to the different large-scale structure of the flow and the toroidal component of the kinetic energy.

Long relaxation times and tilt sensitivity in Rayleigh Bénard turbulence

Abstract: We study the influence of a small tilt angle ( $0 < \theta < 3 \times 10^{-2}$ rd) on the Nusselt number in a 1/2 aspect ratio Rayleigh-Benard cell, at high Rayleigh number (5 x 1011 < Ra < 4 x 1012). The small decrease observed is interpreted as revealing a two rolls structure of the flow. Transitions between different global flows are also observed, on very long times, comparable to the diffusion time on the whole cell. The consequence is that the Nusselt number observed in most high Ra experiments should significantly depend on initial conditions.

Thermal convection in faulted extensional sedimentary basins: theoretical results from finite‐element modeling

Abstract: Thermal convection has the potential to be a significant and widespread mechanism of fluid flow, mass transport, and heat transport in rift and other extensional basins. Based on numerical simulation results, large-scale convection can occur on the scale of the basin thickness, depending on the Rayleigh number for the basin. Our analysis indicates that for syn-rift and early post-rift settings with a basin thickness of 5 km, thermal convection can occur for basal heat flows ranging from 80 to 150 mW m−2, when the vertical hydraulic conductivity is on the order of 1.5 m year−1 and lower. The convection cells have characteristic wavelengths and flow patterns depending on the thermal and hydraulic boundary conditions. Steeply dipping extensional faults can provide pathways for vertical fluid flow across large thicknesses of basin sediments and can modify the dynamics of thermal convection. The presence of faults perturbs the thermal convective flow pattern and can constrain the size and locations of convection cells. Depending on the spacing of the faults and the hydraulic properties of the faults and basin sediments, the convection cells can be spatially organized to align with adjacent faults. A fault-bounded cell occurs when one convection cell is constrained to occupy a fault block so that the up-flow zone converges into one fault zone and the down-flow zone is centred on the adjacent fault. A fault-bounded cell pair occurs when two convection cells occupy a fault block with the up-flow zone located between the faults and the down-flow zones centred on the adjacent faults or with the reverse pattern of flow. Fault-bounded cells and cell pairs can be referred to collectively as fault-bounded convective flow. The flow paths in fault-bounded convective flow can be lengthened significantly with respect to those of convection cells unperturbed by the presence of faults. The cell pattern and sense of circulation depend on the fault spacing, sediment and fault permeabilities, lithologic heterogeneity, and the basal heat flow. The presence of fault zones also extends the range of conditions for which thermal convection can occur to basin settings with Rayleigh numbers below the critical value for large-scale convection to occur in a basin without faults. The widespread potential for the occurrence of thermal convection suggests that it may play a role in controlling geological processes in rift basins including the acquisition and deposition of metals by basin fluids, the distribution of diagenetic processes, the temperature field and heat flow, petroleum generation and migration, and the geochemical evolution of basin fluids. Fault-bounded cells and cell pairs can focus mass and heat transport from longer flow paths into fault zones, and their discharge zones are a particularly favourable setting for the formation of sediment-hosted ore deposits near the sea floor.

Interaction between natural convection and radiation in a square cavity heated from below

Abstract: The radiation effect of gray surfaces on multiple steady-state solutions obtained in a square enclosure filled with air has been investigated numerically, by a finite-difference procedure. The bottom and top surfaces of the cavity are, respectively, heated and cooled at constant temperatures, while its vertical walls are adiabatic. Parameters of the problem are the Rayleigh number ( ) and the surface emissivity ( ). The results obtained show that surface radiation alters significantly the existence range of the solution and the average heat transfer through the horizontal walls of the cavity. The parameter is found to reduce considerably the critical Rayleigh number characterizing the transition toward the oscillatory convection.

Velocity oscillations in turbulent Rayleigh–Bénard convection

Abstract: A systematic study of velocity oscillations in turbulent thermal convection is carried out in small aspect-ratio cells filled with water. Local velocity fluctuations and temperature-velocity cross-correlation functions are measured over varying Rayleigh numbers and spatial positions across the entire convection cell. These structural measurements reveal how the thermal plumes interact with the bulk fluid in a closed cell and provide an interesting physical picture for the dynamics of the temperature and velocity oscillations in turbulent convection.

Thermomagnetic convection in a square enclosure using a line dipole

Abstract: We have simulated thermomagnetic convection in a differentially heated square cavity with an infinitely long third dimension. The cavity is under the influence of an imposed two-dimensional magnetic field that conforms to the Maxwell’s equations. Our objective is to characterize the thermomagnetic convection in terms of the geometric length scales, magnetic fluid properties, temperature differences, and strengths of the imposed magnetic fields. Fluid motion occurs due to both the gradients of the magnetic field and the temperature. Colder fluid that has a larger magnetic susceptibility is attracted toward regions with larger field strength during thermomagnetic convection, which displaces warmer fluid of lower susceptibility. The height-averaged Nusselt number Nuav increases with increasing magnetic dipole strength and temperature but decreases with increasing fluid viscosity. Thermomagnetic heat transfer increases when the length scale decreases if the dipole strength of the source magnet is constant. Th...

Simulation of mixed convection heat transfer to carbon dioxide at supercritical pressure

Abstract: Computational simulations of turbulent mixed convection heat transfer experiments using carbon dioxide at supercritical pressure have been performed by solving the Reynolds averaged transpo...

On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

Abstract: This paper presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number $R$ for the onset of oscillatory and stationary convection are close to each other, and we develop and analyse a nonlinear theory in such a parameter regime which takes into account those mixed stationary and oscillatory modes of convection with common wavenumber vectors. Under a near-eutectic approximation and in the limit of large far-field temperature, we first determine a number of weakly nonlinear solutions, and then the stability of these solutions is investigated. The most interesting result is the preference for a mixed solution composed of standing and stationary hexagonal modes over a relatively wide range of the parameter values and for $R$ just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. ( Nature , vol. 359, 1992, pp. 406–408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes.