Showing papers on "Rayleigh number published in 2005"

Convection Heat Transfer

Abstract: Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions

Abstract: Previous studies of fluid convection in porous media have considered the onset of convection in isotropic systems and the steady convection in anisotropic systems. This paper bridges between these and develops new results for the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. These results are relevant to sedimentary formations where the average vertical permeability is some fraction γ of the average horizontal permeability. Linear and global stability analyses are used to define the critical time tc at which the instability occurs as a function of γ and the dimensionless Rayleigh-Darcy number Ra* for both thermal and solute-driven convection in an infinite horizontal slab. Numerical results and approximate analytical solutions are obtained for both a slab of finite thickness and the limit of large slab thickness. For a thick slab, the increase in tc as γ decreases is approximately given by (1+γ)4∕(16γ2). One important application is to the geological storage of carbon dioxide where it is shown that the use of an effective vertical permeability in estimating the critical time is only valid for low permeabilities. The time scale for the onset of convection in geological storage can range from less than a year (for high-permeability formations) to decades or centuries (for low-permeability ones).

Heat transport by turbulent Rayleigh-Bénard 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 σ=4.38) with diameters D = 49.7, 24.8, and 9.2 cm, all with aspect ratio F≡D/L≃1 (L is the sample height). In addition, we present data for D=49.7 and r=1.5,2,3, and 6. For each sample the data cover a range of a little over a decade of R. For Γ≃1 they jointly span the range 10 7 ≤ R ≤ 10 11 . Where needed, the data were corrected for the influence of the finite conductivity of the top and bottom plates and of the sidewalls on the heat transport in the fluid to obtain estimates of N∞ for plates with infinite conductivity and sidewalls of zero conductivity

A Two-Temperature Model for Solid-Liquid Phase Change in Metal Foams

Abstract: Transient solid-liquid phase change occurring in a phase-change material (PCM) embedded in a metal foam is investigated. Natural convection in the melt is considered. Volume-averaged mass and momentum equations are employed, with the Brinkman-Forchheimer extension to the Darcy law to model the porous resistance. Owing to the difference in the thermal diffusivities between the metal foam and the PCM, local thermal equilibrium between the two is not assured. Assuming equilibrium melting at the pore scale, separate volume-averaged energy equations are written for the solid metal foam and the PCM and are closed using an interstitial heat transfer coefficient. The enthalpy method is employed to account for phase change. The governing equations are solved implicitly using the finite volume method on a fixed grid. The influence of Rayleigh, Stefan, and interstitial Nusselt numbers on the temporal evolution of the melt front location, wall Nusselt number, temperature differentials between the solid and fluid, and the melting rate is documented and discussed. The merits of incorporating metal foam for improving the effective thermal conductivity of thermal storage systems are discussed.

Natural Convection in a Vertical Microchannel

Abstract: It is highly desirable to understand the fluid flow and the heat transfer characteristics of buoyancy-induced micropump and microheat exchanger in microfluidic and thermal systems. In this study, we analytically investigate the fully developed natural convection in an open-ended vertical parallel-plate microchannel with asymmetric wall temperature distributions. Both of the velocity slip and the temperature jump conditions are considered because they have countereffects both on the volume flow rate and the heat transfer rate. Results reveal that in most of the natural convection situations, the volume flow rate at microscale is higher than that at macroscale, while the heat transfer rate is lower. It is, therefore, concluded that the temperature jump condition induced by the effects of rarefaction and fluid-wall interaction plays an important role in slip-flow natural convection.

Heat transport in turbulent Rayleigh-Bénard convection: Effect of finite top- and bottom-plate conductivities

Abstract: We describe three apparatus, known as the large, medium, and small apparatus, used for high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for cylindrical samples of fluid and present results illustrating the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid. We used water samples at a mean temperature of 40°C (Prandtl number σ=4.4). The samples in the large apparatus had a diameter D of 49.69cm and heights L≃116.33, 74.42, 50.61, and 16.52cm. For the medium apparatus we had D=24.81cm, and L=90.20 and 24.76cm. The small apparatus contained a sample with D=9.21cm and L=9.52cm. For each aspect ratio Γ≡D∕L the data covered a range of a little over a decade of R. The maximum R≃1×1012 with Nusselt number N≃600 was reached for Γ=0.43. Measurements were made with both aluminum (conductivity λp=161W∕mK) and copper (λp=391W∕mK) top and bottom plates of nominally identical size and shape. For the large and medium apparatus the...

Convective heat transfer as a function of wavelength: Implications for the cooling of the Earth

Abstract: [1] Attempting to reconstruct the thermal history of the Earth from a geophysical point of view has for a long time been in disagreement with geochemical data. The geophysical approach uses parameterized models of mantle cooling. The rate of cooling of the Earth at the beginning of its history obtained in these models is generally too rapid to allow a sufficient present-day secular cooling rate. Geochemical estimates of radioactive element concentrations in the mantle then appear too low to explain the observed present mantle heat loss. Cooling models use scaling laws for the mean heat flux out of the mantle as a function of its Rayleigh number of the form Q / Ra b . Recent studies have introduced very low values of the exponent b, which can help reduce the cooling rate of the mantle. The present study instead focuses on the coefficient C in the relation Q = CR a b and, in particular, on its variation with the wavelength of convection. The heat transfer strongly depends on the wavelength of convection. The length scale of convection in Earth’s mantle is that of plate tectonics, implying convective cells of wide aspect ratio. Taking into account the long wavelength of convection in Earth’s mantle can significantly reduce the efficiency of heat transfer. The likely variations of this wavelength with the Wilson cycle thus imply important variations of the heat flow out of the Earth on a intermediate timescale of 100 Ma, which renders parameterized models of thermal evolution inaccurate for quantitative predictions.

Local thermal non-equilibrium in porous medium convection

Abstract: Many papers exist which either derive or use equations which govern local thermal nonequilibrium phenomena in porous medium convection, where the intrinsic average of the temperatures of the solid and fluid phases may be regarded as being different. We compile and present the most commonly used of these model equations. Attention is then focused on describing some of the most recent research using these equations. Attention is focussed primarily on free and forced convection boundary layers, and on free convection within cavities.

Free-Convective Heat Transfer: With Many Photographs of Flows and Heat Exchange

Abstract: Basic Statements and Equations of Free Convection.- Free Convection on a Plane.- Free Convection on Curved Surfaces.- Natural Convection in Enclosures.- Free Convection in Tubes and Channels, on Ribbed Surfaces and in Tube Bundles.- Nonstationary Processes of Free Convection.- Heat Transfer by Mixed Convection.- Heat Transfer in Media with Special Properties.

Heat transport by turbulent Rayleigh–Bénard convection in 1 m diameter cylindrical cells of widely varying aspect ratio

Abstract: High-precision measurements of the Nusselt number Nu as a function of the Rayleigh number Ra have been made in water-filled 1 m diameter cylindrical cells of aspect ratio r=0.67, 1, 2, 5, 10 and 20. The measurements were conducted at the Prandtl number Pr ≃4 with Ra varying from 1 × 10 7 to 5 × 10 12 . When corrections for the finite conductivity of the top and bottom plates are made, the estimates obtained of Nu∞ for perfectly conducting plates may be described by a combination of two power laws Nu∞ =C 1 (Γ)Ra β 1 + C 2 (Γ)Ra β2 for all the aspect ratios

Turbulent thermal convection at high Rayleigh numbers for a Boussinesq fluid of constant Prandtl number

Abstract: The results from direct numerical simulations of turbulent Boussinesq convection are briefly presented. The flow is computed for a cylindrical cell of aspect ratio 1∕2 in order to compare with the results from recent experiments. The results span eight decades of Ra from 2×106 to 2×1014 and form the baseline data for a strictly Boussinesq fluid of constant Prandtl number (Pr=0.7). A conclusion is that the Nusselt number varies nearly as the 1∕3 power of Ra for about four decades towards the upper end of the Ra range covered.

Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence

Abstract: The Ra and Pr number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh–Benard turbulence, i.e., Rayleigh–Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh–Benard flow and therefore as model for the so-called “ultimate regime of thermal convection.” With respect to the Ra dependence of Nu and Re we confirm our earlier results [ D. Lohse and F. Toschi, “The ultimate state of thermal convection,” Phys. Rev. Lett. 90, 034502 (2003) ] which are consistent with the Kraichnan theory [ R. H. Kraichnan, “Turbulent thermal convection at arbitrary Prandtl number,” Phys. Fluids 5, 1374 (1962) ] and the Grossmann–Lohse (GL) theory [ S. Grossmann and D. Lohse, “Scaling in thermal convection: A unifying view,” J. Fluid Mech. 407, 27 (2000) ; “Thermal convection for large Prandtl number,” Phys. Rev. Lett. 86, 3316 (2001) ; “Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection,” Phys. Rev. E 66, 016305 (2002) ; “Fluctuations in turbulent Rayleigh–Benard convection: The role of plumes,” Phys. Fluids 16, 4462 (2004) ], which both predict Nu ∼ Ra1/2 and Re ∼ Ra1/2. However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory Nu ∼ Pr1/2, Re ∼ Pr−1/2. For the thermal and kinetic dissipation rates we find ϵθ/(κΔ2L−2) ∼ (Re Pr)0.87 and ϵu/(ν3L−4) ∼ Re2.77, both near (but not fully consistent) the bulk dominated behavior, whereas the temperature fluctuations do not depend on Ra and Pr. Finally, the dynamics of the heat transport is studied and put into the context of a recent theoretical finding by Doering et al. [“Comment on ultimate state of thermal convection” (private communication)].

Convective-conductive transitions and sensitivity of a convecting ice shell to perturbations in heat flux and tidal-heating rate: Implications for Europa

Abstract: We investigate the response of conductive and convective ice shells on Europa to variations of heat flux and interior tidal-heating rate. We present numerical simulations of convection in Europa's ice shell with Newtonian, temperature-dependent viscosity and tidal heating. Modest variations in the heat flux supplied to the base of a convective ice shell, �F , can cause large variations of the ice-shell thickness �δ . In contrast, for a conductive ice shell, largeF involves relatively small �δ . We demonstrate that, for a fluid with temperature-dependent viscosity, the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number Racr. This jump implies that, for a range of heat fluxes relevant to Europa, two equilibrium states—corresponding to a thin, conductive shell and a thick, convective shell—exist for a given heat flux. We show that, as a result, modest variations in heat flux near the critical Rayleigh number can force the ice shell to switch between the thin, conductive and thick, convective configurations over a ∼10 7 -year interval, with thickness changes of up to ∼10-30 km. Depending on the orbital and thermal history, such switches might occur repeatedly. However, existing evolution models based on parameterized- convection schemes have to date not allowed these transitions to occur. Rapid thickening of the ice shell would cause radial expansion of Europa, which could produce extensional tectonic features such as fractures or bands. Furthermore, based on interpretations for how features such as chaos and ridges are formed, several authors have suggested that Europa's ice shell has recently undergone changes in thickness. Our model provides a mechanism for such changes to occur.

Heat transport in turbulent Rayleigh-Benard convection: Effect of finite top- and bottom-plate conductivity

Abstract: We describe three apparatus, known as the large, medium, and small apparatus, used for high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for cylindrical samples of fluid and present results illustrating the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid. We used water samples at a mean temperature of 40 degrees C (Prandtl number sigma = 4.4). The samples in the large apparatus had a diameter D of 49.69 cm and heights L = 116.33, 74.42, 50.61, and 16.52 cm. For the medium apparatus we had D = 24.81 cm, and L = 90.20 and 24.76 cm. The small apparatus contained a sample with D = 9.21 cm, and L = 9.52 cm. For each aspect ratio Gamma = D/L the data covered a range of a little over a decade of R. The maximum R = 10^12 with Nusselt numbers N = 600 was reached for Gamma = 0.43. Measurements were made with both Aluminum and Copper top and bottom plates of nominally identical size and shape. For the large and medium apparatus the results with Aluminum plates fall below those obtained with Copper plates, thus confirming qualitatively the prediction by Verzicco that plates of finite conductivity diminish the heat transport in the fluid. The Nusselt number N_infinity for plates with infinite conductivity was estimated by fitting simultaneously Aluminum- and Copper-plate data sets to an effective powerlaw for N_infinity multiplied by a correction factor f(X) = 1 - exp[-(aX)^b] that depends on the ratio X of the thermal resistance of the fluid to that of the plates as suggested by Verzicco. Within their uncertainties the parameters a and b were independent of Gamma for the large apparatus and showed a small Gamma-dependence for the medium apparatus. The correction was larger for the large, smaller for the medium, and negligible for the small apparatus.

The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall

Abstract: The lattice Boltzmann equation method in two dimensions was used to analyse natural convective flows. The method was validated with experiments in an open cavity with one of the vertical walls divided into two parts, the lower part conductive, the upper part and all the other walls adiabatic. An upward thermal boundary layer formed near the conductive wall. This layer gave way to a wall plume. The numerical results compared well with experiments in the laminar (. The wall plume grows in three stages: in the first with constant acceleration, in the second with constant ascending velocity and in the third with negative acceleration due to the presence of the top boundary layer. The acceleration of the first stage and the velocity of the second both scale with the Rayleigh number.