Showing papers on "Rayleigh number published in 2011"

Multiple scaling in the ultimate regime of thermal convection

Abstract: Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse, Phys. Rev. Lett. 86, 3316 (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu∝Ra0.14, Nu∝Ra0.22, and Nu∝Ra0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the ...

Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Abstract: Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-B\'enard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as $0.588Pr^{-1/2}$ for $Pr\ll Pr^*$ and as $0.982 Pr^{-1/3}$ for $Pr^*\ll\Pr$, with $Pr^*= 0.046$. Comparison of the Prandtl--Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl--Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-B\'enard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than $\sim\Ra^{0.15}$. This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations.

Laminar natural convection of power-law fluids in a square enclosure with differentially heated side walls subjected to constant temperatures

Abstract: Two-dimensional steady-state simulations of laminar natural convection in square enclosures with differentially heated sidewalls subjected to constant wall temperatures have been carried out where the enclosures are considered to be completely filled with non-Newtonian fluids obeying the power-law model. The effects of power-law index n in the range 0.6 ⩽ n ⩽ 1.8 on heat and momentum transport are investigated for nominal values of Rayleigh number (Ra) in the range 103–106 and a Prandtl number (Pr) range of 10–105. It is found that the mean Nusselt number Nu ¯ increases with increasing values of Rayleigh number for both Newtonian and power-law fluids. However, Nu ¯ values obtained for power-law fluids with n 1 ( n > 1 ) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to strengthening (weakening) of convective transport. With increasing shear-thickening (i.e. n > 1) the mean Nusselt number Nu ¯ settles to unity ( Nu ¯ = 1.0 ) as heat transfer takes place principally due to thermal conduction. The effects of Prandtl number have also been investigated in detail and physical explanations are provided for the observed behaviour. New correlations are proposed for the mean Nusselt number Nu ¯ for both Newtonian and power-law fluids which are shown to satisfactorily capture the correct qualitative and quantitative behaviour of Nu ¯ in response to changes in Ra, Pr and n.

Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection

Abstract: Results from direct numerical simulation for three-dimensional Rayleigh–Benard convection in samples of aspect ratio and up to Rayleigh number are presented. The broad range of Prandtl numbers is considered. In contrast to some experiments, we do not see any increase in with increasing , neither due to an increasing , nor due to constant heat flux boundary conditions at the bottom plate instead of constant temperature boundary conditions. Even at these very high , both the thermal and kinetic boundary layer thicknesses obey Prandtl–Blasius scaling.

Natural convection of Cu–water nanofluid in a cavity with partially active side walls

Abstract: The buoyancy-driven fluid flow and heat transfer in a square cavity with partially active side walls filled with Cu–water nanofluid is investigated numerically. The active parts of the left and the right side walls of the cavity are maintained at temperatures T h and T c , respectively, with T h > T c . The enclosure’s top and bottom walls as well as the inactive parts of its side walls are kept insulated. The governing equations in the two-dimensional space are discretized using the control volume method. A proper upwinding scheme is employed to obtain stabilized solutions. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, the locations of the active parts of the side walls, and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the cavity are investigated. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nanoparticles. Moreover, the maximum average Nusselt number for the high and the low Rayleigh numbers occur for the bottom–middle and the middle–middle locations of the thermally active parts, respectively.

Non-oscillatory and oscillatory nanofluid bio-thermal convection in a horizontal layer of finite depth

Abstract: The onset of bio-thermal convection in a suspension containing both nanoparticles and gyrotactic microorganisms, such as algae, is considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are included in the model. The suspension occupies a horizontal layer of finite depth. The lower boundary of the layer is assumed rigid while at the upper boundary both cases of either rigid or stress-free top boundaries are considered. A linear instability analysis is performed and the resulting eigenvalue problem is solved analytically using the Galerkin method. The cases of oscillatory and non-oscillatory convection are studied. Investigation of the dependence of the thermal Rayleigh number on the nanoparticle Rayleigh number and the bioconvection Rayleigh number is performed. The boundaries of oscillatory and non-oscillatory instability are established. The effect of nanoparticles can be either stabilizing or destabilizing, depending on whether the basic nanoparticle distribution is bottom-heavy or top-heavy. The effect of upswimming microorganisms is generally destabilizing.

Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries.

Abstract: Rigorous upper limits on the vertical heat transport in two-dimensional Rayleigh-B\'enard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $\mathrm{Nu}\ensuremath{\le}0.2891{\mathrm{Ra}}^{5/12}$ uniformly in the Prandtl number Pr. This scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.

Connecting flow structures and heat flux in turbulent Rayleigh-Bénard convection

Abstract: The aspect ratio (Γ) dependence of the heat transfer (Nusselt number Nu in dimensionless form) in turbulent (two-dimensional) Rayleigh-Benard convection is numerically studied in the regime 0.4⩽Γ⩽1.25 for Rayleigh numbers 107⩽Ra⩽Ra9 and Prandtl numbers Pr=0.7 (gas) and 4.3 (water). Nu(Γ) shows a very rich structure with sudden jumps and sharp transitions. We connect these structures to the way the flow organizes itself in the sample and explain why the aspect ratio dependence of Nu is more pronounced for small Pr. Even for fixed Γ different turbulent states (with different resulting Nu) can exist, between which the flow can or cannot switch. In the latter case the heat transfer thus depends on the initial conditions.

Numerical Analysis of Al2O3/Water Nanofluids Natural Convection in a Wavy Walled Cavity

Abstract: In the present work, heat transfer enhancement of Al2O3-water nanofluids in natural convection applied to differentially heated wavy cavities is investigated numerically The governing equations are written in stream function-vorticity form and solved by using the finite volume technique The effective thermal conductivity and viscosity of the nanofluid are approximated by the Maxwell–Garnetts and Brinkman models, respectively The solutions are presented by streamlines, isotherms, local and mean Nusselt numbers, and velocity profiles for different Rayleigh numbers (103 ≤ Ra ≤ 105), amplitude of wavy wall (085 ≤ a ≤ 11), and different volume fraction of nanofluids (ϕ = 005 and 01) It is observed that the addition of a nanoparticle of Al2O3 into the base fluid increases the mean Nusselt number It is also more effective on flow field than that of temperature distribution The geometry parameter or surface waviness can be decided to heat transfer regime even for the same Rayleigh number

The impact of geochemistry on convective mixing in a gravitationally unstable diffusive boundary layer in porous media: CO2 storage in saline aquifers

Abstract: The storage of carbon dioxide and acid gases in deep geological formations is considered a promising option for mitigation of greenhouse gas emissions. An understanding of the primary mechanisms such as convective mixing and geochemistry that affect the long-term geostorage process in deep saline aquifers is of prime importance. First, a linear stability analysis of an unstable diffusive boundary layer in porous media is presented, where the instability occurs due to a density difference between the carbon dioxide saturated brine and the resident brine. The impact of geochemical reactions on the stability of the boundary layer is examined. The equations are linearised, and the obtained system of eigenvalue problems is solved numerically. The linear stability results have revealed that geochemistry stabilises the boundary layer as reaction consumes the dissolved carbon dioxide and makes the density profile, as the source of instability, more uniform. A detailed physical discussion is also presented with an examination of vorticity and concentration eigenfunctions and streamlines' contours to reveal how the geochemical reaction may affect the hydrodynamics of the process. We also investigate the effects of the Rayleigh number and the diffusion time on the stability of a boundary layer coupled with geochemical reactions. Nonlinear direct numerical simulations are also presented, in which the evolution of density-driven instabilities for different reaction rates is discussed. The development of instability is precisely studied for various scenarios. The results indicate that the boundary layer will be more stable for systems with a higher rate of reaction. However, our quantitative analyses show that more carbon dioxide may be removed from the supercritical free phase as the measured flux at the boundary is always higher for flow systems coupled with stronger geochemical reactions.