Showing papers on "Rayleigh number published in 2010"

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 (BL) thicknesses in turbulent Rayleigh–Benard (RB) convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl–Blasius BL equations, we calculate the ratio between the thermal and kinetic BL thicknesses, which depends on the Prandtl number only. It is approximated as for and as for , with . Comparison of the Prandtl–Blasius velocity BL thickness with that evaluated in the direct numerical simulations by Stevens et al (2010 J. Fluid Mech. 643 495) shows very good agreement between them. Based on the Prandtl–Blasius-type considerations, we derive a lower-bound estimate for the minimum number of computational mesh nodes required to conduct accurate numerical simulations of moderately high (BL-dominated) turbulent RB convection, in the thermal and kinetic BLs close to the bottom and top plates. It is shown that the number of required nodes within each BL depends on and and grows with the Rayleigh number not slower than . This estimate is in excellent agreement with empirical results, which were based on the convergence of the Nusselt number in numerical simulations

Convective dissolution of carbon dioxide in saline aquifers

Abstract: [1] Geological carbon dioxide (CO2) storage is a means of reducing anthropogenic emissions. Dissolution of CO2 into the brine, resulting in stable stratification, increases storage security. The dissolution rate is determined by convection in the brine driven by the increase of brine density with CO2 saturation. We present a new analogue fluid system that reproduces the convective behaviour of CO2‐enriched brine. Laboratory experiments and high‐resolution numerical simulations show that the convective flux scales with the Rayleigh number to the 4/5 power, in contrast with a classical linear relationship. A scaling argument for the convective flux incorporating lateral diffusion from downwelling plumes explains this nonlinear relationship for the convective flux, provides a physical picture of high Rayleigh number convection in a porous medium, and predicts the CO2 dissolution rates in CO2 accumulations. These estimates of the dissolution rate show that convective dissolution can play an important role in enhancing storage security. Citation: Neufeld,J.A.,M.A .Hesse,A.Riaz,M. A.H allworth, H. A. Tchelepi, and H. E. Huppert (2010), Convective dissolution of carbon dioxide in saline aquifers, Geophys. Res. Lett., 37, L22404, doi:10.1029/2010GL044728.

Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection

Abstract: Results from direct numerical simulation (DNS) for three-dimensional Rayleigh–Benard convection in a cylindrical cell of aspect ratio 1/2 and Prandtl number Pr=0.7 are presented. They span five decades of Rayleigh number Ra from 2 × 106 to 2 × 1011. The results are in good agreement with the experimental data of Niemela et al. (Nature, vol. 404, 2000, p. 837). Previous DNS results from Amati et al. (Phys. Fluids, vol. 17, 2005, paper no. 121701) showed a heat transfer that was up to 30% higher than the experimental values. The simulations presented in this paper are performed with a much higher resolution to properly resolve the plume dynamics. We find that in under-resolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the better-resolved ones, because of insufficient thermal dissipation mainly close to the sidewall (where the grid cells are largest), and therefore the Nusselt number in under-resolved simulations is overestimated. Furthermore, we compare the best resolved thermal boundary layer profile with the Prandtl–Blasius profile. We find that the boundary layer profile is closer to the Prandtl–Blasius profile at the cylinder axis than close to the sidewall, because of rising plumes close to the sidewall

Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model

Abstract: The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.

Topological design of heat dissipating structure with forced convective heat transfer

Abstract: This paper discusses the use of the topology optimization formulation for designing a heat dissipating structure that utilizes forced convective heat transfer. In addition to forced convection, there is also natural convection due to natural buoyancy forces induced by local heating inside fluid. In the present study, the temperature distribution due to forced convection, neglecting buoyancy and viscous dissipation inside fluid, was simulated and optimized. In order to analyze the heat transfer equation with forced convective heat loss and the Navier-Stokes equation, a common sequential computational procedure for this thermo/hydraulic characteristic was implemented. For topology optimization, four material properties were interpolated with respect to spatially defined density design variables: the inverse permeability in the Navier-Stokes equation, the conductivity, density, and the specific heat capacity of the heat transfer equation. From numerical examples, it was found that the balance between the conduction and convection of fluid is of central importance to the design of heat dissipating structures.

Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection

Abstract: The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh–Benard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra. The Prandtl number is Pr = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 107 and 109. The Nusselt number Nu is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law Nu = A × Raβ shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of Nu(Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 107 and Ra = 108, respectively. This is the point where the LSC undergoes a transition from a single-roll to a double-roll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ ≳ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen–Loeve or proper orthogonal decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale – the Kolmogorov scale ηK – and the largest scale – the cell height H – are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ ≳ 8).

Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls

Abstract: In this study, two-dimensional steady-state simulations of laminar natural convection in square enclosures with differentially heated sidewalls have been carried out where the enclosures are considered to be completely filled with a yield stress fluid obeying the Bingham model. Yield stress effects 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 0.1–100. It is found that the mean Nusselt number N u ¯ increases with increasing values of Rayleigh number for both Newtonian and Bingham fluids. However, N u ¯ values obtained for Bingham fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to weakening of convective transport. The mean Nusselt number N u ¯ in the case of Bingham fluids is found to decrease with increasing Bingham number, and, for large values of Bingham number Bn, the value settles to unity ( N u ¯ = 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 N u ¯ for both Newtonian and Bingham fluids which are shown to satisfactorily capture the correct qualitative and quantitative behaviour of N u ¯ in response to changes in Ra, Pr and Bn.

Buoyancy Driven Heat Transfer of Nanofluids in a Tilted Enclosure

Abstract: Buoyancy driven heat transfer of water-based nanofluids in a differentially heated, tilted enclosure is investigated in this study. The governing equations (obtained with the Boussinesq approximation) are solved using the polynomial differential quadrature method for an inclination angle ranging from 0 deg to 90 deg, two different ratios of the nanolayer thickness to the original particle radius (0.02 and 0.1), a solid volume fraction ranging from 0% to 20%, and a Rayleigh number varying from 104 to 106. Five types of nanoparticles, Cu, Ag, CuO, Al2O3, and TiO2 are taken into consideration. The results show that the average heat transfer rate from highest to lowest is for Ag, Cu, CuO, Al2O3, and TiO2. The results also show that for the particle radius generally used in practice ( 0.1 or 0.02), the average heat transfer rate increases to 44% for Ra 104, to 53% for Ra 105, and to 54% for Ra 106 if the special case of 90 deg, which also produces the minimum heat transfer rates, is not taken into consideration. As for 90 deg, the heat transfer enhancement reaches 21% for Ra 104, 44% for Ra 105, and 138% for Ra 106. The average heat transfer rate shows an increasing trend with an increasing inclination angle, and a peak value is detected. Beyond the peak point, the foregoing trend reverses and the average heat transfer rate decreases with a further increase in the inclination angle. Maximum heat transfer takes place at 45 deg for Ra 104 and at 30 deg for Ra 105 and 106. DOI: 10.1115/1.4000744

Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection

Abstract: The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh-B\'{e}nard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio $\Gamma$ of a closed cylindrical cell and the Rayleigh number $Ra$. For small and moderate aspect ratios, the global heat transfer law $Nu=A\times Ra^{\beta}$ shows a power law dependence of both fit coefficients $A$ and $\beta$ on the aspect ratio. A minimum Nusselt number coincides with the point where the LSC undergoes a transition from a single-roll to a double-roll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations. The aspect ratio dependence of the turbulent heat transfer for small and moderate $\Gamma$ is in line with a varying amount of energy contained in the LSC, as quantified by the Proper Orthogonal Decomposition analysis. For $\Gamma\gtrsim 8$ the heat transfer becomes independent of the aspect ratio.

Numerical modelling of finite-amplitude electro-thermo-convection in a dielectric liquid layer subjected to both unipolar injection and temperature gradient

Abstract: In this paper, we solve numerically the entire set of equations associated with the electro-thermo-convective phenomena that take place in a planar layer of dielectric liquid heated from below and subjected to unipolar injection. For the first time the whole set of coupled equations is solved: Navier–Stokes equations, electrohydrodynamic (EHD) equations and the energy equation. We first validate the numerical simulation by comparing the electro-convection stability criteria with ones obtained with a stability approach. The numerical solution of the electro-thermo-convection problem is then presented entirely with a detailed analysis of stability parameters. In particular, the relation between fluid velocity, non-dimensional electrical parameter T, Rayleigh number Ra and Prandtl number Pr is given. An analytical model is presented in order to understand the flow behaviour at some critical conditions. The way that the onset of motion passes from purely electrical convection to purely thermal convection is, in particular, investigated and explained in detail. Finally, a result on the heat transfer enhancement due to electro-convection is exhibited and compared with data from experimental works available in this field.

A community benchmark for 2-D Cartesian compressible convection in the Earth's mantle

Abstract: Benchmark comparisons are an essential tool to verify the accuracy and validity of computational approaches to mantle convection. Six 2-D Cartesian compressible convection codes are compared for steady-state constant and temperature-dependent viscosity cases as well as time-dependent constant viscosity cases. In general we find good agreement between all codes when comparing average flow characteristics such as Nusselt number and rms velocity. At Rayleigh numbers near 106 and dissipation numbers between 0 and 2, the results differ by approximately 1 per cent. Differences in discretization and use of finite volumes versus finite elements dominate the differences. There is a small systematic difference between the use of the anelastic liquid approximation (ALA) compared to that of the truncated ALA. In determining the onset of time-dependence, there was less agreement between the codes with a spread in the Rayleigh number where the first bifurcation occurs ranging from 7.79 × 10^5 to 1.05 × 10^6.

Long wavelength convection, Poiseuille–Couette flow in the low-viscosity asthenosphere and the strength of plate margins

Abstract: SUMMARY Mixed heated 3-D mantle convection simulations with a low-viscosity asthenosphere reveal relatively short and long wavelength regimes with different scalings in terms of surface velocity and surface heat flux and show that mantle flow in the lithosphere–asthenosphere region is a Poiseuille–Couette flow. The Poiseuille/Couette velocity magnitude ratio, D/U, allows us to characterize solid-state flow in the asthenosphere and to predict the regime transition. The transition from dominantly pressure-driven Poiseuille flow at shorter wavelengths to dominantly shear-driven Couette flow at long wavelengths depends on the relative strength of lithosphere and asthenosphere and is associated with a switch in the dominant resistance to convective motion. In the Poiseuille regime significant resistance is provided by plate-bending, whereas in the Couette regime most resistance is due to vertical shear in the bulk mantle. The Couette case corresponds to classical scaling ideas for mantle convection whereas the Poiseuille case, with asthenospheric velocities exceeding surface velocities, is an example of a sluggish lid mode of mantle convection that has more recently been invoked for thermal history models of the Earth. Our simulations show that both modes can exist for the same level of convective vigour (i.e. Rayleigh number) but at different convective wavelengths. Additional simulations with temperature- and yield-stress dependent viscosity show consistent behaviour and suggest an association of the regime crossover with the relative strength of plate margins. Our simulations establish a connection between the strength of plate margins, solid-state flow in the asthenosphere and the wavelength of mantle convection. This connection suggests that plate tectonics in the sluggish lid mode is wavelength dependent and potentially more robust than previously envisioned.

Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh-Bénard convection

Abstract: In turbulent thermal convection in cylindrical samples with an aspect ratio Γ≡D/L (D is the diameter and L the height), the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number 1/Roc, with 1/Roc∝1/Γ. We present a Ginzburg-Landau–like model that explains the existence of a bifurcation at finite 1/Roc as a finite-size effect. The model yields the proportionality between 1/Roc and 1/Γ and is consistent with several other measured or computed system properties.

Prandtl-Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh-Bénard convection

Abstract: The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Benard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense