Showing papers on "Nusselt number published in 2006"

Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields

Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet

Abstract: An analysis is made for the steady mixed convection boundary layer flow near the two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in its own plane. The stretching velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation-point. Two equal and opposite forces are impulsively applied along the x-axis so that the wall is stretched, keeping the origin fixed in a viscous fluid of constant ambient temperature. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using a very efficient numerical scheme known as the Keller-box method. The features of the flow and heat transfer characteristics are analyzed and discussed in detail. Both cases of assisting and opposing flows are considered. It is observed that, for assisting flow, both the skin friction coefficient and the local Nusselt number increase as the buoyancy parameter increases, while only the local Nusselt number increases but the skin friction coefficient decreases as the Prandtl number increases. For opposing flow, both the skin friction coefficient and the local Nusselt number decrease as the buoyancy parameter increases, but both increase as Pr increases. Comparison with known results is excellent.

Thermal-hydraulic performance of MEMS-based pin fin heat sink

Abstract: An experimental study on heat transfer and pressure drop of de-ionized water over a hank of shrouded staggered micro pin fins 243 μm long with hydraulic diameter of 99.5 μm has been performed. Average heat transfer coefficients have been obtained for effective heat fluxes ranging from 3.8 to 167 W/cm 2 and Reynolds numbers from 14 to 112. The results were used to derive the Nusselt numbers, total thermal resistances, and friction factors. It has been found that for Reynolds numbers below ∼50 long tube correlations overpredicted the experimental Nusselt number, while at higher Reynolds numbers existing correlations predicted the results moderately well. Endwall effects, which diminish at high Reynolds numbers, and a delay in flow separation for compact pin fins were attributed to the obtained trend.

Numerical simulation of an asymptotically reduced system for rotationally constrained convection

Abstract: For rotationally constrained convection, the Taylor–Proudman theorem enforces an organization of nonlinear flows into tall columnar or compact plume structures. While coherent structures in convection under moderate rotation are exclusively cyclonic, recent experiments for rapid rotation have revealed a transition to equal populations of cyclonic and anticyclonic structures. Direct numerical simulation (DNS) of this regime is expensive, however, and existing simulations have yet to reveal anticyclonic vortical structures. In this paper, we use a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation to perform numerical simulation of Rayleigh–B´ enard convection in an infinite layer rotating uniformly about the vertical axis. Visualization indicates the existence of cyclonic and anticyclonic vortical populations for all parameters examined. Moreover, it is found that the flow evolves through three distinct regimes with increasing Rayleigh number (Ra). For small, but supercritical Ra ,t he fl ow is dominated by a cellular system of hot and cold columns spanning the fluid layer. As Ra increases, the number density of these columns decreases, the up- and downdrafts within them strengthen and the columns develop opposite-signed ‘sleeves’ in all fields. The resulting columns are highly efficient at transporting heat across the fluid layer. In the final regime, lateral mixing plays a dominant role in the interior and the columnar structure is destroyed. However, thermal plumes are still injected and rejected from the thermal boundary layers. We identify the latter two regimes with the vortex-grid and geostrophic turbulence regimes, respectively. Within these regimes, we investigate convective heat transport (measured by the Nusselt number), mean temperature profiles, and root-mean-square profiles of the temperature, vertical velocity and vertical vorticity anomalies. For all Prandtl numbers investigated, the mean temperature saturates in a non-isothermal profile as Ra increases owing to intense lateral mixing.

Non-Oberbeck-Boussinesq effects in strongly turbulent Rayleigh-Bénard convection

Abstract: Non-Oberbeck–Boussinesq (NOB) effects on the Nusselt number $Nu$ and Reynolds number $\hbox{\it Re}$ in strongly turbulent Rayleigh–Benard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the temperature at the horizontal midplane were measured. Three cells of different heights $L$, all filled with water and all with aspect ratio $\Gamma$ close to 1, were used. For each $L$, about 1.5 decades in $Ra$ were covered, together spanning the range $10^8 \,{\le}\, Ra \,{\le}\, 10^{11}$. For the largest temperature difference between the bottom and top plates, $\Delta \,{=}\, 40$K, the kinematic viscosity and the thermal expansion coefficient, owing to their temperature dependence, varied by more than a factor of 2. The Oberbeck–Boussinesq (OB) approximation of temperature-independent material parameters thus was no longer valid. The ratio $\chi$ of the temperature drops across the bottom and top thermal boundary layers became as small as $\chi\,{=}\,0.83$, which may be compared with the ratio $\chi \,{=}\, 1$ in the OB case. Nevertheless, the Nusselt number $Nu$ was found to be only slightly smaller (by at most 1.4%) than in the next larger cell with the same Rayleigh number, where the material parameters were still nearly height independent. The Reynolds numbers in the OB and NOB case agreed with each other within the experimental resolution of about 2%, showing that NOB effects for this parameter were small as well. Thus $Nu$ and $\hbox{\it Re}$ are rather insensitive against even significant deviations from OB conditions. Theoretically, we first account for the robustness of $Nu$ with respect to NOB corrections: the NOB effects in the top boundary layer cancel those which arise in the bottom boundary layer as long as they are linear in the temperature difference $\Delta$. The net effects on $Nu$ are proportional to $ \Delta^2$ and thus increase only slowly and still remain minor despite drastic material-parameter changes. We then extend the Prandtl–Blasius boundary-layer theory to NOB Rayleigh–Benard flow with temperature-dependent viscosity and thermal diffusivity. This allows calculation of the shift in the bulk temperature, the temperature drops across the boundary layers, and the ratio $\chi$ without the introduction of any fitting parameter. The calculated quantities are in very good agreement with experiment. When in addition we use the experimental finding that for water the sum of the top and bottom thermal boundary-layer widths (based on the slopes of the temperature profiles at the plates) remains unchanged under NOB effects within the experimental resolution, the theory also gives the measured small Nusselt-number reduction for the NOB case. In addition, it predicts an increase by about 0.5% of the Reynolds number, which is also consistent with the experimental data. By studying theoretically hypothetical liquids for which only one of the material parameters is temperature dependent, we are able to shed further light on the origin of NOB corrections in water: while the NOB deviation of $\chi$ from its OB value $\chi \,{=}\, 1$ mainly originates from the temperature dependence of the viscosity, the NOB correction of the Nusselt number primarily originates from the temperature dependence of the thermal diffusivity. Finally, we give predictions from our theory for the NOB corrections if glycerol were used as the operating liquid.

Bejan’s Heatlines and Masslines for Convection Visualization and Analysis

Abstract: Heatlines were proposed in 1983 by Kimura and Bejan (1983) as adequate tools for visualization and analysis of convection heat transfer. The masslines, their equivalent to apply to convection mass transfer, were proposed in 1987 by Trevisan and Bejan. These visualization and analysis tools proved to be useful, and their application in the fields of convective heat and/or mass transfer is still increasing. When the heat function and/or the mass function are made dimensionless in an adequate way, their values are closely related with the Nusselt and/or Sherwood numbers. The basics of the method were established in the 1980(s), and some novelties were subsequently added in order to increase the applicability range and facility of use of such visualization tools. Main steps included their use in unsteady problems, their use in polar cylindrical and spherical coordinate systems, development of similarity expressions for the heat function in laminar convective boundary layers, application of the method to turbulent flow problems, unification of the streamline, heatline, and massline methods (involving isotropic or anisotropic media), and the extension and unification of the method to apply to reacting flows. The method is now well established, and the efforts made towards unification resulted in very useful tools for visualization and analysis, which can be easily included in software packages for numerical heat transfer and fluid flow. This review describes the origins and evolution of the heatlines and masslines as visualization and analysis tools, from their first steps to the present. DOI: 10.1115/1.2177684

Magnetohydrodynamic Convective Flow of a Micropolar Fluid Past a Continuously Moving Vertical Porous Plate in the Presence of Heat Generation/Absorption

Abstract: Magnetohydrodynamic convective flow and heat transfer of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption with constant suction has been analyzed numerically. With appropriate transformations the boundary layer equations are transformed into nonlinear ordinary differential equa- tions. The local similarity solutions of the transformed dimensionless equations for the flow, microrotation, and heat transfer characteristics are evaluated using Nachtsheim- Swigert shooting iteration technique. Numerical results are presented in the form of velocity, microrotation, and temperature profiles within the boundary layer for different parameters entering into the analysis. Also the effects of the pertinent parameters on the local skin friction coefficient and rate of the heat transfer in terms of the local Nusselt number are also discussed. DOI: 10.1115/1.2136918

Turbulent convection at high Rayleigh numbers and aspect ratio 4

Abstract: We report measurements of the Nusselt number, Nu, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was Ra=2×10 13 . Except for the last half a decade or so of Ra, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the log Nu-log Ra slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios. The increasing slope over the last half a decade of Ra is inconclusive because the corresponding conditions are non-Boussinesq. Finally, we report a modified scaling relation between the plume advection frequency and Ra that collapses data for different aspect ratios.

Metal Foam and Finned Metal Foam Heat Sinks for Electronics Cooling in Buoyancy-Induced Convection

Abstract: In this paper, we present our recent experimental results on buoyancy-induced convection in aluminum metal foams of different pore densities [corresponding to 5, 10, 20, and 40 pores per in. (PPI)] and porosities (0.89-0.96). The results show that compared to a heated surface, the heat transfer coefficients in these heat sinks are five to six times higher. However, when compared to commercially available heat sinks of similar dimensions, the enhancement is found to be marginal. The experimental results also show that for a given pore size, the heat transfer rate increases with porosity, suggesting the dominant role played by conduction in enhancing heat transfer. On the other hand, if the porosity is held constant, the heat transfer rate is found to be lower at higher pore densities. This can be attributed to the higher permeability with the larger pores, which allows higher entrainment of air through the porous medium. New empirical correlations are proposed for the estimation of Nusselt number in terms of Rayleigh and Darcy numbers. We also report our results on novel finned metal foam heat sinks in natural convection. Experiments were conducted on aluminum foams of 90% porosity with 5 and 20 PPI with one, two, and four aluminum fins inserted in the foam. All of these heat sinks were fabricated in-house. The results show that the finned metal foam heat sinks are superior in thermal performance compared to the normal metal foam and conventional finned heat sinks. The heat transfer increases with an increase in the number of fins. However, the relative enhancement is found to decrease with each additional fin. The indication is that there exists an optimum number of fins beyond which the enhancement in heat transfer, due to increased surface area, is offset by the retarding effect of overlapping thermal boundary layers. Similar to normal metal foams, the 5 PPI samples are found to give higher values of h compared to the 20 PPI samples due to higher permeability of the porous medium. Future work is planned to arrive at the optimal heat sink configuration for even larger enhancement in heat transfer.