Dimensionless Rayleigh Number Research Articles

Математическая модель течения припоя в вертикальной трубке при различных уровнях гравитации с учетом процессов смачивания и плавления

The motion of a solder inside a ceramic tube with an aluminum insert is considered using the phase-field model of multiphase flows. The problem is solved in the non-isothermal formulation, which allows analyzing the two-phase flow dynamics and the kinetics of the contact line driven by wetting. The total time required for the solder to heat up, melt and then move inside the tube is calculated accounting for its position in the insert and action of wetting forces. The melting heat of the solder is taken into account in the system through the introduction of effective heat capacity as a function of temperature. The values of the dimensionless Bond, Rayleigh, Grasgoff and Marangoni numbers are calculated, which made it possible to analyze the contribution of various physical phenomena to the behavior of the system. It was found that the effect of gravity forces on the shape of the upper and lower free surfaces of the melt is not significant because of the small weight of the solder and the small diameter of the tube. The graphs showing the variation in the center of mass of the solder are obtained. The model predicts the solder leakage from the insert in the presence of gravity, while under microgravity this does not happen. The velocity fields, which develop in a liquid solder at the gravity levels of 1g and µg, are analyzed. Under microgravity conditions, the maximum velocities are caused by the movement of the melt due to wetting forces, while in Earth gravity the average velocities are two orders of magnitude higher since convection currents are present near the walls of the tube. A small thermocapillary effect on the average flow velocity was noted as a result of low temperature gradients.

Effect of an Adiabatic Obstacle on the Symmetry of the Temperature, Flow, and Electric Charge Fields during Electrohydrodynamic Natural Convection

This study explores the impact of an adiabatic obstacle on the symmetry of temperature, flow, and electric charge fields during electrohydrodynamic (EHD) natural convection. The configuration studied involves a square, differentially heated cavity with an adiabatic obstacle subjected to a destabilizing thermal gradient and a potential difference between horizontal walls. A numerical analysis was performed using the finite volume method combined with Patankar’s “blocked-off-regions” technique, employing an in-house FORTRAN code. The study covers a range of dimensionless electrical Rayleigh numbers (0 to 700) and thermal Rayleigh numbers (102 to 105), with various obstacle positions. Key findings indicate that while the obstacle reduces heat transfer, this can be counterbalanced by electric field effects, achieving up to 165% local heat transfer improvement and 100% average enhancement. Depending on the obstacle’s position and size, convective transfer can increase by 27% or decrease by 21%. The study introduces five multiparametric mathematical correlations for rapid Nusselt number determination, applicable to numerous engineering scenarios. This work uniquely combines passive (adiabatic obstacle) and active (electric field) techniques to control heat transfer, providing new insights into the flow behaviour and charge distribution in electro-thermo-hydrodynamic systems.

Influence of a magnetic field on double‐diffusive convection in an inclined porous layer

AbstractThe present study investigates the impact of a magnetic field on double‐diffusive convection in an inclined porous layer, employing linear instability theory. The perturbed state is solved using the normal mode technique, and the stability eigenvalue problem is numerically analyzed for longitudinal and traveling rolls using the Runge‐Kutta method coupled with the shooting method. Various dimensionless physical parameters, including solutal and thermal Rayleigh numbers, inclination angle, Hartmann number, and Lewis number, are examined for their influence on system stability. The findings reveal that, for , the Hartmann number, solute Rayleigh number, and inclination angle act as stabilizing factors, with greater stability observed for traveling rolls compared to longitudinal rolls. In the case of , the critical Rayleigh number shows a monotonic relationship with the solute Rayleigh number and inclination angle, while its relationship with the Hartmann number is non‐monotonic for traveling rolls. Moreover, for , the Hartmann number stabilizes the system by raising the onset threshold value, favouring longitudinal modes. The solute Rayleigh number also contributes to system stability. The impact of the inclination angle on system stability is contingent upon its magnitude, with small angles favouring the stability of longitudinal rolls and larger angles leading to an initial transition from traveling to longitudinal rolls, indicating a complex non‐monotonic behavior.

Stacked Heterogeneous Ensemble Learning Model in Mixed Convection Heat Transfer from a Vertically Oscillating Flat Plate

In this study, the effects of mixed convection heat transfer from a moving vertical flat plate with an experimental and stacked heterogeneous ensemble learning approach are analyzed. In the experimental work, the effects on both natural and forced convection of dimensionless oscillation amplitude (Ao), dimensionless oscillation frequency (Wo) and Rayleigh number (Ra) are investigated. In the experiments, the vertical movement of the plate is provided by a flywheel-motor assembly. The average Nusselt numbers (Nu) on the fixed plate and the moving plate surface were obtained. Additionally, this study is focused on the prediction of heat transfer of a moving flat plate using single-based algorithms (Gradient Boosting, AdaBoost, Multilayer Per-ceptron) and a stacked heterogeneous ensemble learning model. The statistical per-formance of the single-based algorithms and the stacked ensemble model is meas-ured in the prediction of mixed convection heat transfer. The results show that the stacked-based ensemble learning model yielded the MSE = 2.01, RMSE = 1.42, MAE = 1.1 and R2 = 0.99 values. Overall, this study reveals that the proposed stacked en-semble machine learning model can be used successfully for modeling convection heat transfer of a moving plate.

Effect of porous partition height on thermal performance of a ventilated cavity using LBMMRT

The objective of this work is to study the effect of the thickness of a porous separation on the thermal performance in a cavity with displacement ventilation. The cold air jet enters and exits through two openings located in the lower and upper parts of the left wall and the right wall respectively. The other horizontal walls are also adiabatic. The hydrodynamic and thermal characteristics of the transfer were studied for three configurations with the same aspect ratio L/H=2. The height Hp of the porous separation was varied between 0.2 and 0.8 where is placed in the center of the cavity. The transfer rates on the active wall for the thicknesses were studied for different permeability therefore different Darcy numbers varying over an interval:10-6≤Da≤10. The dimensionless Rayleigh and Reynolds numbers were taken from the rows: 10≤Ra≤106 and 50≤Re≤500. The governing equations of momentum and energy were solved by the Lettice Boltzmann Multiple Relaxation Time Method (LB-MRT) D2Q9 for the velocity field and D2Q5 for the temperature field. In order to take into account the introduction of the porous medium, an additional term is added to the standard LB equations based on the generalized model (Darcy model extended to Brinkman-Forchheimer).

Lattice Boltzmann Simulation of Magnetic Field Effect on Electrically Conducting Fluid at Inclined Angles in Rayleigh-B閚ard Convection

The magneto-hydrodynamics (MHD) effect is studied at different inclined angles in Rayleigh-Bénard (RB) convection inside a rectangular enclosure using the lattice Boltzmann method (LBM). The enclosure is filled with electrically conducting fluids of different characteristics. These characteristics are defined by Prandtl number, Pr. The considered Pr values for this study are 10 and 70. The influence of other dimensionless parameters Rayleigh numbers <i>Ra</i> = 10³; 10⁴; 10⁵; 10⁶ and Hartmann numbers <i>Ha</i> = <i>0</i>, <i>10</i>, <i>25</i>, <i>50</i>, <i>100</i>, on fluid flow and heat transfer, are also investigated considering different inclined angles <i>φ</i> of magnetic field by analyzing computed local Nusselt numbers and average Nusselt numbers. The results of the study show the undoubted prediction capability of LBM for the current problem. The simulated results demonstrate that the augmentation in heat transfer is directly related to <i>Ra</i> values, but it is opposite while observing the characteristics of <i>Ha</i> values. However, it is also found that <i>φ</i> has a significant impact on heat transfer for different fluids. Besides, isotherms are found to be always parallel to the horizontal axis at <i>Ra</i> = 10³ as conduction overcomes the convection in the heat transfer, but this behaviour is not seen at <i>Ra</i> = 10⁴ when <i>Ha</i> ≥ 25. Furthermore, at <i>Ra</i> = 10⁶, oscillatory instability appears but LBM is still able to provide a complete map of this predicted behavior. An appropriate validation with previous numerical studies demonstrates the accuracy of the present approach.

Rivalry of diffusion, external field and gravity in micro-convection of magnetic colloids

Magnetic fields and magnetic materials have promising microfluidic applications. For example, magnetic micro-convection can enhance mixing considerably. However, previous studies have not explained increased effective diffusion during this phenomenon. Here we show that enhanced interface smearing comes from a gravity induced convective motion within a thin microfluidic channel, caused by a small density difference between miscible magnetic and non-magnetic fluids. This motion resembles diffusive behavior and can be described with an effective diffusion coefficient. We explain this with a theoretical model, based on a dimensionless gravitational Rayleigh number, and verify it by numerical simulations and experiments with different cell thicknesses. Results indicate the applicability and limitations for microfluidic applications of other colloidal systems. Residual magnetic micro-convection follows earlier predictions.

Magnetic source effect on EG-CuO nanofluid in a semi-annulus using RBFs

In this study, numerical simulation of hydrothermal behaviour of ethylene glycol based CuO nanofluid in a semi-annulus with a magnetic source is investigated. The non-dimensional governing equations are discretized by radial basis functions in spatial derivatives, and by backward-Euler in temporal derivatives. The effects of dimensionless parameters Hartmann number, Rayleigh number, magnetic number, Eckert number and solid volume fraction are presented both in terms of streamlines, isotherms and vorticity contours, and average Nusselt number through the heated wall. The results show that augmentation in magnetic number accelerates the convective heat transfer at a small Rayleigh number, and at a larger heated inner circle. Convective heat transfer is inhibited with the rise of Hartmann number, and with the increase in Eckert number while it is enhanced with the increase in concentration of nanoparticles and Rayleigh number.

Electro-thermo-convection in dielectric liquid subjected to partial unipolar injection between two eccentric cylinders

Purpose The purpose of this work is to highlight the effects of partial unipolar injection on electro-thermo-convection (ETC) in dielectric liquid contained between two eccentric cylinders. Design/methodology/approach A finite volume method was used to solve governing equations. The study is performed for different parameters, such as radius ratio (0.2 ≤ Γ ≤ 0.6), dimensionless electric Rayleigh number (0 ≤ T ≤ 900), eccentricity (−0.4 ≤ e ≤ 0.4) and thermal Rayleigh number (10 ≤ Ra ≤ 5.105). Findings It is found that heat transfer increases with increase in dimensionless electric Rayleigh number and eccentricity ratio. Originality/value The originality of this work is to analyze the ETC in dielectric liquid subjected to partial unipolar injection between two eccentric cylinders

Heat transfer intensification induced by electrically generated convection between two elliptical cylinders

A computational study has been performed to see the heat transfer intensification induced by electrically generated convection in a two co-focal elliptical cylinder layers for dielectric liquid. Temperature of the inner cylinder is higher than of outer one. Finite volume method is used to solve the governing equations by writing a homemade computer code. The study is performed for dimensionless electric Rayleigh number (0 ≤ T ≤ 700), Rayleigh numbers (102≤ Ra ≤105), eccentricity ratio (0.4 ≤ e ≤ 1). It is observed that the unipolar charge injection by the internal electrode changes significantly the topology of the fluid flow. Moreover, the computations highlight an important effect of the eccentricity value on the intensity and circulation cell number. Finally, it is found that heat transfer is improved with increasing dimensionless electric Rayleigh number.

Axisymmetric lattice Boltzmann simulation of the heat-exchanger method-based sapphire crystal growth

The crystal-growth process in a heat-exchanger method (HEM)-based system was simulated using the lattice Boltzmann technique. The two-dimensional enthalpy-based lattice Boltzmann model was extended to treat the axisymmetric solid-liquid phase change problems. The proposed model demonstrates high accuracy over a large range of Biot and Stefan numbers and exhibits superiority over the first-order perturbation method, especially, in cases involving large Stefan numbers. Macroscopic crystal-growth simulations primarily focus on flow patterns in the melt and heat-transfer processes inside the crucible. Effects of dimensionless convective boundaries and Rayleigh numbers (Ra) on transport phenomena and the crystal/melt interface have also been discussed. It was observed that a decrease in the superheating temperature causes an increase in crystal fraction and decline in convection intensity. The area of the cooling zone mainly influences crystal growth rate during the initial stages of the process. In the later stages, however, the Biot number corresponding to the heating boundary assumes a more significant role. At low Rayleigh numbers, heat transfer via conduction is dominant in the melt. With increase in Ra values, melt convection tends to strengthen, thereby tending to deform the isothermal lines and phase interface.

Evolution of Dimensionless Numbers in Geodynamo Models

The evolution of the dimensionless Ekman, Rossby, and Rayleigh numbers in the model of composite convection in the Earth’s, which that determine the force balance and magnetic field evolution, is considered on the basis of known estimates of the rate of solid core growth and variations in the day length. Various scenarios of this evolution, both in the past and future, are presented over time intervals comparable with the age of the core. It is shown, in particular, that the current intensity of convection in the fluid core continues to increase. This fact corresponds to the increase in the frequency of geomagnetic field inversions and the transition from a dipole field to a multipole one. A decrease in the volume of the liquid core leads to a decrease in the magnetic field strength.

Analysis of natural convection heat transfer through staggered pin finned horizontal base plate within a rectangular enclosure

An experimental study of natural convection heat transfer from heat staggered Pin fin array within air filled rectangular enclosure has been performed experimentally. It has been done to analyze the effects of several inducing parameters on system performance. The study has been done at wide ranges of influencing parameters such as Rayleigh number (328,042 ≤ Ra ≤ 794,323) and fin spacing (25 mm ≤ S ≤ 100 mm) and at a given fin height (L = 25 mm). The present study reported that the average Nusselt number (Nu) always increases by increasing Ra. While, by decreasing in fin spacing Nu increases initially up to an extreme value then tends to decrease. The fin effectiveness decreases continuously by increasing in Rayleigh number. It has been found that by decreasing in fin spacing, fin effectiveness increases initially up to a maximum value, beyond which it tends to decrease with further decrease in fin spacing. The correlations are developed for Nu as a function of dimensionless number (S/H) and Rayleigh numbers (Ra) and verified with experimental data.

How Properties that Distinguish Solids from Fluids and Constraints of Spherical Geometry Suppress Lower Mantle Convection

The large magnitude of the dimensionless Rayleigh number (Ra ∼108) for Earth’s ∼3 000 km thick mantle is considered evidence of whole mantle convection. However, the current formulation assumes behavior characteristic of gases and liquids and also assumes Cartesian geometry. Issues arising from neglecting physical properties unique to solids and ignoring the spherical shapes for planets include: (1) Planet radius must be incorporated into Ra, in addition to layer thickness, to conserve mass during radial displacements. (2) The vastly different rates for heat and mass diffusion in solids, which result from their decoupled transport mechanisms, promote stability. (3) Unlike liquids, substantial stress is needed to deform solids, which independently promotes stability. (4) High interior compression stabilizes the mantle in additional minor ways. Therefore, representing conditions for convection in solid, self-gravitating spheroids, requires modifying formulae developed for bottomheated fluids near ambient conditions under an invariant gravitational field. To derive stability criteria appropriate to solid spheres, we use dimensional analysis, and consider the effects of geometry, force competition, and microscopic behavior. We show that internal heating has been improperly accounted for in the Ra. We conclude that the lower mantle is stable for two independent reasons: heat diffusion far outpaces mass diffusion (creep) and yield strength of solids at high pressure exceeds the effective deviatoric stress. We discuss the role of partial melt in lubricating plate motion, and explain why the Ra is not applicable to the multi-component upper mantle. When conduction is insufficient to transport heat in the Earth, melt production and ascent are expected, not convection of solid rock.

NUMERICAL SIMULATION OF NATURAL CONVECTION IN A POROUS CAVITY FILLED WITH FERROFLUID IN PRESENCE OF MAGNETIC SOURCE

In this study, numerical simulation of natural convection in a porous square cavity filled with Fe3O4-water is investigated. A magnetic source through the left wall of the cavity is also taken into account. Radial basis function based pseudo spectral (RBF-PS) method is applied. The effects of dimensionless parameters Darcy (Da), Hartmann (Ha), Rayleigh (Ra) numbers and solid volume fraction 𝝓 are presented both in terms of streamlines, isotherms and vorticity contours and average Nusselt number through the heated wall. Convective heat transfer is inhibited with the rise of Ha, and with the decrease in Da while it is enhanced with the increase in 𝝓 and Ra.

On the representation of subsea aquitards in models of offshore fresh groundwater

Fresh groundwater is widespread globally in offshore aquifers, and is particularly dependent on the properties of offshore aquitards, which inhibit seawater-freshwater mixing thereby allowing offshore freshwater to persist. However, little is known of the salinity distribution in subsea aquitards, especially in relation to the offshore freshwater distribution. This is critical for the application of recent analytical solutions to subsea freshwater extent given requisite assumptions about aquitard salinity. In this paper, we use numerical simulation to explore the extent of offshore freshwater in simplified situations of subsea aquifers and overlying aquitards, including in relation to the upward leakage of freshwater. The results show that available analytical solutions significantly overestimate the offshore extent of upwelling freshwater due to the presumption of seawater in the aquitard, whereas the seawater wedge toe is less sensitive to the assumed aquitard salinity. We also explore the use of implicit, conductance-based representations of the aquitard (i.e., using the popular SEAWAT code), and find that SEAWAT's implicit approach (i.e., GHB package) can represent the offshore distance of upwelling freshwater using a novel parameterization strategy. The results show that an estimate of the upward freshwater flow that is required to freshen the aquitard is associated with the dimensionless Rayleigh number, whereby the critical Rayleigh number that distinguishes fresh and saline regions (based on the position of the 0.5 isochlor) within the aquitard is approximately 2.

Stratification in hot water pipe-flows

Abstract: In district heating and cooling networks, hot or cold fluid is transported in pipe systems over long distances. Heat loss (or gain) from the fluid-carrying pipes to the surrounding is a primary driving mechanism that can lead to various thermal dynamics effects in the fluid. Here, we investigate the thermal dynamics of low-Reynolds number hot water flow in horizontal pipes. In such flows, density differences between hot and cold water generate buoyancy effects. Further, the coupling between heat transfer and momentum transfer along with the temperature-dependent viscosity of water can result in stratified asymmetric flow profiles where fluid is transported with high velocities in the top region of the pipe cross-section and can be almost stagnant in the bottom region. In the flow laboratory at Kamstrup A/S, we carried out experiments to analyze thermally stratified flow profiles with laser-Doppler velocimetry. In this article, we present results for the downstream development of stratified flow profiles for different flow parameters including temperature, pipe diameter, and flow rate (or dimensionless Reynolds number and Rayleigh number, respectively.) Further, we discuss the impact of thermally stratified flow profiles on metering applications in district heating and cooling networks.

Influence of Local Heating on Marangoni Flows and Evaporation Kinetics of Pure Water Drops.

The effect of localized heating on the evaporation of pure sessile water drops was probed experimentally by a combination of infrared thermography and optical imaging. In particular, we studied the effect of three different heating powers and two different locations, directly below the center and edge of the drop. In all cases, four distinct stages were identified according to the emerging thermal patterns. In particular, depending on heating location, recirculating vortices emerge that either remain pinned or move azimuthally within the drop. Eventually, these vortices oscillate in different modes depending on heating location. Infrared data allowed extraction of temperature distribution on each drop surface. In turn, the flow velocity in each case was calculated and was found to be higher for edge heating, due to the one-directional nature of the heating. Additionally, calculation of the dimensionless Marangoni and Rayleigh numbers yielded the prevalence of Marangoni convection. Heating the water drops also affected the evaporation kinetics by promoting the "stick-slip" regime. Moreover, both the total number of depinning events and the pinning strength were found to be highly dependent on heating location. Lastly, we report a higher than predicted relationship between evaporation rate and heating temperature, due to the added influence of the recirculating flows on temperature distribution and hence evaporation flux.

Interacting convection modes in a saturated porous medium of nearly square planform: a special case

A multitude of convection modes may occur within a confined rectangular box of saturated porous medium when the associated dimensionless Rayleigh number R is above some critical value. For particular sizes of box, however, it is possible for multiple modes (typically three) to share a common critical Rayleigh number. For box shapes close to these geometries, modes can interact and compete nonlinearly near the onset of convection. The generic examples of this phenomenon can be conveniently classified as belonging to one of two distinctive classes distinguished by whether or not fixed points are possible with all three modes having a non-zero amplitude. It transpires that this classification is not quite exhaustive for there is one particular case which falls outside this pattern. This last case, which is described by a system of evolution equations that is structurally different from that applying in the generic situation, is explored in this paper. Some rich dynamical behaviours are uncovered and, in particular, it is shown that at sufficiently large R stable states arise in which all three modes persist. This contrasts to the typical generic behaviour where a single mode is preferred. The bifurcation sequence that develops as the Rayleigh number grows is mapped out from primary through to tertiary stages. In addition, it is found that a cusp bifurcation is present and that the details of the ordering of the various bifurcations are shown to be sensitive to the precise geometry of the box.

Experimental measurements in melting ingots in the melt of the same material

This study concerns the melting of ingots of different materials in melt of the same material. We investigated the pure materials ice, lead, tin, and zinc, the magnesium alloys AZ91 and AM50, and the aluminum alloy A226. We used melting pots made from steel (for Pb, Sn, Zn, AZ91, AM50) and clay graphite (for A226) with a volume of 16 L, inserted into a resistance furnace. Some experiments with AZ91 were also carried out in a 2500 kg industrial furnace. The ice ingots were melted in a 20 L beaker. The temperature profile adjacent to the melting ingot was recorded over time. From this profile, the mean temperature of the melt adjacent to the ingots was calculated. Together with the geometrical and thermophysical properties of the investigated materials, the dimensionless Nusselt, Rayleigh, Prandtl, and Stefan numbers were calculated and interpreted as an empirical function, Nu=0.114·(Ra·Pr)0.291·Ste0.754. This function describes the melting behavior of all of the materials considered. This partly agrees with results from the literature, but considerable deviations were also determined. Once the mean temperature is known, the time needed to melt the different materials in different geometrical shapes can be estimated along with the maximum melting rate. This simple model helps understand technical processes where melting of materials is relevant, for example when calculating energy consumption in the foundry industry.