Time-dependent Natural Convection Research Articles

Numerical Analysis and Computation of a Type of IMEX Method for the Time-Dependent Natural Convection Problem

Abstract A new numerical regularization method for the natural convection problem is presented, which is based on a type of implicit-explicit (IMEX) second-order time-stepping schemes in temporal discretization and stabilized mixed finite element in spatial discretization. This method deals with a non-linear advection term in both the momentum equation and the energy equation by linearization. We only need to solve a linear problem at each time step and the discrete curvature of the solutions is added as a stabilization term for the velocity, the pressure and the temperature, respectively. Unconditional stability is proved and an a priori error estimate is derived. Finally, a series of numerical experiments are also given to confirm the theoretical analysis and to demonstrate the efficiency of the new method.

Experimental investigation of the effect of inclination angle on convection-driven melting of phase change material in a rectangular enclosure

This paper investigates the dynamic thermal behavior of phase change material (PCM) melting in a rectangular enclosure at various inclination angles. Lauric acid as a PCM with high Prandtl number (Pr≈100) is used. The enclosure is heated isothermally from one side while the other walls are thermally insulated. Experiments were performed with hot wall temperatures of 55, 60 and 70°C (3.6×108⩽Ra⩽8.3×108) for different inclination angles of 0°, 45° and 90°. Image processing of melt photographs along with recorded temperatures were used to calculate the melt fractions, Nusselt numbers and the local interfacial heat transfer rates at the solid–liquid interface. Qualitative time-dependent natural convection flow structures were deduced indirectly from the instantaneous shape of the solid–liquid interface which were confirmed by quantitative data from temperature measurements. The results reveal that the enclosure inclination has a significant effect on the formation of natural convection currents and consequently on the heat transfer rate and melting time of the PCM. As the inclination angle is decreased from 90° to 0°, the convection currents in the enclosure increases and chaotic flow structures appear. When melting commences in the horizontally inclined enclosure, the solid–liquid interface line becomes wavy which implies the formation of Benard convection cells in the liquid PCM. For the same hot wall temperatures, a decrease in inclination angle leads to a considerable enhancement in energy transport from the hot wall of the enclosure to the PCM. It is found that the heat transfer enhancement ratio for the horizontal enclosure is more than two times higher than that of the vertical enclosure.

Lattice Boltzmann simulations of a time-dependent natural convection problem

A two-dimensional double Multiple-Relaxation-Time thermal lattice Boltzmann method is used to simulate natural convection flows in differentially heated cavities. The buoyancy effects are considered under the Boussinesq assumption. Flow and temperature fields are respectively solved with nine and five discrete velocities’ models. Boundary conditions are implemented with the classical bounce-back or an “on-node” approach. The latter uses the popular Zou and He and Counter-Slip formulations. This paper evaluates the differences between the two implementations for steady and time-dependent flows as well as the space and time convergence orders.

Verification and experimental validation of a numerical simulation of natural convection in a slender cylinder

Steady and time dependent natural convection of water confined in a vertical cylindrical container heated from below is studied experimentally and numerically. The container has a circular cross-section and an aspect ratio (height/diameter) of 1.25. Convective motions are analyzed for a constant Prandtl number of 6.7 and a range of Rayleigh numbers from 3.0×105 to 2.0×106. These conditions include steady and time-dependent flows and in all cases, flow patterns present complex three-dimensional structures. Experimental observations were made with a composed PIV system capable of simultaneously obtaining velocity distributions in two mutually perpendicular planes. The numerical model comprises the solution of the three-dimensional time-dependent Boussinesq equations in cylindrical coordinates. Experimental observations are compared with the numerical model and a satisfactory agreement is obtained for steady flow and averaged values in unsteady flow. The general structure of the present study follows the procedure required for the verification of the numerical model and its validation with experimental observations.

Onset of Multilayer Convection in a Partially Heated Circular Cavity

In this article, the time-dependent double-diffusive natural convection is investigated inside a circular cavity containing stably stratified brine solution. The container is heated partially through its sidewalls by a constant heat flux rate. By using Galerkin finite element method, numerical solutions to the Navier-Stokes equations under the Boussinesq flow assumptions have been systematically organized and developed. The evolutions of the stream function, temperature and concentration fields are presented in each evaluated case. Steep and sharp concentration variation in the interfaces together with wavy distribution of temperature within the layers is observed. At first a length scale, which is the vertical displacement of a heated element of the fluid, in a stably stratified solution under Neumann boundary condition is defined. Then a special form of Rayleigh number based on the length scale is obtained and named as universal Rayleigh number. A categorization based on single, two and multilayer formation is established. Temperature distribution along the cavity in the different cases shows a correlation between heat transfer coefficient and the rate of formation and degradation of the layers.

Benchmark Solution for Time-Dependent Natural Convection Flows with an Accelerated Full-Multigrid Method

A computational procedure is presented with an accelerated full-multigrid scheme for an efficient modeling of time-dependent buoyancy-driven flows. The smoother is the iterative red-and-black successive overrelaxation (RBSOR) scheme. In order to improve the convergence, an acceleration parameter, Γ, is implemented in the classical full-multigrid procedure. It is shown that an optimal value of Γ = 3.75 minimizes the number of iterations needed for convergence. Numerical results are presented and compared with available investigations for an 8:1 differentially heated enclosure and a square heated cavity. Solutions for Prandtl number Pr = 0.71, Rayleigh number Ra = 3.4 × 105 for the 8:1 heated enclosure, and 105 ≤ Ra ≤ 109 for the square cavity are presented and show excellent agreement.

Hypoaigic influences on groundwater flux to a seasonally saline river

Hypoaigic zones are aquifer volumes close to and beneath the shores of saline surface water bodies, and are characterized by the presence of time-dependent natural convection and chemical stratification. When transient and cyclic processes are involved there is significant potential for complex flow and reaction in the near-shore aquifer, presenting a unique challenge to pollutant risk assessment methodologies. This work considers the nature of some hypoaigic processes generated by the seasonally saline Canning River of Western Australia near a site contaminated by petroleum hydrocarbons. A dissolved hydrocarbon plume migrates within the shallow superficial aquifer to the nearby bank of the Canning River. Beneath the river bank a zone of complex fluid mixing is established by seasonal and tidal influences. Understanding this complexity and the subsequent ramifications for local biogeochemical conditions is critical to inferring the potential for degradation of advecting contaminants. A range of modelling approaches throws light on the overall topographic controls of discharge to the river, on the saline convection processes operating under the river bank, on the potential for fluid mixing, and on the various important time scales in the system. Saline distributions simulated within the aquifer hypoaigic zone are in at least qualitative agreement with previous field measurements at the site and are strongly affected by seasonal influences. Groundwater seepage velocities at the shoreline are found to be positively correlated with river salinity. Calculations of fluid age distributions throughout the system show sensitivity to dispersivity values; however, maximum fluid ages under the river appear to be diffusion limited to a few decades. The saline convection cell in the aquifer defines a zone of strong dispersive dilution of aged (many decades) deep aquifer fluids with relatively young (several months) riverine fluids. Seasonal recharge and river salinity cycles induce regular perturbations to the convection cell, yielding intra-annual variations of 50% in seepage velocity and almost 30% in wedge penetration distance at the plume location.

Numerical study of transient flow within a horizontal cylinder heated from below and cooled from the top

The time dependent natural convection in a horizontal circular cylinder, heated from below and cooled from above, is computed using the stream function-vorticity formulation for the two-dimensional equations of motion and energy. The fluid studied has a Prandtl number of 7. The range of Rayleigh numbers (Ra) presented in this paper is 580 ≤ Ra ≤ 100,000. For the range of Ra presented, the initial motion consists of four cells. As Ra is increased, a large single cell with two weak cells is formed. At low Rayleigh numbers, steady state solutions are obtained. At higher values of Ra, the flow becomes unsteady. The temperature, streamlines, relative total Kinetic Energy (KE) and Nusselt number are presented as functions of time.

Unsteady numerical simulation of double diffusive convection heat transfer in a pulsating horizontal heating annulus

A numerical study is conducted on time-dependent double-diffusive natural convection heat transfer in a horizontal annulus. The inner cylinder is heated with sinusoidally-varying temperature while the outer cylinder is maintained at a cold constant temperature. The numerical procedure used in the present work is based on the Galerkin weighted residual method of finite-element formulation by incorporating a non-uniform mesh size. Comparisons with previous studies are performed and the results show excellent agreement. In addition, the effects of pertinent dimensionless parameters such as the thermal Rayleigh number, Buoyancy ratio, Lewis number, and the amplitude of the thermal forcing on the flow and heat transfer characteristics are considered in the present study. Furthermore, the amplitude and frequency of the heated inner cylinder is found to cause significant augmentation in heat transfer rate. The predictions of the temporal variation of Nusselt and Sherwood numbers are obtained and discussed.

Mixing with time dependent natural convection

Chaotic mixing inside a two dimensional cavity can be achieved with time dependent natural convection if the motion of a fluid is generated by imposing alternating hot and cold wall temperatures. With this set up no moving walls are required to mix the fluid inside the container. In this comment we illustrate this idea by numerically solving the governing equations of natural convection in a two dimensional square cavity with sections of its upper and lower horizontal walls cooled and heated in a periodic manner. These conditions generate a vortex of time dependent intensity that moves its center in a closed loop around the geometrical center of the container. The mixing properties of the flow are illustrated by Lagrangian tracking of a collection of points originally located in a line.

Natural convection in a horizontal layer of fluid with a periodic array of square cylinders in the interior

This study of thermal convection uses the following geometry: a horizontal layer of fluid heated from below and cooled from above, with a periodic array of evenly spaced square cylinders occupying the center of the layer (whose aspect ratio is 6). Periodic boundary conditions are employed in the horizontal direction to allow for lateral freedom for the convection cells. A two-dimensional solution for unsteady natural convection is obtained, using an accurate and efficient Chebyshev spectral multi-domain methodology, for different Rayleigh numbers varying over the range of 103 to 106. In order to assess the impact of different geometric approximations on the flow structure, dynamics and overall heat transfer, the results for the above case, denoted WC (wide aspect ratio cell with periodic side boundaries and six internal bodies), are compared with those for the cases of UCNS (unit cell with no-slip adiabatic side boundaries with a single internal body), UCPC (unit cell with periodic side boundaries with a single internal body) and pure Rayleigh–Bénard convection. The results for the case of WC with six adiabatic bodies are also compared to those with six neutral isothermal bodies to consider the effects of the thermal boundary condition on time-dependent natural convection in the enclosure.

Time-dependent buoyant convection in an enclosure with discrete heat sources

A numerical study is performed for the time-dependent laminar natural convection air cooling in a vertical rectangular enclosure with three discrete flush-mounted heaters. The lowest-elevation heater changes the thermal condition between the ‘on’ and ‘off’ modes. In Case 1 (Case 2), the lowest-elevation heater is abruptly switched on (off) from the ‘off’ (‘on’) state. In Case 3, the ‘on’ and ‘off’ modes are repeated periodically. Numerical simulations were conducted for a range of non-dimensional period and for the Rayleigh number based on the cavity width between 10 5 and 10 7 with given heater size and locations. The results show the influence of the time-dependent thermal condition of the lowest-elevation heater on the temperatures of the other heaters. At low Rayleigh number, the cycle-averaged temperatures of all heaters are little affected by the periodic change. At high Rayleigh number, the temperatures of the heaters reach peak values when the non-dimensional period of the change in thermal condition takes intermediate values. The evolutions of global flow and temperature fields are exemplified to provide physical interpretations. The study emphasizes that the transient-stage temperatures at the heaters can exceed the corresponding steady-state values. This is relevant to practical design of electronic devices.

Unsteady fluid flow and temperature fields in a horizontal enclosure with an adiabatic body

A two-dimensional solution for unsteady natural convection in an enclosure with an adiabatic square body is obtained using an accurate and efficient Chevyshev spectral collocation method. A spectral multi-domain methodology is used to handle an adiabatic body located at the center of computational domain. The physical model considered here is that an adiabatic body is located at the center between the bottom hot and top cold walls. In order to see the effects of the presence of an adiabatic body on time-dependent natural convection between the hot and cold walls, we investigated the detail structure of fluid flow and heat transfer as a function of time for different Rayleigh numbers varying in the range of 103 to 106. When Ra=103, streamlines and isotherms reach the steady state without any oscillatory transients, and the flow and temperature distribution around the body in the enclosure shows a four-fold symmetrical pattern. At Ra=4×103, 104, and 105, streamlines and isotherms reach the steady state after starting oscillatory transients, and the flow and temperature fields change their shape from the symmetrical pattern to the nonsymmetrical one as time goes by. When Ra=106, streamlines and isotherms become time-dependent, and the fluid flow and temperature fields also change their shape in a regularly periodic fashion as a function of time.

Rotating magnetic fields: Fluid flow and crystal growth applications

Rotating or alternating magnetic fields are widely used in the industrial steel casting process or in metallurgical manufacturing. For the growth of single crystals, these techniques attracted a rapidly increasing attention within the last years: a well defined melt flow leads to a more homogeneous temperature and concentration distribution in the melt and consequently improves the growth process. Rotating magnetic fields (RMF) might be used instead of crucible and/or crystal rotation avoiding mechanically induced disturbances or might be added to conventional rotation mechanisms to gain a further flow control parameter. Compared to static magnetic fields, rotating ones are distinguished by a much lower energy consumption and technical effort. Furthermore, there are no reports on detrimental effects such as the generation of thermoelectromagnetic convection or coring effects in the grown crystals. One advantage of rotating magnetic fields is the possibility to apply them even to melts with a rather low electrical conductivity like e.g. aqueous solutions. High flow velocities are already generated with moderate fields. Therefore the field strength has to be adjusted with care because otherwise undesirable Taylor vortices might be induced. In the last years, the potential of rotating magnetic fields for crystal growth processes was demonstrated for model arrangements using e.g. gallium or mercury as a test liquid as well as for a variety of growth techniques like Float Zone, Czochralski, Bridgman, or Travelling Heater Methods: Fluctuations of the heat transport due to time-dependent natural convection have could be reduced by more than an order of magnitude or the mass transport could be improved with respect to the a better radial symmetry and/or a more homogeneous microscopic segregation.

On the structure of an unsteady convecting mushy layer

Nonlinear time dependent natural convection in a mushy layer during solidification of a binary alloy is investigated. Asymptotic and scaling analyses are applied to convective flow within the mushy layer and in cylindrical chimneys with small radiusa. Various results in four distinctly identified regimes, corresponding to high or low Prandtl number melt and strongly or weakly time dependent flow, are determined. In particular, it is found that strongly time dependent flow can provide non-vertical chimneys, and for weakly time dependent flow of a low Prandtl number melt vertical chimneys are possible only for sufficiently smalla.

Experiments on the oscillatory convection of low Prandtl number liquid in Czochralski configuration for crystal growth with cusp magnetic field

This paper presents results of experiments on the oscillatory convection of mercury in a Czochralski configuration with cusp magnetic field. Temperature fluctuation measurements are carried out to determine the critical Rayleigh number for the onset of time dependent natural convection. The effects of a cusp magnetic field on the supercritical natural convection coupled with rotation of crystal disk are investigated. In the presence of a rotating flow it is found that a cusp magnetic field can induce a new long wave instability and can amplify the temperature fluctuation depending on the magnitude of the relevant flow similarity parameters and the melt aspect ratios. A flow regime diagram for the amplification and damping of the temperature fluctuations is presented to provide an experimental data base for finding optimum growth conditions in the cusp magnetic field Czochralski process.

Time-dependent natural convection in a glass melting furnace

The main purpose of this study is to determine bifurcation as the primary instability of a glass melting furnace. Steady-state and unsteady characteristics of natural convection in the partially open cavity as appeared in a glass melting furnace is investigated by using numerical analysis. Three types of convection, such as steady laminar, unsteady periodic or unsteady quasi-periodic convection may occur according to the temperature difference between upper two isothermal surfaces along the depth of cavity in a glass melting furnace. In the temperature difference of 150-900 K between batch and free surface, the larger the temperature difference, the weaker the convection strength and unsteadiness. Since the glass viscosity is increasing exponentially in the lower temperature, the batch freezes the thermofluidic field especially below the surface of it. If the depth of cavity is 0.5 m, the bifurcation to time-dependent natural convection may occur in the range of 60-650 K. If that is 1.0 m, it may occur in the whole range of temperature difference.

NUMERICAL SIMULATION OF UNSTEADY THERMAL CONVE-CTIVE PROCESS IN CZOCHRALSK1 CRYSTAL GROWTH

A numerical study for representing time-dependent natural convection of Czochralski melts, while the growing crystal and crucible do not rotate, is presented in this paper. Using a finite control volume algorithm with central-diference discretization of advective and diffusive fluxes, the Czochralski flow which is governed by the rwo-dimensional N-S equations based on Bous-sinesq approximation is computed. The calculations present some interesting streamlines and isotherms with typical time, total kinetic and thermal energy as function of time. The time-de-pendent transport phenomena in Czochralski melt extend into oscillatory for Gr = 1.6×106 in Pr=0.015. This results obviously influence the thermal fluxes at the crystal-liquid interface. The unsteady simulation in this paper is much helpful for the recognition of natural convection and actural crystal growth in Czochralski melts. The program are also available to other models of Czochralski crvstal growth.

Parallel algorithms for the numerical simulation of three-dimensional natural convection

In this paper we deal with the numerical simulation of time-dependent three-dimensional natural convection on parallel computers working in Single Instruction Multiple Data (SIMD) mode. Applying finite differences on a staggered grid in combination with a pressure correction method to the underlying nonlinear system of partial differential equations, we reduce the numerical solution of the problem to the solution of a sequence of sparse linear systems. Using polynomial preconditioned conjugate gradient methods for the solution of these systems results in a highly parallel algorithm for the simulation of the considered flows on SIMD computers. Numerical experiments on an array processor illustrate the capabilities of the proposed algorithm.

The problem of time-dependent natural convection melting with conduction in the solid

This paper reports a theoretical and experimental study of the effect of solid-side subcooling on the phenomenon of melting by natural convection at the vertical interface between a solid body and a pool of its own liquid. The first part of the study consists of a conjugate boundary layer analysis of the transient melting process, in which the effect of solid subcooling leads to the formation of a time-dependent conduction boundary layer in the solid. The theoretical solution documents not only the effect of solid subcooling on the melting rate, but also the effects of liquid superheating (Stefan number) and thermal diffusivity ratio. The second part of the study consists of melting experiments conducted in a rectangular enclosure heated at a constant rate from the side. The solid subcooling parameter B covers the range 0.3–6.3, as the Rayleigh number remains fixed. The experiments show that in the conduction-dominated regime the Nusselt number increases as B increases. In the convection-dominated regime, the effect of the subcooling parameter B is insignificant. It is shown that these experimental observations agree with the trends anticipated theoretically in the first part of the study.