Moderate Grashof Numbers Research Articles

Transient mixed convection in a cylindrical cavity with a rotating finned cylinder

This article presents a numerical study of horizontal ring mixed convection with a heated inner cylinder equipped with two fins. Thermal buoyancy forces are maintained by considering the inner and outer cylinders at different uniform temperatures. The flow is generated by the rotation of the inner cylinder with a constant angular velocity (ω) in the counter-clockwise direction while the outer cylinder is held fixed. A finite volume scheme is adopted using the SIMPLE algorithm to solve the transport equations for continuity, momentum and energy. Simulations are made for various combinations of rotating Reynolds (Re = 0 to 1500) and Grashof numbers (Gr = 10 to 106). First, we focused on the criteria and parameters controlling the fully developed periodic regime. Thereafter, the heat transfer rates analyzed through the averaged Nusselt number (instant and over a period) highlights a complex phenomenon. For moderate Grashof numbers (Gr ≤ 103) the overall Nusselt number is not affected by Gr further, Re = 750 constitute a limit for the improvement of the heat transfer beyond which it decreases strongly. For larger Gr (>103), the reverse phenomenon occurs, i.e., the overall mean Nusselt number grows with Gr as Re < 750, then decreases for higher values.

Structure of the natural convective flow above to the horizontal surface with localized heating

Pure thermal plumes have been the subject of study for the past decades. The active development of experimental and numerical research methods allowed to identify the flow characteristics in the entire region of the plume formation. Nevertheless, the bifurcation discovered in the last ten years remains to be studied, in the authors' opinion, poorly. Until now, there is no general conception of the existence of several stable flow regimes and the key features of the unsteady regime. In this paper results of an experimental and numerical study of pure thermal plume structure in the near-hot-wall region at moderate Grashof numbers are presented. Experimental data on the general flow structure and temperature field are supplemented by the results of numerical simulation. The analysis of changes in instantaneous and average values is carried out, in particular, the geometric characteristics of the near-wall layer are determined. Additionally, data on the characteristics of heat transfer, information on which in modern studies is usually limited and is fragmentary, are presented.

Thermal convection in a toroidal duct of a liquid metal blanket. Part II. Effect of axial mean flow

The work continues the exploration of the effect of thermal convection on flows in toroidal ducts of a liquid metal blanket. This time we consider the effect of the mean flow along the duct and of the associated heat transfer diverting the heat deposited by captured neutrons. Numerical simulations are conducted for a model system with two-dimensional (streamwise-uniform) fully developed flow, purely toroidal magnetic field, and perfectly electrically and thermally insulating walls. Realistically high Grashof (up to 1011) and Reynolds (up to 106) numbers are used. It is found that the flow develops thermal convection in the transverse plane at moderate Grashof numbers. At large Grashof numbers, the flow is dominated by the top-bottom asymmetry of the streamwise velocity and stable stratification of temperature, which are caused by the buoyancy force due to the mean temperature growing along the duct. This leads to suppression of thermal convection, weak mixing, and substantial gradients of wall temperature. Further analysis based on more realistic models is suggested.

Numerical study of molten magnesium convection in a titanium reduction apparatus

The structure of a convective flow of molten magnesium in a metallothermic titanium reduction apparatus has been studied numerically for various retort heating and cooling configurations. The mathematical model is based on the thermogravitational convection equations for a singlephase fluid in the Boussinesq approximation. A nonuniform computational grid with a total of 5 million grid points was used. The LES (Large Eddy Simulation) technique was applied to take into account the turbulence. The problem was considered in a three-dimensional nonstationary formulation, which allowed one to construct the instantaneous and average characteristics of the process and to analyze the velocity and temperature pulsation fields. It has been found that steady axisymmetric flows occur at moderate Grashof numbers (Gr ~ 107–108), while unsteady turbulent flows take place at Grashof numbers corresponding to the actual titanium reduction process (Gr ~ 1012). The influence of the degree of nonhomogeneity of the heat release due to the titanium reduction reaction that runs mainly on the magnesium surface has been studied. The computations have been performed for two configurations of the system for maintaining the apparatus heating regime: with furnace heaters operating at full power and with switched-off furnace heaters. Fundamental differences in convective flow structure for these two heating methods have been revealed. It has been established that the most intense velocity and temperature pulsations arise in the region adjacent to the interface between the cooled and heated parts of the retort side surface.

506 平行平板間流れにおける乱流遷移域の大規模間欠構造(GS5 流体工学(3),研究討論セッション)

Direct numerical simulation of a spectral method was performed to study the turbulent structure in Poiseuille flow under stably stratification. Turbulence was attenuated by enhancing the effect of stratification, and friction coefficient decreased systematically. The quasi-laminar regions where the low-speed streaks tend to elongate to the streamwise direction expand with increasing the stratification effect. At moderate Grashof number, turbulent stripes were clearly observed. The flow structure of the turbulent stripe was similar to ones in turbulent Poiseuille flow in transitional region without any external forces. However, the quasi-laminar region diminished in the flow under strong stratification.

Natural convection in power-law fluids from two square cylinders in tandem arrangement at moderate Grashof numbers

In this work, free convective flow and heat transfer in power-law fluids from two heated square cylinders in tandem arrangement is studied. The governing differential equations have been solved numerically over wide ranges of Grashof number, 10 ≤ Gr ≤ 1,000, Prandtl number, 0.71 ≤ Pr ≤ 50 and power-law index, 0.4 ≤ n ≤ 1.8. In order to elucidate the extent of inter-cylinder interaction, the non-dimensional inter-cylinder spacing, L/d is varied in the range, 2 ≤ L/d ≤ 6. The results are interpreted in terms of streamline and isotherm contours in the proximity of two cylinders to gain physical insights into the nature of flow. At the next level, the distribution of the local Nusselt number along the surface of the cylinders is presented. At the minimum inter-cylinder spacing due to the intense interference, the downstream cylinder contributes much less to the overall heat transfer whereas it experiences much higher hydrodynamic drag than the upstream cylinder. Broadly, the local and average Nusselt number for both cylinders show a positive dependence on both Grashof and Prandtl numbers. Also, all else being equal, shear-thinning fluid behaviour promotes the rate of heat transfer and shear-thickening fluid behaviour impedes it. Finally, the present numerical results have been correlated by using simple forms of equations thereby enabling the estimation of Nusselt number in a new application.

A numerical study of three-dimensional laminar mixed convection past an open cavity

A numerical study has been carried out to analyze the effects of mixed convective flow over a three-dimensional cavity that lies at the bottom of a horizontal channel. The vertical walls of the cavity are isothermal and all other walls are adiabatic. The cavity is assumed to be cubic in geometry and the flow is laminar and incompressible. A direct numerical simulation is undertaken to investigate the flow structure, the heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and the buoyancy-induced flow from the heated wall over a wide range of the Grashof number (10 3–10 6) and two Reynolds numbers Re = 100 and 1000. The computed thermal and flow fields are displayed and discussed in terms of the velocity fields, streamlines, the temperature distribution and the averaged Nusselt number at the heated and cooled walls. It is found that the flow becomes stable at moderate Grashof number and exhibit a three-dimensional structure, while for both high Reynolds and Grashof numbers the mixed convection effects come into play, push the recirculating zone further upstream and the flow becomes unsteady with Kelvin–Helmholtz instabilities at the shear layer.

Influence of Soret and Dufour effects on entropy generation in transient double diffusive convection

The influence of Soret and Dufour effects on entropy generation in transient double diffusive convection of a binary gas mixture for the special case of opposing buoyancy forces with equal intensity is numerically studied. It is an interesting case since it corresponds to minimum entropy generation. For moderate thermal Grashof number (GrT = 104), Soret and Dufour parameters induce a slight increase of entropy generation, for relatively higher thermal Grashof number (GrT = 5 × 104), oscillatory behaviour of entropy generation is obtained. Contributions of Soret and Dufour effects on entropy generation, on heat and mass transfer are discussed.

Analysis of the flow and heat transfer characteristics for assisting incompressible laminar flow past an open cavity

A numerical study has been carried out to analyze the effects of mixed convective assisting flow past three-dimensional open cavity over a wide range of Reynolds (100–1000) and Richardson (0.001–10) numbers. The vertical walls in the inflow and outflow sides are isothermal while all other walls are adiabatic. The cavity is assumed to be cubic in geometry and the flow is laminar. A direct numerical simulation is undertaken to investigate the flow structure, the heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and the buoyancy-induced flow from the heated wall. It is found that the flow becomes stable at moderate Grashof number and exhibits a three-dimensional structure, while for high Richardson number the mixed convection effects come into play and push the recirculating zone further upstream and the flow may becomes unstable.

The effect of wall heating on instability of channel flow

A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Bénard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

Modelling of thermogravitation convection in closed volume with local sources of heat release

A mathematical model is presented for unsteady conjugate convective-conductive heat transfer in a closed volume with local sources of heat release under the conditions of a convective-radiation heat exchange on one of external faces of the solution region. The thermogravitation convection regime was analysed for moderate Grashof numbers. The typical temperature and velocity fields were obtained, and the fields of desired quantities were compared for the planar and three-dimensional models in one of the typical sections of the solution region.

Gravitational Convection of a Liquid Mixture in a Horizontal Cylindrical Gap at Moderate Grashof Numbers

Weak concentration convection which arises in the process of diffusion of impurities into the solvent filling a gap between two coaxial cylinders is studied experimentally. It is found that convective motion in the range of Grashof numbers 103–5 × 104 has a clear boundary-layer character. Near the inner porous cylinder, which is a source of impurity, a diffusion boundary layer passing into a two-dimensional convective plume is formed. The data on the structure and thickness of this layer are presented depending on the integral flux of impurity. The prospects of making an experiment in order to discover concentration convection onboard an orbital station are discussed.

FREE CONVECTION INSIDE AN L-SHAPED ENCLOSURE

We give buoyancy induced flow pattern and heat transfer characteristics inside an L-shaped enclosure with differentially heated isothermal walls. The integral form of the governing equations are solved by control volume based finite volume method using SIMPLE algorithm and Stone's SIP solver. Flow and thermal fields are presented in the form of streamlines and isothermal lines for Grashof number ranges 10 to 10 6 . Three different values of interwall spacing are selected. Prandtl number of fluid is kept constant (=0.7). Unicellular flow pattern is observed at very low Grashof number. Corners of the enclosed space divide the unicell into two distinct cells (one horizontal and one vertical) at moderate Grashof numbers. Multicellular flow is observed at high Grashof number. Interwall spacing greatly affects the growth of multicellular pattern at high Grashof number. Local and average Nusselt number distribution present the heat transfer rate

Double diffusion in a vertical fluid layer: Onset of the convective regime

This paper considers the onset of double diffusive natural convection in a vertical layer of a binary fluid submitted to horizontal thermal and compositional gradients. The analysis deals with the particular situation where the resulting buoyancy forces (the Grashof numbers corresponding to the thermal and solutal effects) are opposing and of equal intensity. The stability analysis for the infinite layer shows that the purely diffusive (motionless) solution prevails at moderate Grashof numbers, and an analytical expression of the critical Rayleigh number as a function of the Lewis number is obtained. These results are then compared to the critical values which are determined from numerical simulations in an enclosure. Numerical calculations in the transient regime are used to give an interpretation of the stability of the steady state diffusive regime when the buoyancy forces are below the critical value.

Spectral simulations of oscillatory convection at low Prandtl number

AbstractPseudospectral methods are used for the computation of the time‐dependent convective flows which arise in shallow cavities filled with low‐Prandtí‐number liquids when submitted to a horizontal temperature gradient. In similar situations several former numerical results have been shown to disagree about the determination of the threshold of oscillations and about the subsequent supercritical regimes. Two different tau–Chebyshev methods based on the vorticity–streamfunction formulation and using multistep time schemes are considered. Their results are discussed to assess the validity of the solutions. The physical problems concern rectangular cavities which involve either a rigid or a stress‐free top wall and either conducting or insulating horizontal walls. Aside from the prediction of the onset of oscillations, which is discussed in the various situations with respect to the results of linear and non‐linear analyses and to other computational results, the present study exhibits some bifurcation sequences and a hysteresis cycle at moderate Grashof numbers which are associated to the occurrence of multiple solutions.

Three-dimensional numerical study of convection in a cylindrical thermal diffusion cell: Its influence on the separation of constituents

The convection arising in a cylindrical container, heated from the ends and designed for a measurement of Soret coefficient, is examined. A three-dimensional method using a pseudostationary scheme with a finite differences technique is used. Attention is focused on a horizontal cylinder with an aspect ratio of 6, a Prandtl number of 0.6, and a Schmidt number of 60. The influence of convection on the separation and on the mass fraction profiles is examined for moderate Grashof numbers (0.01≤GrH≤10) and realistic Soret parameters (−0.75≤S≤1). A domain is found where the flow has no influence on the separation, corresponding to a ‘‘separation’’ regime. Extensions of the results to different Prandtl and Schmidt numbers and to larger aspect ratios are proposed.

Effects of sound on local transport from a heated cylinder

The steady transport of heat from a horizontal circular cylinder supported in a standing transverse sound field was investigated. Experimental conditions were chosen to allow ready comparison with solutions of the momentum and energy equations. Local heat transport was studied, it being more sensitive to sound fields than overall transport. Experiments were performed with a heated cylinder in an anechoic chamber and a laser schlieren-interferometer was used to obtain local measurements of temperature gradients in the air surrounding the cylinder. Sound field intensities ranged up to 142 db. The convection was analyzed in the equations for mean motion using the Reynolds stresses generated by the sound to represent the effect of the sound field. The velocity and temperature fields of the mean motion were represented as an azimuthal series of functions of the radial coordinate, and the solutions computed at finite Grashof and streaming Reynolds numbers for conditions at the bottom stagnation region. Substantial agreement was found between experimental transport and the computed solutions for horizontal sound fields. The analysis showed that the effect of the sound field is significantly greater at the moderate Grashof numbers (0 (10 3)) used in the experiments than predicted by a boundary layer analysis ( Gr → ∞) for otherwise similar conditions. The results support the hypothesis that effects of sound and vibrations can be accounted as the influence on the mean motion of Reynolds stresses associated with the oscillations.

Numerical simulation of free convective heat transfer from a sphere

A numerical study of time-dependent free convective heat transfer from a solid sphere to an incompressible Newtonian fluid has been carried out for Grashof numbers between 0.05 and 12 500 and for Prandtl numbers of 0.72, 10 and 100. The energy and vorticity transport equations were solved using Peaceman and Rachford's ADI method and the stream function equation was solved using point iterative successive over-relaxation. From the late-time steady-state solutions it was observed that even at extremely low Grashof numbers, weak convection processes were present in the region close to the outer boundary. However, it was found that even at moderate Grashof numbers the dominant mode of vorticity transport close to the surface was by diffusion.

Numerical investigation of convective heat transfer in liquid hydrocarbons stored in underground reservoirs

In planning for the underground storing of liquid hydrocarbons and calculating the technological parameters of underground reservoirs formed by leeching from a rock salt massif, it is necessary to understand the hydrodynamic and heat transfer processes produced by natural convection. The paper is devoted to numerical study of the initial stage of convective heat transfer in a vertical cylindrical cavity completely filled with a liquid hydrocarbon. It is assumed that at the initial time the temperature of the liquid, which is at rest, is homogeneous, Convective flow develops in part due to the initial temperature difference between the liquid and the massif and partly due to the geothermal temperature gradient in the latter. The problem is regarded as coupled, the convective process in the liquid being determined simultaneously with the solution of the heat-conduction problem in the surrounding rock. The Grashof numbers characterizing the intensity of the real process are very large: G ≈ lO12−lO15. A numerical solution was obtained for moderate Grashof numbers G ≈ lO7−lO11. The asymptotic dependences for the integral characteristics of the unsteady process obtained in a series of calculations were extrapolated to the real values of the Grashof number.

Numerical Solution for Free Convection Between Concentric Spheres at Moderate Grashof Numbers

Numerical Solution for Free Convection Between Concentric Spheres at Moderate Grashof Numbers