Current trends and future directions in turbulent thermal convection

The system of turbulent thermal convection is introduced. Progresses in recent decades in the four major areas of research in turbulent convection are briefly reviewed. Some of the recent trends of the field are then discussed, which also serve to point out that the future directions in this important field of fluid mechanics lie in the extension to the non-standard or non-classical Rayleigh-Bénard configuration.

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H1 norm error estimates are given.

Thermal Convection in a Tilted Rectangular Cell with Aspect Ratio 0.5

Thermal convection in a three-dimensional tilted rectangular cell with aspect ratio 0.5 is studied using direct nu- merical simulations within both Oberbeck-Boussinesq （OB） approximation and strong non-Oberbeck-Boussinesq （NOB） effects. The considered Rayleigh numbers Ra range from 105 to 107, the working fluid is air at 30OK, and the corresponding Prandtl number Pr is 0.71. Within the OB approximation, it is found that there exist multiple states for Ra = 105 and hysteresis for Ra = 106. For a relatively small tilt angle/3, the large-scale circulation can either orient along one of the the vertical diagonal planes （denoted by Ma mode） or orient parallel to the front wall （denoted by Mp mode）. Which of the two modes transports heat more efficiently is not definitive, and it depends on the Rayleigh number Ra. For/Ta = 107 and β = 0°, the time-averaged flow field contains four rolls in the upper half and lower half of the cell, respectively, Md and Mp modes only developing in tilted cells. By investigating NOB effects in tilted convection for fixed/Ta = 106, it is found that the NOB effects on the Nusselt number Nu, the Reynolds number Re and the central temperature Tc for different β ranges are different. NOB effects can either increase or decrease Nu, Re and Tc when β is varied.

EXPERIMENTAL INVESTIGATION ON THE CHAOTIC PHENOMENA IN THE WAKE OF A NATURAL THERMAL CONVECTION FLOW

Chaotic phenomena in the wake of thermal convection flow fields above a heating flat plate were investigated experimentally. A newly developed electron beam fluorescence technique (EBF) was used to simultaneously measure density fluctuation at 7 points in a cross section above the plate. Correlation dimensions, intermittence coefficients, Fourier spectrum have been obtained for different Grashof numbers. Spatial distribution of correlation dimensions are presented. The experimental result shows that there is a certain relationship between the density fluctuation and the Gr number. And time-spacial characteristic of chaos evolution is also given.

Thermal Convection Mechanics

Assuming thermal convection takes place continuously inside the North Pacific High, an application of physics to the rising sea-warmed air plumes increases understanding of the relatively high air pressure at the sea surface and at the bases of the plumes. Since the ocean is warmer than the air under the NPH, heat is transferred upward decreasing the air density, which should then accelerate upward when no other forces are around to stop it. By Newton’s 3rd law the upward accelerating air will cause an equal but opposite (downward) reaction resulting in high pressure in the air under the rising column. That is the central proposal offered, which is consistent with available observations. New data that confirm the upward acceleration of the heated air are especially needed. Perhaps dye or neutrally buoyant particles could be released from a ship or buoy to make the upward air motion visible.

Effect of Thermal Convection on Density Segregation in Binary Granular Gases with Dissipative Lateral Walls

Molecular dynamics simulations are employed to investigate the effect of thermal convection induced only by dissipative lateral walls on density segregation of the strongly driven binary granular gases under low gravity conditions. It is found that the thermal convection due to dissipative lateral walls has significant influence on the segregation intensity of the system. The dominant factor in determining the degree of segregation achieved by the system is found to be the relative convection rate between differing species. Moreover, a qualitative explanation is proposed for the relationship between the thermal convection due to dissipative lateral walls and the observed segregation intensity profiles.

Thermal Convection in the North Pacific High Pressure Cell

If it is accepted that thermal convection consistently takes place inside the North Pacific High, as proposed here, then the existence of the NPH, as well as its seasonal variation, will be explained simultaneously, building on an earlier attempt. More observations than available at present would help prove that thermal convection happens and pin down its characteristics, since it is not visible. Also the physics of how thermal convection produces relatively high pressure at sea level needs work.

Nonlinear free thermal convection in a spherical shell:A variable viscosity model

In mantle convection models, the mantle viscosity is generally assumed constant or dependent on depth. In this paper, a laterally variable viscosity is introduced into mantle convection model in which the mantle viscosity consists of a constant background and latitude-dependent viscosity with small fluctuations. The features of toroidal field dependent on depth and Rayleigh number are discussed under two boundary conditions, i.e., the top rigid and bottom stress-free boundaries (R-F boundary for short) and both rigid ones (R-R boundary for short), respectively. The results show that the energy of toroidal field mainly concentrates in the middle and upper parts of a spherical shell, and the ratio of toroidal to total velocities amounts to only a few percents and hardly depends on Rayleigh number, while the convection patterns of toroidal field have been strongly affected by Rayleigh number. It is found that the convection patterns and velocities of toroidal field have obvious differences in latitudinal direction, which clearly reflects the effects of lateral mantle viscosity variations on the convection patterns. These preliminary results give us a possible hint to study some global tectonic phenomena, e.g. the asymmetry of the southern and northern hemispheres and the Earth's differential rotation.

Three-Dimensional Convection in an Inclined Porous Layer Subjected to a Vertical Temperature Gradient

In this paper,we study the onset and development of three-dimensional convection in a tilted porous layer saturated with a liquid.The layer is subjected to a gravitational field and a strictly vertical temperature gradient.Typically,problems of thermal convection in tilted porous media saturated with a liquid are studied by assuming constant different temperatures at the boundaries of the layer,which prevent these systems from supporting conductive(non-convective)states.The boundary conditions considered in the present work allow a conductive state and are representative of typical geological applications.In an earlier work,we carried out a linear stability analysis of the conductive state.It was shown that at any layer tilt angles,the most dangerous type of disturbances are longitudinal rolls.Moreover,a non-zero velocity component exists in z-direction.In the present work,threedimensional non-linear convection regimes are studied.The original three-dimensional problem is reduced to two-dimensional one with an analytical expression for the velocity z-component v_(z)=v_(z)(x,y).It is shown that the critical Rayleigh number values obtained through numerical solutions of the obtained 2D problem by a finite difference method for different layer inclination angles,are in a good agreement with those predicted by the linear theory.The number of convective rolls realized in nonlinear calculations also fits the linear theory predictions for a given cavity geometry.Calculations carried out at low supercriticalities show that a direct bifurcation takes place.With increasing supercriticality,no transitions to other convective regimes are detected.The situation studied in this problem can be observed in oil-bearing rock formations under the influence of a geothermal temperature gradient,where the ensuing fluid convection can affect the distribution of oil throughout the layer.

The viscoelastic effects on thermal convection of an Oldroyd-B fluid in open-top porous media

The effects of two viscoelastic parameters on the thermal convection of a viscoelastic Oldroyd-B fluid in an open-top porous square box with constant heat flux are investigated. The results show that the increase of relaxation time is able to destabilize the fluid flow leading to a higher heat transfer rate, while the increase of retardation time tends to stabilize the flow and suppress the heat transfer. The flow bifurcation appears earlier with the increase of the relaxation time and the decrease of the retardation time, re- suiting in more complicated flow patterns in the porous medium.

Thermal Convection in a Spherical Shell with Melting/Freezing at either or both of Its Boundaries

In a number of geophysical or planetological settings, including Earth＇s inner core, a sili- cate mantle crystallizing from a magma ocean, or an ice shell surrounding a deep water ocean--a situa- tion possibly encountered in a number of Jupiter and Saturn＇s icy satellites--a convecting crystalline layer is in contact with a layer of its melt. Allowing for melting/freezing at one or both of the boundaries of the solid layer is likely to affect the pattern of convection in the layer. We study here the onset of thermal convection in a viscous spherical shell with dynamically induced melting/freezing at either or both of its boundaries. It is shown that the behavior of each interface---permeable or impermeable-- depends on the value of a dimensionless number P （one for each boundary）, which is the ratio of a melting/freezing timescale over a viscous relaxation timescale. A small value of P corresponds to perme- able boundary conditions, while a large value of P corresponds to impermeable boundary conditions. Linear stability analysis predicts a significant effect of semi-permeable boundaries when the number P characterizing either of the boundary is small enough： allowing for melting/freezing at either of the boundary allows the emergence of larger scale convective modes. The effect is particularly drastic when the outer boundary is permeable, since the degree 1 mode remains the most unstable even in the case of thin spherical shells. In the case of a spherical shell with permeable inner and outer boundaries, the most unstable mode consists in a global translation of the solid shell, with no deformation. In the limit of a full sphere with permeable outer boundary, this corresponds to the ＂convective translation＂ mode recently proposed for Earth＇s inner core. As another example of possible application, we discuss the case of thermal convection in Enceladus＇ ice shell assuming the presence of a global subsurface ocean, and found that melting/freezing could have an important effect on the pattern of convection in the ice shell.

Numerical simulation of thermal convection of viscoelastic fluids in an open-top porous medium with constant heat flux

This paper makes a numerical study of the buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box under the constant heat flux boundary condition. The effects of the relaxation and retardation times on the onset of the oscillatory convection, the convection heat transfer rate and the flow pattern transition are investigated. It is shown that a large relaxation time can destabilize the fluid flow leading to an early onset of the thermal convection and a high heat transfer rate, while a large retardation time tends to stabilize the flow and suppress the convection onset and the heat transfer. After the convection sets in, the flow bifurcation appears earlier with the increase of the relaxation time and the decrease of the retardation time, resulting in more complicated flow patterns in the porous medium. Furthermore, with the increase of the ratio of the relaxation time to the retardation time, the fluid may be blocked from flowing through the open-top boundary, which may be caused by the viscoelastic effect. Finally, the comparison of our results with those under isothermal heating boundary conditions reveals that the heat transfer rate correspo- nding to a constant heat flux boundary is always higher.

Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition, thermal radiation, and mixed convection

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer as- sumptions are taken into account to govern the mathematical model of the flow analy- sis. Some suitable similarity variables are introduced to transform the partial differen- tial equations into ordinary differential systems. fifth-order techniques with the shooting method The Runge-Kutta-Fehlberg fourth- and are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the non- linear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.

Heat transfer and plume statistics in turbulent thermal convection with sparse fractal roughness

Rough-surface Rayleigh-Benard convection is investigated using direct numerical simulations in two-dimensional convection cells with aspect ratioГ=2.Three types of fractal roughness elements,which are marked as nl,n2 and n3,are constructed based on the Koch curve and sparsely mounted on both the plates,where n denotes the level of the roughness.The considered Rayleigh numbers Ra range from 10^(7)to 10^(11)with Prandtl number Pr=1.Two regimes are identified for cases nl,n2.In Regime I,the scaling exponentsβin the effective Nusselt number Nu vs Ra scaling Nu~Ra^(β)reach up to about 0.4.However,when Ra is larger than a critical value Ra_(c),the flow enters RegimeⅡ,with p saturating back to a value close to the smooth-wall case(0.3).Rac is found to increase with increasing n,and for case n3,only Regime I is identified in the studied Ra range.The extension of Regime I in case n3 is due to the fact that at high Ra,the smallest roughness elements can play a role to disrupt the thermal boundary layers.The thermal dissipation rate is studied and it is found that the increasedβin Regime I is related with enhanced thermal dissipation rate in the bulk.An interesting finding is that no clear convection roll structures can be identified for the rough cases,which is different from the smooth case where well-organized convection rolls can be found.This difference is further quantified by the detailed analysis of the plume statistics,and it is found that the horizontal profiles of plume density and velocity are relatively flattened due to the absence of clear convection rolls.

Hybrid Effects of Thermal and Concentration Convection on Peristaltic Flow of Fourth Grade Nanofluids in an Inclined Tapered Channel: Applications of Double-Diffusivity

This article brings into focus the hybrid effects of thermal and concentration convection on peristaltic pumping of fourth grade nanofluids in an inclined tapered channel.First,the brief mathematical modelling of the fourth grade nanofluids is provided along with thermal and concentration convection.The Lubrication method is used to simplify the partial differential equations which are tremendously nonlinear.Further,analytical technique is applied to solve the differential equations that are strongly nonlinear in nature,and exact solutions of temperature,volume fraction of nanoparticles,and concentration are studied.Numerical and graphical findings manifest the influence of various physical flow-quantity parameters.It is observed that the nanoparticle fraction decreases because of the increasing values of Brownian motion parameter and Dufour parameter,whereas the behaviour of nanoparticle fraction is quite opposite for thermophoresis parameter.It is also noted that the temperature profile decreases with increasing Brownian motion parameter values and rises with Dufour parameter values.Moreover,the concentration profile ascends with increasing thermophoresis parameter and Soret parameter values.

The patterns of high-degree thermal free convection and its features in a spherical shell

A high-degree (degree l = 6 and order m = 0, 1, 2, [midline ellipsis] , l. High-order model for short) and steady thermal free convective motion of an infinite Prandtl number and Boussinesq fluid in a spherical shell is calculated by a Galerkin method. Convection is driven by an imposed temperature drop across top rigid and bottom stress-free isothermal boundaries only heated from below of the shell. In this paper, the scalar poloidal and fluctuating temperature fields are expanded into associated Legendre polynomials with degree l = 6 and order m = 0, 1, 2, [midline ellipsis] , l. Compared with zero-order model (degree l = 6 and order m = 0), from which 2-D longitudinal (r-θ) profiles can be obtained, high-order model can provide a series of southerly (r-θ), easterly (r-φ) and radial (θ-φ) velocity profiles, which probably reveal more detail features of mass motion in the mantle. It is found that Rayleigh number has great effects on the patterns and velocities of thermal free convection and controls the relative ratio of hot and cold plume in the shell. Probably, the present results mainly reveal the mass motion in the lower mantle, while the striking differences of convection patterns from velocities at different positions have important geodynamical significances.

Parameterized thermal model of a mixed mantle convection

Simple parameterized models, either whole mantle convection or layered mantleconvection, cannot explain the tectonic characteristics of the Earth's evolution history, therefore a mixed mantle convection model has been carried out in this paper. We introduce a time-dependent parameter F, which denotes the ratio betWeen the mantle material involved in whole mantle convection and the material of the entire mantle, and introduce a local Rayleigh number Raloc as well as two critical numbers Ra1 and Ra2. These parameters are used to describe the stability of the phase boundary between the upper and lower mantle. The result shows that the mixed mantle convection model is able to simulate the episodic tectonic evolution of the Earth.

TEMPERATURE PROFILES OF LOCAL THERMAL NONEQUILIBRIUM FOR THERMAL DEVELOPING FORCED CONVECTION IN POROUS MEDIUM PARALLEL PLATE CHANNEL

Based on the two-energy equation model, taking into account viscous dissipation due to the interaction between solid skeleton and pore fluid flow, temperature expressions of the solid skeleton and pore fluid flow are obtained analytically for the thermally developing forced convection in a saturated porous medium parallel plate channel, with walls being at constant temperature. It is proved that the temperatures of the two phases for the local thermal nonequilibrium approach to the temperature derived from the one-energy equation model for the local thermal equilibrium when the heat exchange coefficient goes to infinite. The temperature profiles are shown in figures for different dimensionless parameters and the effects of the parameters on the local thermal nonequilibrium are revealed by parameter study.

Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper.A linear stability analysis and a Chebyshevτ-QZ algorithm are employed to solve the thermal mixed convection.Unlike the case in a single layer,the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers.We find that the longitudinal rolls(LRs)only depend on the depth ratio.With the existence of the shear flow,the effects of the depth ratio,the Reynolds number,the Prandtl number,the stress relaxation,and strain retardation times on the transverse rolls(TRs)are also studied.Additionally,the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system.In contrast to the case for Newtonian fluids,the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases.

Water-Tank Experiment on the Thermal Circulation Induced by the Bottom Heating in an Asymmetric Valley

Water tank experiments were carried out to investigate the thermal convection due to the bottom heating in an asymmetrical valley under neutral and stably stratified approach flows with the Particle Image Velometry (PIV) visualization technique. In the neutral stratification approach flow, the ascending draft induced by bottom heating is mainly located in the center of the valley in calm ambient wind. However, with ambient wind flow, the thermal convection is shifted leeward, and the descending draft is located on the leeward side of the valley, while the ascending draft is located on the windward side. The descending draft is minorly turbulent and organized, while the ascending draft is highly turbulent. With the increase of the towing speed, the descending and ascending drafts induced by the mechanical elevation begin to play a more dominant role in the valley flow, while the role of the thermal convection in the valley airflow becomes limited. In the stable stratification approach flow, the thermal convection is limited by the stable stratification and no distinct circulation is formed in calm ambient wind. With ambient wind, agravity wave appears in the upper layer in the valley. With the increase of the ambient wind speed, a gravity wave plays an important role in the valley flow, and the location and intensity of the thermal convection are also modulated by the gravity internal waves. The thermal convection has difficulty penetrating the upper stable layer. Its exchange is limited between the air in the upper layer and that in the lower layer in the valley, and it is adverse to the diffusion of pollutants in the valley.