Showing papers on "Rayleigh number published in 1995"
TL;DR: In this article, a finite element multigrid scheme was employed for large viscosity variations and convection with up to 1014 contrasts was systematically investigated in a 2D square cell with free slip boundaries.
Abstract: Previous experimental studies of convection in fluids with temperature‐dependent viscosity reached viscosity contrasts of the order of 105. Although this value seems large, it still might not be large enough for understanding convection in the interiors of Earth and other planets whose viscosity is a much stronger function of temperature. The reason is that, according to theory, above 104–105 viscosity contrasts, convection must undergo a major transition—to stagnant lid convection. This is an asymptotic regime in which a stagnant lid is formed on the top of the layer and convection is driven by the intrinsic, rheological, temperature scale, rather than by the entire temperature drop in the layer. A finite element multigrid scheme appropriate for large viscosity variations is employed and convection with up to 1014 viscosity contrasts has been systematically investigated in a 2D square cell with free‐slip boundaries. We reached the asymptotic regime in the limit of large viscosity contrasts and obtained s...
509 citations
TL;DR: In this article, the authors investigated the idea of formulating free convection in large-scale models as a special case of forced convection and showed that the wind speed in the surface transfer law is related to the velocity of the large eddies in the mixed layer and is proportional to the convection velocity scale.
Abstract: This paper investigates the idea of formulating free convection in large-scale models as a special case of forced convection. In free convection, the wind speed in the surface transfer law is related to the velocity of the large eddies in the mixed layer and is proportional to the convection velocity scale. The empirical coefficient is estimated with the help of large eddy simulation data by Sykes et al. (1993) and the resulting formulation is compared with field data by Stull (1994). This concept is shown to be applicable to smooth ocean surfaces as well as rough land surfaces. It is argued that within this framework, free convection is a natural extension of forced convection and only needs a minor modification of traditional transfer laws.
499 citations
TL;DR: In this paper, a finite difference scheme consisting of modified ADI (Alternating Direction Implicit) method and SLOR (Successive Line Over Relaxation) method is used to solve the vorticity-stream function formulation of the problem.
484 citations
TL;DR: In this article, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo, and it is concluded that compositional convection may not dominate thermal convection, as had previously been argued by Braginsky.
Abstract: Convection in Earth's fluid core is regarded as a small deviation from a well-mixed adiabatic state of uniform chemical composition. The core is modeled as a binary alloy of iron and some lighter constituent, whose precise chemical composition is unknown but which is here assumed to be FeAd, where Ad = Si, O or S. The turbulent transport of heat and light constituent is considered, and a simple ansatz is proposed in which this is modeled by anisotropic diffusion. On this basis, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo. The dual (thermal + compositional) nature of core convection is reconsidered. It is concluded that compositional convection may not dominate thermal convection, as had previously been argued by Braginsky (Soviet Phys. Dokl., v. 149, p. 8, 1963; Geomag, and Aeron., v. 4, p. 698, 1964), but that the two mechanisms are most probably comparable in importance. The key parameters leading to this conclusion are isolated...
483 citations
Book•
01 Feb 1995
TL;DR: In this article, Navier-Stokes and Pohlhausen provided an energy equation for Laminar Boundary-Layer Equations of the Boundary Layer and Energy Equation Problems.
Abstract: Foundations of Heat Transfer Nomenclature Introductory Remarks Modes of Heat Transfer Continuum Concept Some Definitions and Concepts of Thermodynamics General Laws Particular Laws Problems References Suggested Reading Governing Equations of Convective Heat Transfer Nomenclature Introduction Continuity Equation Momentum Equations Energy Equation Discussion of the Fundamental Equations Similarities in Fluid Flow and Heat Transfer Problems References Boundary-Layer Approximations for Laminar Flow Nomenclature Introduction Momentum Equations of the Boundary Layer Boundary-Layer Energy Equation Problems References Heat Transfer in Incompressible Laminar External Boundary Layers: Similarity Solutions Nomenclature Introduction Laminar Velocity Boundary Layer Thermal Boundary Layer Fluid Friction and Heat Transfer Flows with Pressure Gradients Problems References Integral Method Nomenclature Introduction Momentum Integral Equation Energy Integral Equation Laminar Forced Flow over a Flat Plate Thermal Boundary Layer on an Isothermal Flat Plate Thermal Boundary Layer on a Flat Plate with Constant Surface Heat Flux Flat Plate with Varying Surface Temperature Flows with Pressure Gradient Problems References Laminar Forced Convection in Pipes and Ducts Nomenclature Introduction Laminar and Turbulent Flows in Ducts Some Exact Solutions of Navier-Stokes Equations Friction Factor Noncircular Cross-Sectional Ducts Laminar Forced Convection in Ducts Thermal Boundary Conditions Laminar Forced Convection in Circular Pipes with Fully Developed Conditions Laminar Forced Convection in the Thermal Entrance Region of a Circular Duct Laminar Flow Heat Transfer in the Combined Entrance Region of Circular Ducts Laminar Convective Heat Transfer between Two Parallel Plates Integral Method Asymptotic Values of Heat-Transfer Coefficients in Ducts Effect of Circumferential Heat-Flux Variation Heat Transfer in Annular Passages Problems References Forced Convection in Turbulent Flow Nomenclature Introduction Governing Equations with Steady Turbulent Flow Turbulence Models Velocity Distribution in Turbulent Flow Friction Factors for Turbulent Flow Analogies between Heat and Momentum Transfer Further Analogies in Turbulent Flow Turbulent Heat Transfer in a Circular Duct with Variable Circumferential Heat Flux Turbulent Heat Transfer in Annular Passages Effect of Boundary Conditions on Heat Transfer Turbulent Flow on a Flat Plate Problems References Unsteady Forced Convection in Ducts Nomenclature Introduction Transient Laminar Forced Convection in Ducts Transient Turbulent Forced Convection in Ducts Analysis of Transient Forced Convection for Timewise Variation of Inlet Temperature Problems References Empirical Correlations for Single-Phase Forced Convection in Ducts Nomenclature Introduction Dimensional Analysis of Forced Convection Laminar Forced Convection Effects of Variable Physical Properties Turbulent Forced Convection Turbulent Flow in Smooth Straight Noncircular Ducts Effects of Variable Physical Properties in Turbulent Forced Convection Liquid Metal Heat Transfer Summary Problems References Heat Transfer in Natural Convection Nomenclature Introduction Basic Equations of Laminar Boundary Layer Pohlhausen Solution for Laminar Boundary Layer over a Constant Temperature Vertical Flat Plate Exact Solution of Boundary-Layer Equations for Uniform Heat Flux Inclined and Horizontal Surfaces Property Variation in Free Convection Approximate Solution of Laminar Free Convection on a Vertical Plate: Von Karman-Pohlhausen Integral Method Turbulent Heat Transfer on a Vertical Plate Dimensional Analysis in Natural Convection Interferometric Studies Natural Convection in Enclosed Spaces Correlations for Natural Convection in Enclosures Combined Free and Forced Convection Problems References Heat Transfer in High-Speed Flow Nomenclature Introduction Stagnation Temperature Adiabatic Wall Temperature and Recovery Factor Governing Equations in High-Velocity Flow Thermal Boundary Layer over a Flat Plate in High-Speed Flow Heat Transfer in 2D Turbulent Boundary Layers Problems References Convective Heat Transfer in Microchannels Nomenclature Introduction Definitions in Microchannels Convective Heat Transfer for Gaseous Flow in Microchannels Effects of Temperature Jump Effects of Viscous Dissipation Effects of Channel Roughness Effects of Variable Fluid Properties Empirical Correlations for Gaseous Forced Convection in Microchannels Empirical Correlations for Liquid Forced Convection in Microchannels Problems References Enhancement of Convective Heat Transfer with Nanofluids Nomenclature Introduction Nanofluid Convective Heat-Transfer Modeling Empirical Correlation for Single-Phase Forced Convection with Nanofluids Problems References Appendices Index
251 citations
TL;DR: In this article, the steady natural convection in an enclosure heated from below and symmetrically cooled from the sides is studied numerically, using a stream function-vorticity formulation.
237 citations
TL;DR: In this paper, the effects of temperature-dependent viscosity and of the shape of a conductive lid on the formation of large-scale convection cells in the Earth's mantle were investigated.
Abstract: At the Earth's surface, continents and oceans impose different thermal boundary conditions at the top of the mantle. Laboratory experiments are used to investigate the consequences of this for mantle convection. The upper boundary of the experimental tank was made of copper plates enforcing a fixed temperature and had a conductive lid of finite width in the middle. Beneath this lid, the thermal boundary condition was of the "mixed" type, with a Biot number depending on the dimensions and thermal conductivity of the lid. Experimental values of the Biot number were scaled to Earth values. Experiments were run for a large range of Rayleigh numbers, from 104 to 107, and for several lid widths. The effects of temperature-dependent viscosity and of the shape of the lid were investigated. At steady state, in all cases, there is an upwelling beneath the conductive lid, which feeds two symmetrical and elongated convective cells. Three different dynamic regimes were identified as a function of Rayleigh number, independently of the lid width. At Rayleigh numbers lower than 1.2 105, the upwelling is steady both in geometry and temperature structure. At Rayleigh numbers between 1.2 105 and 2 106, this central upwelling is intermittent. At larger values of the Rayleigh number, there is no longer a simple upwelling structure, but a set of small plumes rising together and distorted by a cellular circulation of large horizontal extent. Thus the conductive lid always imposes a large-scale flow pattern. The length of these convective cells is a function of lid width. It is equal to the lid width at large values and decreases to the Rayleigh-Benard value as the lid width decreases to zero. A fluid loop model explains the most important features of this form of convection. The cell length is such that the upwelling temperature is minimized for a given Rayleigh number and lid width and is an increasing function of lid width and a decreasing function of Rayleigh number. Transient experiments demonstrate that the large-scale flow structure develops rapidly with even small horizontal temperature differences. Implications for the Earth are that large-scale convection cells exist in conditions which, in the absence of continents, would probably lead to a chaotic convection pattern dominated by plumes. At high Rayleigh number, continental breakup is effected by a large-scale line upwelling structure which includes a number of individual plumes.
140 citations
TL;DR: In this article, the effect of a transverse magnetic field on buoyancy-driven convection in a shallow rectangular cavity is numerically investigated (horizontal Bridgman configuration), where the enclosure is insulated on top and bottom walls while it is heated from one side and cooled from the other.
Abstract: The effect of a transverse magnetic field on buoyancy-driven convection in a shallow rectangular cavity is numerically investigated (horizontal Bridgman configuration). The enclosure is insulated on the top and bottom walls while it is heated from one side and cooled from the other. Both cases of a cavity with all rigid boundaries and a cavity with a free upper surface are considered. The study covers the range of the Rayleigh number, Ra, from 10 2 to 10 5 , the Hartmann number, Ha, from 0 to 10 2 , the Prandtl number, Pr, from 0.005 to 1 and aspect ratio of the cavity, A, from 1 to 6. Comparison is made with an existing analytical solution (Garandet et al.), based on a parallel flow approximation, and its range of validity is delineated. Results are presented for the velocity and temperature profiles and heat transfer in terms of Ha number. At high Hartmann numbers, both analytical and numerical analyses reveal that the velocity gradient in the core is constant outside the two Hartmann layers at the vicinity of the walls normal to the magnetic field.
118 citations
TL;DR: In this article, the authors investigated a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls and showed that the flow structure increasingly departs from the well-known laminar one.
Abstract: Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 10 10 . The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Rayleigh number investigated they grow to such a point that they result in the formation of large unsteady eddies that totally disrupt the boundary layers. These eddies throw hot and cold fluid into the upper and lower parts of the core region, resulting in thermally more homogeneous top and bottom regions that squeeze a region of increased stratification near the mid-cavity height. It is also shown that these large unsteady eddies keep the internal waves in the stratified core region excited. These simulations also give access to the second-order statistics such as turbulent kinetic energy, thermal and viscous dissipation, Reynolds stresses and turbulent heat fluxes.
114 citations
TL;DR: In this article, a numerical study is made of double-diffusive natural convection in a rectangular fluid-saturated vertical porous enclosure, where the flows are driven by conditions of uniform heat and mass fluxes imposed along the two vertical side walls of the cavity where the two buoyancy effects can either augment or counteract each other.
113 citations
TL;DR: In this paper, the effects of Rayleigh numbers from 1.5 to 1.7, inclination angles from 10° to 90°, and aspect ratios of 0.5, 1.0, and 2.0 are investigated for a fixed Prandtl number (0.7).
Abstract: The present work is concerned with natural convection from open cavities or heated plates attached with parallel vertical strips. The bottom of the cavity is heated, and the vertical walls are assumed adiabatic. Numerical results are presented for steady, laminar natural convection for the geometry described. Effects of Rayleigh numbers from 1 × 10 3 to 1 × 10 7 , inclination angles from 10° to 90°, and aspect ratios of 0.5, 1.0, and 2.0 are investigated for a fixed Prandtl number (0.7). It is found that the average Nusselt number is not very sensitive to the inclination angle. Flow becomes unstable at high Rayleigh numbers and at low inclination angles. Flow pattern and heat transfer results are presented and discussed.
TL;DR: In this article, the phenomenon of natural convection in trapezoidal enclosures where upper and lower walls are not parallel, in particular a triangular geometry, is re-examined over a parameter domain in which the aspect ratio of the enclosure ranges from 0.1 to 1.0, the Rayleigh number varies between 102 and 105, and the Prandtl numbers correspond to air and water.
TL;DR: In this article, a comparison is made between the numerical results obtained by 10 groups for a standard test case, which was defined for turbulent natural-convection flow in enclosures.
Abstract: A comparison is made between the numerical results obtained by 10 groups for a standard test case, which was defined for turbulent natural-convection flow in enclosures. The standard case considers air at a Rayleigh number of 5 × 1010in a square, differentially heated enclosure with adiabatic horizontal walls. The turbulence is modeled by a standard k-ϵ model. Some contributions find a strong grid dependence of the transition point in the vertical boundary layers, which prevents finding the grid-independent solution. In some contributions the boundary condition for e on the wall leads to convergence problems. The solution that shows the best agreement with half of the contributions is taken as the numerical reference solution.
TL;DR: In this paper, the effect of free and forced convection on crystal dissolution was examined both theoretically and experimentally, and well-established relationships for heat and mass transfer were applied to obtain approximate expressions for the dissolution velocity and the associated thickness of the compositional boundary layer.
Abstract: The effect of free and forced convection on crystal dissolution is examined both theoretically and experimentally. Well-established relationships for heat and mass transfer are applied to obtain approximate expressions for the dissolution velocity and the associated thickness of the compositional boundary layer. These expressions are found to be in good agreement with experimental observations of the dissolution of quartz crystals in basalt and NaCl crystal in water. When applied to light felsic crystals in basaltic magmas, the expressions predict that forced convection will produce a boundary layer thickness of about 100 μm and a dissolution velocity of order 10−6 cm s−1. These velocities are too slow for xenocrysts to be dissolved significantly during magma ascent in dykes, but are sufficient for cm-size crystals to dissolve in the interior of a convecting magma chamber. Larger crystals are likely to accumulate at the chamber's roof, where free convection is predicted to dissolve them at velocities of order 10−7 cm s−1. In an Appendix, the dissolution of the chamber's walls is also considered, and a velocity of order 10−8cm s−1 is predicted.
TL;DR: The magnetic fields generated by thermal convection in a rapidly rotating fluid spherical shell are studied in this paper, where the inner core is sandwiched between a finitely conducting solid inner core and a nonconducting mantle.
TL;DR: In this article, the authors review current capabilities for predicting flow in the cooling passages and cavities of jet engines and show that progress is being made, particularly in respect to the flow in serpentine blade-cooling passages.
TL;DR: In this paper, the effect of Rayleigh's vortical acoustic streaming that appears in the region between the plates as a result of the sound wave leading to forced heat convection was analyzed theoretically.
TL;DR: In this article, the effect of medium permeability on thermal convection in micropolar fluids is considered and it is found that the presence of coupling between thermal and micro-fluid effects may introduce oscillatory motions in the system.
TL;DR: In this paper, the authors extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition.
Abstract: In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl number Pr takes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh number R and the Ekman number E for the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation between R and E is derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of small E and Pr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.
TL;DR: Nonlinear convection structures are investigated in quantitative detail as a function of Rayleigh number for several negative and positive Soret coupling strengths (separation ratios) and different Lewis and Prandtl numbers characterizing different mixtures.
Abstract: Nonlinear convection structures are investigated in quantitative detail as a function of Rayleigh number for several negative and positive Soret coupling strengths (separation ratios) and different Lewis and Prandtl numbers characterizing different mixtures. A finite difference method was used to solve the full hydrodynamic field equations in a range of experimentally accessible parameters. We elucidate the important role that the concentration field plays in the nonlinear states of stationary overturning convection (SOC) and of traveling wave (TW) convection. Structural differences in the concentration boundary layers and of the concentration plumes in TW's and SOC's and their physical consequences are discussed. These properties show that the states con- sidered here are indeed strongly nonlinear, as expected from the magnitude of advection and diffusion in the concentration balance. The bifurcation behaviour of the states is analysed using different order parameters such as flow intensity, Nusselt number, a newly defined mixing parameter characterized by the variance of the concentration field, and the TW frequency. For further comparison with experiments, light intensity distributions are determined that can be observed in side-view shadowgraphs. Structural analyses of all fields are made using colour coded isoplots, vertical and lateral field profiles, and lateral Fourier decompositions. Transport properties of TWs are also discussed, in particular the mean lateral concentration current that is caused by the phase difference between concentration wave and velocity wave and that is roughly proportional to the TW frequency. This current plays an important role in the structural dynamics and stability of spatially-localized traveling-wave convection (cf. accompanying paper).
TL;DR: In this paper, a numerical study on the laminar natural convection flow of air in differentially heated, trapezoidal enclosures is presented, where the influence of the inclination angle on the flow and Nusselt number is discussed.
Abstract: Results are presented of a numerical study on the laminar natural convection flow of air in differentially heated, trapezoidal enclosures. Special attention is given to the applied solution method, using a coordinate-invariant formulation of the transport equations. For Rayleigh numbers between 104 and 108, results are presented for inclination angles of the isothermal walls from — 45° ( trapezoidal enclosure) to 0° ( square enclosure) The influence of the inclination angle on the flow and Nusselt number is discussed. Flow patterns and isotherms are shown to give greater understanding of the local heat transfer. The dependence of the averaged Nusselt number on the Rayleigh number is studied.
TL;DR: In this article, the linear stability of the fully developed natural convection flow in a differentially heated tall vertical enclosure under non-Boussinesq conditions was examined and the dependence of the critical Rayleigh number on the temperature difference was investigated.
TL;DR: In this article, the authors studied surface-tension-driven convection in a planar fluid layer by numerical simulation of the three-dimensional time-dependent governing equations in the limit of infinite Prandtl number.
Abstract: Surface-tension-driven convection in a planar fluid layer is studied by numerical simulation of the three-dimensional time-dependent governing equations in the limit of infinite Prandtl number. Emphasis is placed on the spatial scale of weakly supercritical flows and on the generation of small-scale structures in strongly supercritical flows. The decrease of the size of weakly supercritical hexagonal convection cells that we find is in agreement with experimental results. In the case of high Marangoni number, discontinuities of the temperature gradient are formed between convection cells, producing a universal spectrum E - k-3 of the two-dimensional surface temperature field. The possibility of experimental verification is discussed on the basis of shadowgraph images calculated from the predicted hydrodynamic fields.
01 Jan 1995
TL;DR: SAINTS as discussed by the authors is a PC-Aided Numerical Heat Transfer (NHT) system that is used in the SAINTS Load Module "Wind Tunnel Simulator" to simulate the flow and heat transfer in Porous media.
Abstract: Introduction Background PC-Aided Numerical Heat Transfer Outline of the Book Governing Equations for Flow and Heat Transfer Transformation From the System Form to the Control Volume Form Equation of Continuity Momentum Equation Energy Equation Complete Set of Governing Equations and Their Simplified Form General Transport Equation Analytical Treatments for Boundary Layer Equations Numerical Integration of Ordinary Differential Equations Transient Conduction in a Semi-Infinite Solid Boundary Layer Approximation for Heat and Fluid Flow Forced Convection From Concentrated Heat Sources Laminar Forced Convection From Plane Bodies Laminar Forced Convection From Axisymmetric Bodies Asymptotic Solutions for Forced Convection of Small and Large Prandtl Number Fluids Integral Method for Laminar Forced Convection Laminar Free Convection From Plane Bodies Integral Method for Laminar Free Convection Transport Equations for Modeling Turbulence Reynolds-Averaged Navier-Stokes Equation and Energy Equation Effective Viscosity Formulation and Mixing Length Models Wall Laws for Turbulent Shear Flows Turbulent Free jets Reynolds Stress Transport Equation Turbulence Kinetic Energy Transport Equation and Two-Equation Model Low Reynolds Number Model and High Reynolds Number Model Convective Flows in Porous Media Darcy's Law Modified Darcy's Laws Volume-Averaged Navier-Stokes Equation Volume-Averaged Energy Equation Effects of Channeling and Thermal Dispersion Magnitude Analysis on Boundary Layer Equations for Porous Media Darcy-Forchheimer Boundary Layer Equations Simple Flow Cases: Isothermal Flat Plates Modified Peclet Number and Flow Regime Map Unified Treatment for Darcy-Forchheimer Boundary Layer Equations Forced Convection Regime Darcy Free Convection Regime Forchheimer Free Convection Regime Intermediate Flow Regimes Convective Flows Over an Impermeable Horizontal Surface Buoyancy-Induced Flows From Concentrated Heat Sources Boundary Layer Flow and Heat Transfer in Highly Porous Media Description of Numerical Solution Procedure Basic Concept of Discretization Governing Equations and Auxiliary Relationships General Form of Governing Equations: General Transport Equation Coordinate System and Normalization Discretization of General Transport Equation Staggered Grid and Discretized Momentum Equations Pressure Correction Procedure: SIMPLE High Flux Modification: Hybrid Difference Scheme Solution of Discretized Equations PC Program "SAINTS" For Conduction and Convection Problems Overall Aspect of the Program "SAINTS" Classification of Boundaries Specification of Non-Zero Boundary Values Along the Known-Velocity Boundary Description of the Program "SAINTS" Input Procedure: Input Data and Problem-Dependent Subprograms Layout of Output Illustrative Applications of "SAINTS" Applications of the SAINTS Load Module "Wind Tunnel Simulator" Illustrative Applications to Conduction Problems Further Application of SAINTS to Complex Turbulent Flows Applications to Convection Problems in Porous Media Concluding Remarks Appendices Important Dimensionless Numbers Potential Flow Analysis Based on Source-and-Sink Method Listing of Program "SAINTS" Listing of Problem Dependent Subroutine "USERIN" Input Data for Forced Convection in a Tube Sample Output of Program "SAINTS" Program Instructions References Index
TL;DR: In this article, a finite element method employing Galerkin's approach was developed to analyze free convection heat transfer in axisymmetric fluid saturated porous bodies, and the method was used to study the effect of aspect ratio and radius ratio on Nusselt number in the case of a proous cylindrical annulus.
Abstract: A finite element method employing Galerkin’s approach is developed to analyze free convection heat transfer in axisymmetric fluid saturated porous bodies. The method is used to study the effect of aspect ratio and radius ratio on Nusselt number in the case of a proous cylindrical annulus. Two cases of isothermal and convective boundary conditions are considered. The Nusselt number is always found to increase with radius ratio and Rayleigh number. It exhibits a maximum when the aspect ratio is around unity; maximum shifts towards lesser aspect ratios as Rayleigh number increases. Results are compared with those in the literature, wherever available, and the agreement is found to be good.
TL;DR: In this paper, a numerical study examines bifurcation sequences in Rayleigh-Benard convection for small aspect ratio enclosures, where the Boussinesq approximation is invoked with the exception of temperature dependent viscosity of the fluid.
TL;DR: In this paper, the effect of variable viscosity is more pronounced on the friction factor than on Nusselt numbers, and the results indicate that the effects of inlet Rayleigh number and inlet velocity profile only exist in the near-inlet region.
TL;DR: In this article, the authors studied 2D time-dependent convection for a rheology which is both non-Newtonian and temperature-dependent, and found that viscous heating increases strongly with the subduction speed.
Abstract: We have studied 2-D time-dependent convection for a rheology which is both non-Newtonian and temperature-dependent. Strong effects associated with viscous heating are found in the downwelling sheets, which are heated on both sides with an intensity around O(100) times the chondritic value. The magnitude of viscous heating increases strongly with the subduction speed. The slab interior is weakened by viscous heating and slab breakoff then takes place. This process provides a self-regulating mechanism for governing the speed of intact slabs able to reach the deep mantle. Timescales associated with viscous heating are quite short, a few million years. Internal heating by radioactivity decreases the amount of shear heating.
TL;DR: In this paper, the authors explore the dependence of the scale and intensity of convective elements in a rotating fluid on variations in external parameters in a regime relevant to open ocean deep convection.
TL;DR: Nonlinear, spatially localized structures of traveling convection rolls are investigated in quantitative detail as a function of Rayleigh number for two different Soret coupling strengths (separation ratios) with Lewis and Prandtl numbers characterizing ethanol-water mixtures.
Abstract: Nonlinear, spatially localized structures of traveling convection rolls that are surrounded by quiescent fluid in horizontal layers of binary fluids heated from below are investigated in quantitative detail as a function of Rayleigh number for two different Soret coupling strengths (separation ratios) with Lewis and Prandtl numbers characterizing ethanol-water mixtures. A finite-difference method was used to solve the full hydrodynamic field equations numerically in a vertical cross section perpendicular to the roll axes subject to realistic horizontal and laterally periodic boundary conditions with different periodicity lengths. Structure and dynamics of these localized traveling waves (LTW's) are dominated by the concentration field. As in the spatially extended convective states that are investigated in an accompanying paper, the Soret-induced concentration variations strongly influence, via density changes, the buoyancy forces that drive convection. The spatiotemporal properties of this feedback mechanism, involving boundary layers and concentration plumes, show that LTW's are strongly nonlinear states. Light intensity distributions are determined that can be observed in side-view shadowgraphs done with horizontal light along the roll axes. Detailed analyses of all fields are made using color-coded isoplots, among others. In the frame comoving with their drift velocity, LTW's display a nontrivial spatiotemporal symmetry consisting of time translation by one-half an oscillation period combined with vertical reflection through the horizontal midplane of the layer. A time-averaged concentration current is driven by a phase difference between the waves of concentration and vertical velocity in the bulk of the LTW state. The associated large-scale concentration redistribution stabilizes the LTW and controls its drift velocity into the quiescent fluid by generating a buoyancy-reducing concentration "barrier" ahead of the leading LTW front. All considered LTW's drift very slowly into the direction of the phase velocity of the pattern. For weak Soret coupling, $\ensuremath{\psi}=\ensuremath{-}0.08$, LTW's have a small selected width and exist in a narrow band of Rayleigh numbers above the stability threshold for growth of TW's. For stronger coupling, $\ensuremath{\psi}=\ensuremath{-}0.25$, LTW's exist below the bifurcation threshold for extended TW's in a narrow band of Rayleigh numbers. In its lower part, LTW's have a small selected width. For somewhat higher Rayleigh numbers, there exist two LTW attractors with two different widths. For yet higher Rayleigh numbers, there is again only one LTW attractor; however, with a broader width. Dynamical properties and the dependence on the system length are analyzed. Comparisons with experiments are presented.