Showing papers on "Rayleigh number published in 1983"

Natural convection of air in a square cavity: A bench mark numerical solution

Abstract: Details are given of the computational method used to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls. Second-order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 103⩽Ra⩽10 6 which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one-tenth of that at the lowest value.

Convection in Liquids

Abstract: A : Introduction.- I - Fundamental Laws and Basic Concepts.- 1. Balance equations for incompressible fluids.- A. Conservation of mass.- B. Conservation of momentum.- C. Conservation of energy.- 2. Fundamental thermodynamic relations entropy balance equation and second law.- A. Alternative forms of the energy balance equation.- B. The entropy balance equation and the second law of thermodynamics.- 3. Kinetic and constitutive equations.- 4. Systems of coordinates.- A. Rectangular coordinates.- B. Cylindrical coordinates.- C. Special two-dimentional case : the stream function.- 5. Equations for the fluctuations around a steady state.- 6. Definition of stability.- 7. Normal modes.- 8. Dimensionless numbers in fluid dynamics and heat transfer problems.- Exercices.- Bibliographical notes.- II - Mathematical Background and Computational Techniques.- 1. Use of variational principles and/or stationary properties of integrals.- A. Elements of variational calculus. The Euler-Lagrange equations.- B. Variational approach to the conservations laws based on nonequilibrium thermodynamics : the theory of the local potential.- C. The numerical methods associated with the local potential theory.- D. Relation between the local potential and the Galerkin techniques.- 2. Applications to stability problems.- A. The excess local potential.- B. Variational methods for linear eigenvalue problems.- C. Stability criterion based on Lyapounov function.- 3. Purely numerical techniques.- A. Finite differences methods.- B. Conversion of a boundary value problem into an initial value problem.- Exercices.- Bibliographical notes.- B : Fluids at Constant Density, Isothermal Forced Convection.- III - Planar Flows of Newtonian Fluids.- 1. Poiseuille and Couette flow.- A. Plane Poiseuille flow and Poiseuille flow in rectangular channels.- B. Plane Couette flow.- 2. General statements of linear hydrodynamic stability of forced convection.- A. The Orr-Sommerfeld equation.- B. Variational or stationary presentations of the Orr-Sommerfeld equation. Its relation with the Galerkin technique.- C. The Chock-Schechter integration scheme.- D. The Orr and the Prigogine-Glansdorff criterion.- 3. Numerical solutions of the Orr-Sommerfeld equation.- A. Selection of trial functions.- B. Solution for U = constant.- C. Solution for plane Poiseuille flow.- a. Effect of trial functions.- b. High Reynolds numbers.- c. Two and three dimensional perturbations without elimination of variables. Relation to Squire's theorem.- d. Finite difference methods.- e. Solution using the Chock-Schechter method.- f. General discussion, comparison with experiments.- D. Solution for Couette flow.- 4. Nonlinear stability of Poiseuille flow.- A. Introduction.- B. A restricted variational approach to the nonlinear equations.- C. Influence of the initial amplitude of the disturbance.- 5. An oscillatory solution in planar-Poiseuille flow.- A. Introduction.- B. Existence of statistically steady states.- C. Existence of periodic flows.- D. Stability and/or instability of the new periodic flow.- 6. Remarks on the transition to turbulence.- Bibliographical notes.- IV - Cylindrical Flows of Newtonian Fluids.- 1. A. Poiseuille flow in a pipe.- B. Poiseuille flow down an annular pipe.- 2. General statements on linear stability of forced convection in cylindrical coordinates.- A. An equivalent of the Orr-Sommerfeld equation.- B. Non axisymmetric disturbances.- 3. Linear stability of pipe Poiseuille flow.- A. Stability with respect to two-dimensional axisymmetric disturbances.- B. Stability with respect to three-dimensional non axisymmetric disturbances.- Bibliographical notes.- V - Flow Stability of Non-Newtonian Fluids.- 1. Stress-Strain relations for some particular non-newtonian fluids.- A. Introduction.- B. The Coleman-Noll model.- 2. Stability of plane Poiseuille flow for a second order viscoelastic fluid.- A. The generalized Orr-Sommerfeld equation.- B. The solution of the generalized Orr-Sommerfeld equation for plane flow.- C. Plane Poiseuille flow : sufficient condition for stability.- D. Instability of plane Poiseuille flow of a second order fluid : a numerical result.- 3. Stability of pipe Poiseuille flow for a second order fluid..- Bibliographical notes.- C : Non Isothermal One Component Systems.- VI - Free Convection in One Component Fluid.- 1. Introduction.- 2. The linear theory of the Benard problem.- A. The eigenvalue problem. Its solution for simple boundary conditions.- B. Solutions based on approximate numerical calculations.- a. The local potential method.- b. The Chock-Schechter numerical integration.- C. Solution based on the thermodynamic stability criterion.- D. Experimental aspect.- E. Effect of lateral boundaries.- F. Extension of the Benard problem.- a. Surface tension effect.- b. Effect of a magnetic field.- 3. The non-linear theory of the Benard problem.- A. Approximate computational techniques.- B. Global properties of the flow.- a. Variation of the Nusselt number with the Rayleigh number (free boundary conditions).- b. Variation of the Nusselt number with the Rayleigh number (rigid boundary conditions).- c. Variation of the number of convective cells with the Rayleigh number.- C. Fine structure of the flow.- D. Behavior near threshold.- E. Behavior far from the critical point.- a. The Lorenz model.- b. The routes to turbulence.- 4. The thermogravitational process.- A. The steady state profile.- B. The stability of the steady state profile.- Bibliographical notes.- VII - Non Isothermal Forced Convection in a One-Component Fluid.- 1. General aspects of the effect of temperature gradients.- 2. Temperature gradients imposed by the boundary conditions.- 3. Temperature gradients due to viscous heating.- A. Experimental interest.- B. Cylindrical Poiseuille flow with viscous heating.- a. the steady state.- b. stability of cylindrical Poiseuille flow including viscous heating.- 4. Further discussion on the multiplicity of steady states when taking into account viscous heating.- Bibliographical notes.- VIII - Mixed Convection in a One-Component Fluid.- 1. Introduction in the Benard problem with flow.- 2. Relation between two and three dimensional disturbances extension of Squire's theorem.- 3. Experiments on the onset of free convection with a superposed small laminar flow.- 4. Effect of lateral boundaries.- Bibliographical notes.- D : Multicomponent Systems.- IX - Free Convection in a Multicomponent Fluid.- 1. Introduction to the influence of concentration gradients on hydrodynamic stability.- 2. Formulation of the linearized problem.- A. The conservation equations.- B. The thermohaline problem.- C. The effect of thermal diffusion (or Soret effect).- 3. The thermohaline convection : linear stability analysis.- A. The role of boundary conditions.- B. Free boundaries with specified solute concentrations and temperatures.- C. Experimental observations.- 4. Free convection with thermal diffusion : linear analysis.- A. Coupled equations for temperature and mass.- B. Exact solution of the simplified problem for free and pervious boundaries.- C. Variational solution for rigid boundaries.- D. 0.- B. Results for s < 0.- 3. Postface.- Bibliographical notes.- Appendix A.- Appendix B.

Heat transfer and horizontally averaged temperature of convection with large viscosity variations

Abstract: It is pointed out that the understanding of convection in large-Prandtl-number Boussinesq fluids with uniform properties and contained in simple geometries is virtually complete. Present efforts are typically directed towards relaxing some of the original assumptions by going to lower Prandtl number, more complicated geometries, variable material properties, or introducing new dynamical processes such as the Lorentz forces. A description is given of experiments which are concerned with the effect on convection of relaxing the assumption of a uniform viscosity. The reported experiments were designed to measure both the horizontally averaged temperature as a function of depth and the heat transfer of convection over a range of viscosity variations up to 100,000.

The fluid dynamics of an attic space

Abstract: This paper reports a fundamental study of the fluid dynamics inside a triangular (attic-shaped) enclosure with cold upper wall and warm horizontal bottom wall. The study was undertaken in three distinct parts. In the first part, the flow and temperature fields in the cavity are determined theoretically on the basis of an asymptotic analysis valid for shallow spaces (H/L → 0, where H and L are the attic height and length). It is shown that in the H/L → 0 limit the circulation consists of a single elongated cell driven by the cold upper wall. The net heat transfer in this limit is dominated by pure conduction. In the second part of the study, the transient behaviour of the attic fluid is examined, based on a scaling analysis. The transient phenomenon begins with the sudden cooling of the upper sloped wall. It is shown that both walls develop thermal and viscous layers whose thicknesses increase towards steady-state values. The criterion for the existence of distinct thermal layers in the steady state is (H/L)½RaH¼ > 1, where RaH is the Rayleigh number based on attic height. The corresponding criterion for distinct viscous wall jets is (H/L)½RaH¼ Pr−½ > 1, where Pr is the Prandtl number. The third phase of this study focused on a complete sequence of transient numerical simulations covering the ranges H/L = 0.2, 0.4, 1; RaH/Pr = 10, 103, 105; Pr = 0.72, 6. The numerical experiments verify the flow features described theoretically in the first two parts of the study. The effect of thermal convection on the net heat transfer between the bottom and top walls is illustrated numerically. Finally, the transient numerical experiments show that in the present parametric domain the single-cell circulation pattern is stable with respect to the Benard instability expected in fluid layers heated from below.

Radiation-Natural Convection Interactions in Two-Dimensional Complex Enclosures

Abstract: A numerical finite-difference study has been carried out for the two-dimensional radiation-natural convection interaction phenomena in square enclosures with equal vertical finite-thickness partitions located at the centers of the ceiling and floor. Both participating gases (CO2 and NH3 ) and nonparticipating gas (air) are considered. In the radiation calculations, the nongray exponential wide-band models for CO2 and NH3 are used, together with a radial flux method utilizing a more realistic polar description for the radiation exchange in the enclosure. Results on the effects of both surface and gas radiation on the velocity and temperature fields and the overall heat transfer rates as functions of the partition heights at two levels of the Grashof number are presented and discussed in terms of the physical phenomena.

Phase-winding solutions in a finite container above the convective threshold

Abstract: An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L [Gt ] 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band |q| ∼ [(R − R0)/R0]½ in the laterally infinite system to a band |q| ∼ (R − R0)/R0 in a system with sidewalls (R is the Rayleigh number and R0 its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R − R0)/R0]½, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newell & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q0. The applicability of the results to convection in a rectangular container is briefly discussed.

Comprehensive, theoretically based, correlating equations for free convection from isothermal spheres

Abstract: Correlating equations are developed for the local and mean Nusselt number for free convection from an isothermal sphere as a function of the Rayleigh and Prandtl numbers. These expressions are based primarily on theoretical solutions for limiting cases, and hence are presumed to be more reliable than purely empirical correlations. The predictions of the proposed expressions are, however, validated by comparisons with prior experimental data. The expressions for the mean Nusselt number are shown to be applicable for all Ra and Pr. The expressions for the local Nusselt number are limited in applicability to the laminar boundary layer regime. The same equations are applicable to mass transfer and to combined heat and mass transfer in terms of the Sherwood, Schmidt and appropriately modified Rayleigh numbers.

Laminar natural convection from a horizontal plate and the influence of plate-edge extensions

Abstract: Laminar natural convection from a horizontal plate is studied by a finite-difference analysis and by experiments for Rayleigh numbers from 10 to 104. The plate with uniform surface temperature or concentration on one side and insulated on the other is situated in an ‘infinite’ fluid medium. The buoyancy near the surface is directed either outward or inward normal to the active surface – equivalent to a heated plate facing upward or downward. The effect of insulated vertical and horizontal extensions to the plate are also examined.Finite-difference solutions are obtained for a heated strip in a two-dimensional domain for a Prandtl number of 0·7. Mass-transfer experiments are performed with square naphthalene plates in air. Both numerical and experimental results justify a 1/5-power law in the present range of Rayleigh number – i.e. Nusselt number or Sherwood number proportional to the Rayleigh number raised to the 1/5 power. The horizontal extensions cause a limited reduction in the transfer rate for the plate generating ‘outward buoyancy’, and a larger reduction with ‘inward buoyancy’. The vertical walls block the fluid flow directly, and thus greatly lower the transfer rate with either outward or inward buoyancy.

Stability of finite‐amplitude convection

Abstract: Internally generated vertical vorticity enters the lowest‐order amplitude equations for free‐slip boundaries in an essential way when the Prandtl number P is finite and the flow three dimensional. For parallel rolls the band of stable wavenumbers is substantially modified from what was previously believed to be correct. In particular there are no stable states for P<0.301. Numerical simulations for free boundaries and larger P suggest a mechanism through which the box size determines the critical Rayleigh number for noisy time dependent convection. A stability analysis of model amplitude equations for rigid boundaries agrees qualitatively with the numerical results of Cleaver or Busse for P≲O(1). There is now considerable continuity between the stability diagrams for rigid and free boundaries.

Free Convection in a Vertical Annulus With Constant Heat Flux on the Inner Wall

Abstract: Heat transfer measurements are presented for free convection in a vertical annulus wherein the inner cylinder is at constant surface heat flux and the outer cylinder is at constant temperatuare. Overall heat transfer data re corrected for thermal radiation in the annulus. Rayleigh numbers span the conduction, transition and boundary layer regimes of flow, and average heat transfer coefficients are obtained with air and helium as the working fluids. The range of Rayleigh number is 10/sup 3/

Character and stability of axisymmetric thermal convection in spheres and spherical shells

Abstract: The influence of shell size and mode of heating on the behavior and stability of axisymmetric, infinite Prandtl number convection in a spherical geometry is studied. Heating from within and below features convection onset governed by a self-adjoint system of equations and boundary conditions. For heating only from within or from below, linearized equations and boundary conditions are non-self-adjoint. Identification of the parameter which initiates the departure from self-adjointness, together with the properties of the self-adjoint solution, provide a basis for calculating the heat transfer characteristics of the non-self-adjoint situations. The investigations are an effort to develop a model for heat transfer in planetary interiors. Further development of the technique by modifying the Galerkin method by the introduction of diagonal mode truncation is suggested to permit the consideration of higher values of the Rayleigh numbers, i.e., those more commensurate with terrestrial planet mantles.

On the intermittency and crystallization mechanisms of sub-seafloor magma chambers

Abstract: Summary. The conventional view of a magma chamber being an essentially permanent feature of a fast-spreading ridge is not compatible with the physics of either oceanic crustal structure or internal magma convection. Cumulates form a large volumetric fraction of many ophiolites and are deposited at the bottom of magma chambers, so they are not available as an insulating layer between magma and seawater. Extrusives and dykes are cracked and highly permeable to hydrothermal convection, so they also do not offer much thermal resistance to the cooling of magma. Only the layer of ‘plated’ or ‘isotropic’ gabbro, often observed between cumulates and dykes, is limited to conductive heat transport; a 0.5 km thickness implies a chamber lifetime of no more than l0kyr km-’ of magma. The intermittency of the chamber on such a short time-scale requires that the plates move apart to make room much more rapidly than they could at the steady spreading rate. Fluctuating magma pressure can achieve such intermittent movement by stress-change diffusion through an elastic lithosphere overlying a viscous asthenophere. However, the space needed implies stress diffusion almost all around the Earth, and this, in turn, requires that the intermittent spreading be substantially synchronized all along a major segment of ridge. The pressure-dependent slopes of equilibrium temperatures between crystal phases and liquid silicate magmas exceed the adiabatic gradients of the magmas themselves by about 1°C kb-’. If this temperature difference is considered the convective drive, a 3 km high chamber reaches a Rayleigh number of 3 x and a Nusselt number of about 8000. The plated layer should grow until the heat loss rate decreases to that supplied by cumulatedepositing convection, implying a plated layer about 0.5 km thick, in agreement with observations. The boundary-layer/turbulent core structure of convection at high Rayleigh and Reynolds numbers is consistent with the formation of banded cumulates when a new type of fluid circulation is taken into account. If a suspended crystal phase is present in the bulk fluid, a ‘slow-convection’ mode is possible, where flow velocities are restricted by the equilibration rate between crystals and fluid, and the temperature profile is determined by the thermodynamics of crystal-fluid equilibria.

Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below

Abstract: The onset of steady natural convection in a rotating cylindrical volume of fluid completely bounded by rigid surfaces is examined for moderate Taylor numbers (Ta≤2×106) and aspect ratios (A≤2). The critical Rayleigh number for three dimensional disturbances is found to be lower than that for the radially unbounded problem by up to a factor of six. The thermal boundary condition on the lateral walls is shown to have a greater effect here than in the nonrotating case.

Nonlinear convection in a rotating layer: Amplitude expansions and normal forms

Abstract: Two-dimensional convection in a horizontal layer of Boussinesq fluid rotating about a vertical axis is studied. For certain choices of the parameters the dispersion relation describing the stability of the conductive solution has two zero eigenvalues. For nearby parameter values nonlinear solutions are accessible analytically, using either the method of normal forms or an amplitude expansion. The results provide a complete description of the transitions between oscillatory and steady convection as functions of the Rayleigh and Taylor numbers near their critical values.

Local analysis of thermal and magnetic instabilities in a rapidly rotating fluid

Abstract: We examine by means of a local analysis the effects of an azimuthal magnetic field B(r) on the stability of a rapidly-rotating fluid subject to a radial temperature gradient, taking the ratio κ/η of thermal to magnetic diffusivities to be small, as in the Earth's core. According to this theory, previous results for the case B ∝ r are typical of a certain range of magnetic field profiles, but if B decreases with r faster than r −½ we find instead that (i) the critical Rayleigh number increases sharply as the magnetic field strength increases beyond “magnetostrophic” values and (ii) the recently-discovered magnetic instabilities triggered by bottom-heavy density gradients do not occur. If B increases with r faster than r 3/2, on the other hand, the major change to the B∝r picture of events is that the system becomes unstable to comparatively fast magnetic instabilities as soon as the field passes magnetostrophic values.

Numerical Study of High Rayleigh Number Convection in a Vertical Porous Enclosure

Abstract: The results of a numerical simulation of natural convection in a vertical rectangular porous enclosure subjected to a horizontal temperature differential are presented. By use of the stable exponential differencing computation scheme, values of the Darcy-Rayleigh number R have been obtained which are substantially larger than those in previous numerical studies. The present results provide a clear physical picture of the development process as R is increased toward asymptotically high values (R →). Correlations for the heat transfer rate are presented for four different aspect ratios and are compared with earlier experimental and theoretical results.

Effects of Free Convection Currents and Mass Transfer on Flow Past a Vertical Oscillating Plate

Abstract: An exact analysis of the flow caused by an oscillating vertical plate in the presence of free-convection currents and foreign mass has been presented. Solutions have been derived by Laplace-transform technique. Velocity profiles and leading edge effects have been shown for different gases present in air. During the course of discussion, the effects of Gr (Grashof number), Gm (modified Grashof number), Sc (Schmidt number), on the flow have been discussed. It has been observed that at all small values of Sc, transition from conduction to convection exists but at large values of Sc, such a transition is not present.

Planform of convection with strongly temperature-dependent viscosity

Abstract: This paper experimentally investigates the convective planform near critical in a fluid layer whose temperature-dependent viscosity varies from top to bottom by up to a factor of 1500. Convection occurs in three different planforms: rolls, hexagons and squares. The square planform, which appears only for fluids with viscosity variation greater than about 50, replaces the hexagonal convection pattern as the Rayleigh number increases much above critical. The large amplitude of hexagonal convection with strong viscosity variation precludes studying the hexagon-square transition with perturbation methods of the type used to study the hexagon-roll transitions at smaller viscosity variations.