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Three practical problems in conduction/forced convection heat transfer are analyzed using a simplified engineering formulation of convective finite elements. Upwind and conventional finite element solutions are compared for steady-state and transient applications.
Two-dimensional convective flows in shallow cavities with adiabatic.
Numerical experiments are used to study thermally driven flows which occur during vertical Bridgman crystal growth of a single component fluid. The solid-liquid interface was specified as parabolic and flow patterns were calculated for various insulation thicknesses, Grashof, Prandtl, and Biot numbers. When the melt is on top and the gravity vector is axially downward it is shown that flow persists as long as a radial temperature gradient is present. If the interface is convex, as viewed from the liquid, a single cell is observed. A concave interface exhibits multiple counterrotating cells. The insulation thickness and Grashof, Prandtl, and Biot numbers influence the flow in a quantitative manner.
Experiments designed to study the fluid dynamics of buoyancy driven circulations in rotating spherical shells were conducted on the United States Microgravity Laboratory 2 spacelab mission. These experiments address several aspects of prototypical global convection relevant to large scale motions on the Sun, Earth, and on the giant planets. The key feature is the consistent modeling of radially directed gravity in spherical geometry by using dielectric polarization forces. Imagery of the planforms of thermally driven flows for rapidly-rotating regimes shows an initial separation and eventual merger of equatorial and polar convection as the heating (i.e. the Rayleigh number) is increased. At low rotation rates, multiple-states of motion for the same external parameters were observed.
The Galerkin (including single-mode and Lorenz) equations were examined for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. The conclusions were tested by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. It was shown that the transitions from steady state to periodic, then to aperiodic convection depend not only on the Rayleigh number but also very strongly on the horizontal resolution. One of the effects of truncation is to enhance the high wavenumber end of the kinetic energy and thermal variance spectra. The numerical examples indicate that as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution.
The Galerkin equations for convection in a sphere are examined to determine which physical processes are neglected by the severe truncation of the equations of motion. It is demonstrated that the gross features of the flow are affected by truncation in the horizontal direction, with all of the models considered being well resolved in the vertical direction. One of the effects of truncation is to enhance the high-wave number end of the kinetic energy and thermal variance spectra. The examples cited indicate that as long as the kinetic energy spectrum decreases with wave number, a truncation gives a qualitatively correct solution. Conclusions are tested by calculating solutions to the equations of motion for several values of the Rayleigh number and the limit of horizontal spatial resolution.
Experimental results from a test to evaluate interstitial element mass transfer effects on T-111, ASTAR 811C, and ASTAR 1211C after 5000 hours in flowing lithium at 1370 C (2500 F) are presented. No gross corrosion effects were observed. However, hafnium and nitrogen transfer to cooler regions within the loop were noted. Oxygen was in general removed from test specimens, but there was no evidence to indicate that it was a major factor in the mass transfer process. Carbon and hydrogen transfer were not detected.
A Boussinesq system is integrated numerically in three dimensions and time in a study of nonhydrostatic convection in the atmosphere. Simulation of cloud convection is achieved by the inclusion of parametrized effects of latent heat and small-scale turbulence. The results are compared with the cell structure observed in Rayleigh-Benard laboratory conversion experiments in air. At a Rayleigh number of 4000, the numerical model adequately simulates the experimentally observed evolution, including some prominent transients of a flow from a randomly perturbed initial conductive state into the final state of steady large-amplitude two-dimensional rolls. At Rayleigh number 9000, the model reproduces the experimentally observed unsteady equilibrium of vertically coherent oscillatory waves superimposed on rolls.
A Galerkin technique is used to calculate the steady-state axisymmetric nonlinear convective motions in an infinite-Prandtl-number Boussinesq fluid in a relatively thick spherical shell heated from below. A reasonably complete study of the properties of the even and general axisymmetric steady states is carried out for a range of moderately supercritical Rayleigh numbers. In addition, stability analyses are conducted to determine which form of axisymmetric steady convection is the preferred one and whether the axisymmetric steady flows are unstable to azimuthal perturbations.
A Galerkin technique is used to study the finite-amplitude axisymmetric steady convective motions of an infinite Prandtl number Boussinesq fluid in a spherical shell. Two types of heating are considered: in one case, convection is driven both by internal heat sources in the fluid and by an externally imposed temperature drop across the shell boundaries; in the other case, only internal heat sources drive convection and the lower boundary of the shell is adiabatic. Two distinct classes of axisymmetric steady states are found to be possible: states characterized by temperature and radial velocity fields that are symmetric about an equatorial plane; and a class of solutions that does not possess any symmetry properties about the equatorial plane.
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.
Three-dimensional time-dependent convection in a plane layer of fluid, uniformly heated from below and subject to vertical shear and to rotation about an axis tilted from the vertical, was simulated by the numerical solution of the Boussinesq equations, including all Coriolis terms. Rotation about a vertical axis produces smaller convection cells with diminished heat fluxes and considerable vorticity. When the rotation axis is tilted from the vertical to represent tropical latitudes, the convection cells become elongated in a N-S direction. Imposed flows with constant vertical shear produce convective rolls aligned with the mean flow. When the rotation vector is tilted from the vertical, the competing effects due to rotation and shear can stabilize the convective motions.
The flight of the Spacelab 3 microgravity laboratory onboard the Space Shuttle Challenger in May 1985 enabled electroconvection experiments to be conducted using the goephysical fluid flow cell instrument. Experimental results are presented which illustrate the variety of convection achieved by varying the imposed radial and latitudinal temperature gradients, rotation rates, and the strength of the electrostatic gravity. These results are compared with those obtained from nonlinear three-dimensional simulations and good agreement is found.
Nonlinear convective solutions are presented for a shell with properties characteristic of the earth's whole mantle for Rayleigh numbers up to 70,000. The solutions are validated numerically, and two distinct convective patterns (cubic and tetrahedral) are identified which are closely related to the geometric planforms predicted by the analytical theories of slightly supercritical spherical convection. A quantitative analysis of the horizontal and vertical structures of the velocity and temperature fields of the solutions is presented, and their heat transport properties are examined, including the total heat flow and the spatial distribution of the heat flux at the shell boundaries.
Differentially rotating disks of gases and solids occur in several astrophysical systems, in particular in the inner parts of protostellar nebulae, of which our own solar system is thought to be a relic. The objectives of this paper are to: (1) study localized turbulence in circumstances approximating those found in accretion disks using previously existing expertise in performing direct numerical simulation of turbulent, incompressible channel flows with low Reynolds numbers; (2) determine the limitations of such calculations; and (3) extend the type of numerical simulation (e.g., to include density and stratification and compressibility effects and to accommodate higher Reynolds numbers with sub-grid scale modeling) so that the relevant physical effects are realistically captured.
A low-dimensional dynamic model for transitional buoyancy-driven flow in a differentially heated tall enclosure is presented. The full governing partial differential equations with the associated boundary conditions are solved by a spectral element method for a cavity of aspect ratio A=20. Proper orthogonal decomposition is applied to the oscillatory solution at Prandtl number Pr=P tau (omega) = 0.71 and Grashof number G tau (omega) = 3.2 x 10 (exp 4) to construct empirical eigenfunctions. Using the four most energetic empirical eigenfunctions for the velocity and temperature as basis functions and applying Galerkin's method, a reduced model consisting of eight nonlinear ordinary differential equations is obtained. Close to the 'design' conditions (P tau(omega) G tau(omega)), the low-order model (LOM) predictions are in excellent agreement with the predictions of the full model. In particular, the critical Grashof number at the onset of the first temporal flow instability (Hopf bifurcation) was well as the frequency and amplitude of oscillations at supercritical conditions are in excellent agreement with the predictions of the full model. Far from the 'design' conditions, the LOM predicts the existence of multiple stable steady solutions at large values of G tau, and a unique stable steady solution at small values of G tau, and exhibits hysteretic behavior that is qualitatively similar to that observed in direct numerical simulations based on the full model.
Ground-based experiments on g-jitter effects in fluid flow provide insight that complements both theoretical studies and space-based experiments on this problem. We report preliminary results for experiments on Rayleigh-Benard convection subjected to time-dependent accelerations on a shaker table. For sinusoidal modulation, two qualitatively different pattern forming mechanisms come into play: geometry induced wavenumber selection (as in the standard "no-shake" Rayleigh-Benard problem) and dispersion induced wavenumber selection due to parametric instability (as in the Faraday surface-wave problem). We discuss preliminary results on the competition and co-existence of patterns due to these different instability mechanisms. We also discuss the implications of this work on the general question of pattern formation in the presence of noise.
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