Rayleigh-Bénard Convection

The following sections are included:

• CONTENTS

Convection due to nonuniform heating is, without overstatement, the most widespread type of fluid motion in the Universe. It plays also an important role in various engineering setups. All this is sufficient to warrant steadfast and intent interest of investigators in convection. But of late, this interest has also been powerfully stimulated by other circumstances. Convection problems are a rich source of material for the development of new ideas concerning the relation between order and chaos in flows and between simplicity and complexity in the structure and behaviour of hydrodynamic objects. Convective flows can form more or less ordered spatial structures, and their investigation substantially contributes to understanding the general properties of pattern-forming systems, which are the main subject for study in synergetics, an actively developed branch of modern science…

The following sections are included:

• The Boussinesq Approximation

• The Rayleigh–Bénard Problem

• Linear Analysis

• Nonlinear Regimes and Bifurcations

• Planforms of Convection Cells

The following sections are included:

• Experiment

• Theoretical Approaches: A. Expansions in Small Amplitude (Small Reduced Rayleigh Number)

• Theoretical Approaches: B. Two-Dimensional Models of Three-Dimensional Convection
  • Amplitude Equations
  • Manneville's “Microscopic” Equations
  • Model Equations
  • The Lyapunov Functional
  • The Pomeau–Manneville Phase Equation
  • The Cross–Newell Phase Equation

• Theoretical Approaches: C. Numerical Simulation

The following sections are included:

• Two-Dimensional Rolls and Three-Dimensional cells
  • Thermocapillary Effect
  • Temperature Dependence of Viscosity
  • Temperature Dependence of Other Material Parameters of the Fluid
  • Asymmetry of Boundary Conditions
  • Curvature of the Unperturbed Temperature Profile
  • Finite Thermal Conductivity of Horizontal Boundaries
  • Deformation of the Free Surface
  • Time-Dependent Heating
  • The Presence of Suspended Solid Particles
  • Tertiary Flows
  • Hexagonal Cells in a Vertically Symmetric Layer

• Patterns of Quasi-Two-Dimensional Rolls

• Convection Textures. Roll-Pattern Defects
  • Dislocations
  • Disclinations
  • Grain Boundaries

The following sections are included:

• Regime Diagram

• Phase Turbulence

• Spiral-Defect Chaos

The following sections are included:

• Wavenumbers in Experiments with Random Initial Disturbances

• Searches for Universal Selection Criteria

• Stability of Two-Dimensional Roll Flows
  • Theoretical Results
  • Experimental Results

• Lyapunov Functional and Selection

• “Selection Mechanisms”
  • Grain-Boundary Motion
  • Spatial Ramp of Parameters
  • Motion and Equilibrium of Dislocations
  • Axisymmetric Flows
  • Convection-Front Propagation
  • Convection-Front Propagation Combined with Roll Relaxation
  • Relaxation of Rolls in Contact with a Disordered Flow
  • The Effect of Sidewalls
  • The Preferred Wavenumber and Realized Wavenumbers (Discussion and Conclusion)

The following sections are included:

• Effect of Strong Temperature Dependence of viscosity

• Penetrative Convection

• Small-Scale Motions in a Globally Unstable Layer
  • A Layer with a Piecewise-Linear Unperturbed Temperature Profile
  • A Layer with Radiative Energy Transfer

• Astro- and Geophysical Applications
  • Solar Convection Zone
  • The Earth's Mantle

We have considered a number of important aspects of the self-organization of convective flows in a horizontal fluid layer heated from below. Now, after a detailed examination, let us try to give a general outline of the fundamental features of the convection-pattern dynamics as they can be understood on the basis of the presented information…

The following sections are included:

• REFERENCES

• SUBJECT INDEX