Higher Rayleigh numbers

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Table of contents

1 Rayleigh–Bénard instability 
1.1 Rayleigh–Bénard instability
1.1.1 Instability
1.1.2 Rayleigh–Bénard convection
1.1.3 Rayleigh number and convection onset
1.1.4 Secondary flows and Prandtl number
1.1.5 Higher Rayleigh numbers
1.1.6 Investigation tools
1.2 Cylindrical system
1.2.1 Features of the system
1.2.2 Azimuthal wavenumber
1.2.3 Large aspect ratios
1.2.4 Small aspect ratios
1.3 Motivation
2 Governing equations 
2.1 Navier–Stokes equations
2.2 Boundary conditions
2.3 Linear analysis
2.3.1 Linearised equations
2.3.2 Power method
2.3.3 Complex eigenvalues
2.4 Symmetries
2.4.1 Symmetries of Boussinesq equations
2.4.2 Z2 symmetry and inverse patterns
2.4.3 O(2) symmetry
2.4.4 Representations of complex eigenvectors in O(2) symmetry
2.5 Amplitude equations and normal form
3 Numerical integration 
3.1 Spectral discretisation – Galerkin projection
3.1.1 Galerkin decomposition
3.1.2 Fourier series in q
3.1.3 Chebyshev polynomials in z
3.1.4 Chebyshev polynomials in r
3.2 Time discretisation
3.3 Influence matrix
3.3.1 Tau method
3.3.2 Velocity–pressure decoupling
3.3.3 Influence matrix
3.3.4 Tau correction
3.4 Linear simulation
3.4.1 Power method
3.4.2 Arnoldi–Krylov method
3.4.3 Calculating the base state
3.5 Code vectorisation
3.6 Program output
3.7 Visualisation
4 Code validation 
4.1 Choice of resolution
4.2 Choice of timestep
4.3 Choice of parameters for linear runs
4.4 Convergence criteria
4.5 Test cases
5 Standing and travelling waves 
5.1 Oscillatory bifurcation of the axisymmetric state — linear analysis
5.1.1 Steady axisymmetric state
5.1.2 Eigenvalues and eigenvectors
5.2 Nonlinear simulation of time-dependent states
5.2.1 Weakly unstable standing waves
5.2.2 Stable travelling waves
5.2.3 Amplitudes and frequencies
5.2.4 Normal form coefficients
5.3 Conclusion
6 Convective patterns – insulating sidewalls 
6.1 Simulation of the experiment
6.2 Evolution from perturbed conductive state
6.3 Three-roll patterns
6.4 Evolution from four-rolls
6.5 Evolution from pizza pattern
6.6 Evolution from two rolls
6.7 Axisymmetric flows
6.8 Evolution from mercedes pattern
6.9 Evolution from dipole pattern
6.10 Dipole-shaped perturbation
6.11 Summary diagram
7 Convective patterns – conducting sidewalls 
7.1 Start from perturbed conductive state
7.2 Three rolls
7.3 Four rolls
7.4 Evolution from dipole pattern
7.5 Evolution from hourglass pattern
7.6 Evolution from star pattern
7.7 Evolution from da Vinci pattern
7.8 Evolution from Y pattern
7.9 Summary diagram
8 Conclusion