Numerical simulation of super-square patterns in Faraday waves

We report the first simulations of the Faraday instability using the full three-dimensional Navier-Stokes equations in domains much larger than the characteristic wavelength of the pattern. We use a massively parallel code based on a hybrid Front-Tracking/Level-set algorithm for Lagrangian tracking of arbitrarily deformable phase interfaces. Simulations performed in rectangular and cylindrical domains yield complex patterns. In particular, a superlattice-like pattern similar to those of [Douady & Fauve, Europhys. Lett. 6, 221-226 (1988); Douady, J. Fluid Mech. 221, 383-409 (1990)] is observed. The pattern consists of the superposition of two square superlattices. We conjecture that such patterns are widespread if the square container is large compared to the critical wavelength. In the cylinder, pentagonal cells near the outer wall allow a square-wave pattern to be accommodated in the center

Extreme multiplicity in cylindrical Rayleigh-Benard convection: II. Bifurcation diagram and symmetry classification

A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Benard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton's method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk towards parallel rolls, and the requirement imposed by the boundary conditions that primary bifurcations be towards states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; three-fold symmetric patterns called mercedes, mitubishi, marigold and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three-rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number

Patterns in transitional shear flows. Part 2: Nucleation and optimal spacing

Low Reynolds number turbulence in wall-bounded shear flows \emph{en route} to laminar flow takes the form of oblique, spatially-intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatiotemporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent-laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and non-linear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar-turbulent interfaces. This optimal behaviour is primarily due to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.Comment: 27 pages, 14 figure

Patterns in transitional shear flows. Part 1. Energy transfers and mean-flow interaction

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of localised turbulent structures. In plane shear flows, these appear as a regular alternation of turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent-laminar pattern are computed and compared to those of uniform turbulence at low $Re$. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and, secondly, the extraction of energy from turbulent fluctuations (via negative production, associated to a strong energy transfer from small to large scales). These processes are surveyed as uniform turbulence loses its stability. Inverse energy transfers and negative production are also found in the uniformly turbulent state.Comment: 29 pages, 15 figure

The Pack Method for Compressive Tests of Thin Specimens of Materials Used in Thin-Wall Structures

The strength of modern lightweight thin-wall structures is generally limited by the strength of the compression members. An adequate design of these members requires a knowledge of the compressive stress-strain graph of the thin-wall material. The "pack" method was developed at the National Bureau of Standards with the support of the National Advisory Committee for Aeronautics to make possible a determination of compressive stress-strain graphs for such material. In the pack test an odd number of specimens are assembled into a relatively stable pack, like a "pack of cards." Additional lateral stability is obtained from lateral supports between the external sheet faces of the pack and outside reactions. The tests seems adequate for many problems in structural research

Extreme multiplicity in cylindrical Rayleigh-B\'enard convection: I. Time-dependence and oscillations

Rayleigh-Benard convection in a cylindrical container can take on many different spatial forms. Motivated by the results of Hof, Lucas and Mullin [Phys. Fluids 11, 2815 (1999)], who observed coexistence of several stable states at a single set of parameter values, we have carried out simulations at the same Prandtl number, that of water, and radius-to-height aspect ratio of two. We have used two kinds of thermal boundary conditions: perfectly insulating sidewalls and perfectly conducting sidewalls. In both cases we obtain a wide variety of coexisting steady and time-dependent flows

On acceleration of Krylov-subspace-based Newton and Arnoldi iterations for incompressible CFD: replacing time steppers and generation of initial guess

We propose two techniques aimed at improving the convergence rate of steady state and eigenvalue solvers preconditioned by the inverse Stokes operator and realized via time-stepping. First, we suggest a generalization of the Stokes operator so that the resulting preconditioner operator depends on several parameters and whose action preserves zero divergence and boundary conditions. The parameters can be tuned for each problem to speed up the convergence of a Krylov-subspace-based linear algebra solver. This operator can be inverted by the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose to generate an initial guess of steady flow, leading eigenvalue and eigenvector using orthogonal projection on a divergence-free basis satisfying all boundary conditions. The approach, including the two proposed techniques, is illustrated on the solution of the linear stability problem for laterally heated square and cubic cavities

Colored-noise thermostats \`a la carte

Recently, we have shown how a colored-noise Langevin equation can be used in the context of molecular dynamics as a tool to obtain dynamical trajectories whose properties are tailored to display desired sampling features. In the present paper, after having reviewed some analytical results for the stochastic differential equations forming the basis of our approach, we describe in detail the implementation of the generalized Langevin equation thermostat and the fitting procedure used to obtain optimal parameters. We discuss in detail the simulation of nuclear quantum effects, and demonstrate that, by carefully choosing parameters, one can successfully model strongly anharmonic solids such as neon. For the reader's convenience, a library of thermostat parameters and some demonstrative code can be downloaded from an on-line repository