Rayleigh-Benard Convection with a Radial Ramp in Plate Separation

Pattern formation in Rayleigh-Benard convection in a large-aspect-ratio cylinder with a radial ramp in the plate separation is studied analytically and numerically by performing numerical simulations of the Boussinesq equations. A horizontal mean flow and a vertical large scale counterflow are quantified and used to understand the pattern wavenumber. Our results suggest that the mean flow, generated by amplitude gradients, plays an important role in the roll compression observed as the control parameter is increased. Near threshold the mean flow has a quadrupole dependence with a single vortex in each quadrant while away from threshold the mean flow exhibits an octupole dependence with a counter-rotating pair of vortices in each quadrant. This is confirmed analytically using the amplitude equation and Cross-Newell mean flow equation. By performing numerical experiments the large scale counterflow is also found to aid in the roll compression away from threshold but to a much lesser degree. Our results yield an understanding of the pattern wavenumbers observed in experiment away from threshold and suggest that near threshold the mean flow and large scale counterflow are not responsible for the observed shift to smaller than critical wavenumbers.Comment: 10 pages, 13 figure

Projection techniques for iterative solution of A(bar x) equals (bar b) with successive right-hand sides

Two projection techniques for computing approximate solutions to linear systems of the form A(bar x)(sup n) = (bar b)(sup n), for a sequence n = 1, 2, ..., e.g., such as arises from time discretization of a partial differential equation, are presented. The inexpensive approximate solutions can be used as initial guesses for iterative solution of the system, resulting in significantly reduced computational expense. Examples of two- and three-dimensional incompressible Navier-Stokes calculations are presented in which x represents the pressure, and A is a discrete Poisson operator. In flows containing significant dynamic activity, these projection techniques lead to as much as a two-fold reduction in solution time

Extensive chaos in Rayleigh-Bénard convection

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size

Traveling waves in rotating Rayleigh-Bénard convection: Analysis of modes and mean flow

Numerical simulations of the Boussinesq equations with rotation for realistic no-slip boundary conditions and a finite annular domain are presented. These simulations reproduce traveling waves observed experimentally. Traveling waves are studied near threshhold by using the complex Ginzburg-Landau equation (CGLE): a mode analysis enables the CGLE coefficients to be determined. The CGLE coefficients are compared with previous experimental and theoretical results. Mean flows are also computed and found to be more significant as the Prandtl number decreases (from sigma=6.4 to sigma=1). In addition, the mean flow around the outer radius of the annulus appears to be correlated with the mean flow around the inner radius

Rayleigh-Benard Convection in Large-Aspect-Ratio Domains

The coarsening and wavenumber selection of striped states growing from random initial conditions are studied in a non-relaxational, spatially extended, and far-from-equilibrium system by performing large-scale numerical simulations of Rayleigh-B\'{e}nard convection in a large-aspect-ratio cylindrical domain with experimentally realistic boundaries. We find evidence that various measures of the coarsening dynamics scale in time with different power-law exponents, indicating that multiple length scales are required in describing the time dependent pattern evolution. The translational correlation length scales with time as $t^{0.12}$, the orientational correlation length scales as $t^{0.54}$, and the density of defects scale as $t^{-0.45}$. The final pattern evolves toward the wavenumber where isolated dislocations become motionless, suggesting a possible wavenumber selection mechanism for large-aspect-ratio convection.Comment: 5 pages, 6 figure

Generalized scans and tridiagonal systems

AbstractMotivated by the analysis of known parallel techniques for the solution of linear tridiagonal system, we introduce generalized scans, a class of recursively defined length-preserving, sequence-to-sequence transformations that generalize the well-known prefix computations (scans). Generalized scan functions are described in terms of three algorithmic phases, the reduction phase that saves data for the third or expansion phase and prepares data for the second phase which is a recursive invocation of the same function on one fewer variable. Both the reduction and expansion phases operate on bounded number of variables, a key feature for their parallelization. Generalized scans enjoy a property, called here protoassociativity, that gives rise to ordinary associativity when generalized scans are specialized to ordinary scans. We show that the solution of positive-definite block tridiagonal linear systems can be cast as a generalized scan, thereby shedding light on the underlying structure enabling known parallelization schemes for this problem. We also describe a variety of parallel algorithms including some that are well known for tridiagonal systems and some that are much better suited to distributed computation

Large Eddy Simulation of Turbulent Channel Flows by the Rational LES Model

The rational large eddy simulation (RLES) model is applied to turbulent channel flows. This approximate deconvolution model is based on a rational (subdiagonal Pade') approximation of the Fourier transform of the Gaussian filter and is proposed as an alternative to the gradient (also known as the nonlinear or tensor-diffusivity) model. We used a spectral element code to perform large eddy simulations of incompressible channel flows at Reynolds numbers based on the friction velocity and the channel half-width Re{sub tau} = 180 and Re{sub tau} = 395. We compared the RLES model with the gradient model. The RLES results showed a clear improvement over those corresponding to the gradient model, comparing well with the fine direct numerical simulation. For comparison, we also present results corresponding to a classical subgrid-scale eddy-viscosity model such as the standard Smagorinsky model.Comment: 31 pages including 15 figure

The Influence of Horizontal Boundaries on Ekman Circulation and Angular Momentum Transport in a Cylindrical Annulus

We present numerical simulations of circular Couette flow in axisymmetric and fully three-dimensional geometry of a cylindrical annulus inspired by Princeton MRI liquid gallium experiment. The incompressible Navier-Stokes equations are solved with the spectral element code Nek5000 incorporating realistic horizontal boundary conditions of differentially rotating rings. We investigate the effect of changing rotation rates (Reynolds number) and of the horizontal boundary conditions on flow structure, Ekman circulation and associated transport of angular momentum through the onset of unsteadiness and three-dimensionality. A mechanism for the explanation of the dependence of the Ekman flows and circulation on horizontal boundary conditions is proposed.Comment: 23 pages, 7 figures; to be published in the Topical Issue of the Physica Scripta in 200