Vortex simulations of the Rayleigh–Taylor instability

A vortex technique capable of calculating the Rayleigh–Taylor instability to large amplitudes in inviscid, incompressible, layered flows is introduced. The results show the formation of a steady‐state bubble at large times, whose velocity is in agreement with the theory of Birkhoff and Carter. It is shown that the spike acceleration can exceed free fall, as suggested recently by Menikoff and Zemach. Results are also presented for instability at various Atwood ratios and for fluids having several layers

Fast orthogonal derivatives on the star

AbstractIn many numerical problems there is the need for obtaining derivatives in the X and Y directions of m variables at each point on an n×n plane. We consider the case where these derivatives are obtained using spectral methods (i.e. n fast Fourier transforms of length n are taken for each component, multiplied by the wave numbers and reverse transformed).On the CDC STAR-100 all data points corresponding to a plane must be stored in contiguous locations if advantage is to be taken of the powerful pipeline hardware of the machine. This means that derivatives in one direction are obtained very efficiently while derivatives in the orthogonal direction require either the substantial overhead of transposition or the use of scalar operations with no benefits of pipelining.An algorithm is described that overcomes this problem by taking derivatives of all components simultaneously. This is made possible by perfect shuffling of data to effect a pseudo-transposition that permits the FFT routine to take transforms of all m components on a plane at one time. Practical experience with this algorithm for m=5 and n=32 shows a 10% speedup for X-derivatives and a 32% speedup for Y-derivatives over the conventional algorithms (in which X and Y derivatives are taken one component at a time and Y derivatives require transposition of data).A theoretical analysis based on available STAR-100 vector instruction timing data predicts that this algorithm is superior to the conventional algorithm for M ≥ 2, n ≤ 128 (problem sizes of practical interest). We show how further improvement in running time may be obtained if derivatives of several components on more than one plane are required.This analysis is applicable to the new generation of STAR computers (the CDC Cyber 203s) since vector instruction timings are essentially unchanged in the new machines

Analysis of relative dispersion of two particles by Lagrangian stochastic models and DNS methods

Comparisons of the Q1D against the known Lagrangian stochastic well-mixed quadratic form models and the moments approximation models are presented. In the case of modestly large Reynolds numbers turbulence (Re λ ⋍ 240) the comparison of the Q1D model with the DNS data is made. Being in a qualitatively agreemnet with the DNS data, the Q1D model predicts higher rate of separation. Realizability of Q1D model extracted from the transport equation with a quadratic form of the conditional acceleration is shown

Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

We model buoyancy-driven convection with chimneys -- channels of zero solid fraction -- in a mushy layer formed during directional solidification of a binary alloy in two-dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesised that the chimney spacing will adjust to optimise the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sub-linear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carmen-Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.Comment: 20 pages, 10 figures. Changes from previous version correct typos, expand on discussion of the method including new appendix A, and minor changes to the discussion. A modified final version has been accepted for publication in the Journal of Fluid Mechanic

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic, two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlov et al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Generation and Structure of Solitary Rossby Vortices in Rotating Fluids

The formation of zonal flows and vortices in the generalized Charney-Hasegawa-Mima equation is studied. We focus on the regime when the size of structures is comparable to or larger than the deformation (Rossby) radius. Numerical simulations show the formation of anticyclonic vortices in unstable shear flows and ring-like vortices with quiescent cores and vorticity concentrated in a ring. Physical mechanisms that lead to these phenomena and their relevance to turbulence in planetary atmospheres are discussed.Comment: 3 pages in REVTeX, 5 postscript figures separately, submitted to Phys. Rev.