We demonstrate with a new three-dimensional adaptive mesh refinement code that perturbations arising from discretization of the equations of self-gravitational hydrodynamics can grow into fragments in multiple-grid simulations, a process we term "artificial fragmentation." We present star formation calculations of isothermal collapse of dense molecular cloud cores. In simulation of a Gaussian-profile cloud free of applied perturbations, we find artificial fragmentation can be avoided across the isothermal density regime by ensuring the ratio of cell size to Jeans length, which we call the Jeans number, , is kept below 0.25. We refer to the constraint J { Dx/l J that be resolved as the Jeans condition. When an perturbation is included, we again find it necessary l m ϭ 2 J to keep to achieve a converged morphology. Collapse to a filamentary singularity occurs without J ≤ 0.25 fragmentation of the filament, in agreement with the predictions of Inutsuka & Miyama. Simulation beyond the time of this singularity requires an arresting agent to slow the runaway density growth. Physically, the breakdown of isothermality due to the buildup of opacity acts as this agent, but many published calculations have instead used artificial viscosity for this purpose. Because artificial viscosity is resolution dependent, such calculations produce resolution-dependent results. In the context of the perturbed Gaussian cloud, we show that use of artificial viscosity to arrest collapse results in significant violation of the Jeans condition. We also show that if the applied perturbation is removed from such a calculation, numerical fluctuations grow to produce substantial fragments not unlike those found when the perturbation is included. These findings indicate that calculations that employ artificial viscosity to halt collapse are susceptible to contamination by artificial fragmentation. The Jeans condition has important implications for numerical studies of isothermal self-gravitational hydrodynamics problems insofar as it is a necessary but not, in general, sufficient condition for convergence.

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Turbulent Richtmyer-Meshkov instability (RMI) is investigated through a series of high resolution three dimensional smulations of two initial conditions with eight independent codes. The simulations are initialised with a narrowband perturbation such that instability growth is due to non-linear coupling/backscatter from the energetic modes, thus generating the lowest expected growth rate from a pure RMI. By independently assessing the results from each algorithm, and computing ensemble averages of multiple algorithms, the results allow a quantification of key flow properties as well as the uncertainty due to differing numerical approaches. A new analytical model predicting the initial layer growth for a multimode narrowband perturbation is presented, along with two models for the linear and non-linear regime combined. Overall, the growth rate exponent is determined as θ = 0.292 ± 0.009, in good agreement with prior studies; however, the exponent is decaying slowly in time. Also, θ is shown to be relatively insensitive to the choice of mixing layer width measurement. The asymptotic integral molecular mixing measures Θ = 0.792 ± 0.014, Ξ = 0.800±0.014 and Ψ = 0.782±0.013 which are lower than some experimental measurements but within the range of prior numerical studies. The flow field is shown to be persistently anisotropic for all algorithms, at the latest time having between 49% and 66% higher kinetic energy in the shock parallel direction compared to perpendicular and does not show any return to isotropy. The plane averaged volume fraction profiles at different time instants collapse reasonably well when scaled by the integral width, implying that the layer can be described by a single length scale and thus a single θ. Quantitative data given for both ensemble averages and individual algorithms provide useful benchmark results for future research.

The morphology and time-dependent integral properties of the multifluid compressible flow resulting from the shock–bubble interaction in a gas environment are investigated using a series of three-dimensional multifluid-Eulerian simulations. The bubble consists of a spherical gas volume of radius 2.54 cm (128 grid points), which is accelerated by a planar shock wave. Fourteen scenarios are considered: four gas pairings, including Atwood numbers −0.8 < A < 0.7, and shock strengths 1.1 < M ≤ 5.0. The data are queried at closely spaced time intervals to obtain the time-dependent volumetric compression, mean bubble fluid velocity, circulation and extent of mixing in the shocked-bubble flow. Scaling arguments based on various properties computed from one-dimensional gasdynamics are found to collapse the trends in these quantities successfully for fixed A. However, complex changes in the shock-wave refraction pattern introduce effects that do not scale across differing gas pairings, and for some scenarios with A > 0.2, three-dimensional (non-axisymmetric) effects become particularly significant in the total enstrophy at late times. A new model for the total velocity circulation is proposed, also based on properties derived from one-dimensional gasdynamics, which compares favourably with circulation data obtained from calculations, relative to existing models. The action of nonlinear-acoustic effects and primary and secondary vorticity production is depicted in sequenced visualizations of the density and vorticity fields, which indicate the significance of both secondary vorticity generation and turbulent effects, particularly for M > 2 and A > 0.2. Movies are available with the online version of the paper.