John W. Grove

Three-Dimensional Front Tracking

We describe a three-dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two-dimensional computations, we expect three-dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability.

Richtmyer–Meshkov instability growth: experiment, simulation and theory

Richtmyer–Meshkov instability is investigated for negative Atwood number and two-dimensional sinusoidal perturbations by comparing experiments, numerical simulations and analytic theories. The experiments were conducted on the NOVA laser with strong radiatively driven shocks with Mach numbers greater than 10. Three different hydrodynamics codes (RAGE, PROMETHEUS and FronTier) reproduce the amplitude evolution and the gross features in the experiment while the fine-scale features differ in the different numerical techniques. Linearized theories correctly calculate the growth rates at small amplitude and early time, but fail at large amplitude and late time. A nonlinear theory using asymptotic matching between the linear theory and a potential flow model shows much better agreement with the late-time and large-amplitude growth rates found in the experiments and simulations. We vary the incident shock strength and initial perturbation amplitude to study the behaviour of the simulations and theory and to study the effects of compression and nonlinearity.

Robust Computational Algorithms for Dynamic Interface Tracking in Three Dimensions

Front tracking provides sharp resolution of wave fronts through the active tracking of interfaces between distinct materials. A major challenge to this method is to handle changes in the interface topology. We describe two algorithms, implemented in the front tracking code FronTier, to model dynamic changes in three-dimensional interfaces. The two methods can be combined to give a hybrid method that is superior to each individual method. The success of these algorithms is shown by simulations of Rayleigh--Taylor instability, which is an interfacial instability driven by an acceleration directed across a material interface. Our numerical results are validated by comparing the numerical computation of the velocity of a single rising bubble with an analytic model for the bubble velocity.

The bifurcation of tracked scalar waves

The dynamic evolution of tracked waves by a front-tracking algorithm may lead on either numerical or physical grounds to intersections of the waves. The correct resolution of these intersections is described locally by the solution of Riemann problems and requires a bifurcation of the topology defined by the tracked waves. An algorithm is described which is appropriate for the resolution of scalar tracked waves, such as material discontinuities, contact dicontinuities in gas dynamics, or constituent concetration waves including oil-water banks in oil reservoirs Even here the algorithm is not fully general, and the resolution of the intersections of an arbitrary set of curves in the plane for the above range of physical problems remains unsolved. However with the assumption that the set of intersections to be resolved is a small perturbation (resulting for example from a small time step in an evolution) of a valid, non-intersecting front, the algorithm seems to be general. In any case examples will be presented that show that complicated interfaces can be generated automatically from simple ones through successive bifurcations. 15 refs., 9 figs.

Anomalous reflection of a shock wave at a fluid interface

Several wave patterns can be produced by the interaction of a shock wave with a fluid interface. We focus on the case when the shock passes from a medium of high to low acoustic impedance. Curvature of either the shock front or contact causes the flow to bifurcate from a locally self-similar quasi-stationary shock diffraction, to an unsteady anomalous reflection. This process is analogous to the transition from a regular to a Mach reflection when the reflected wave is a rarefaction instead of a shock. These bifurcations have been incorporated into a front tracking code that provides an accurate description of wave interactions. Numerical results for two illustrative cases are described; a planar shock passing over a bubble, and an expanding shock impacting a planar contact.

Front tracking in two and three dimensions

Abstract Front tracking is a method for solving conservation laws in which the evolution of discontinuities is determined through the solution of Riemann problems. This method often does not require highly refined grids, and it has no numerical diffusion. We show the success of this method through a comparison of simulations of the Richtmyer-Meshkov instability, an unstable material interface, with experimental data. Good simulations of such instabilities are notoriously difficult, and we also demonstrate for the same physical problem that grid orientations have no effect on the numerical solution. We also present the first results of our three-dimensional front tracking code by simulating an important aspect of the computer chip manufacturing process: material deposition and etching. Our two- and three-dimensional front tracking code is parallelized for MIMD architectures and runs on our 128 node Intel Paragon.

Numerical investigation of Richtmyer-Meshkov instability using front tracking

Front tracking simulations of the Richtmyer-Meshkov instability produce significantly better agreement with experimentally measured growth rates than obtained in nontracking computations. Careful analysis of the early stages of the shock acceleration process show that nonlinearity and compressibility play a critical role in the behaviour of the shocked interface and are responsible for the deviations from the linear theories. The late-time behaviour of the interface growth rate is compared to a nonlinear potential flow model of Hecht et al .

Experiments to Produce a Hydrodynamically Unstable, Spherically Diverging System of Relevance to Instabilities in Supernovae

Results of the first spherically diverging, hydrodynamically unstable laboratory experiments of relevance to supernovae (SNe) are reported. The experiments are accomplished by using laser radiation to explode a hemispherical capsule, having a perturbed outer surface, which is embedded within a volume of low-density foam. The evolution of the experiment, like that of a supernova, is well described by the Euler equations. We have compared the experimental results to those of two-dimensional simulations using both a radiation-hydrodynamics code and a pure hydrodynamics code with front tracking.

Uncertainty Quantification for Multiscale Simulations

A general discussion of the quantification of uncertainty in numerical simulations is presented. A principal conclusion is that the distribution of solution errors is the leading term in the assessment of the validity of a simulation and its associated uncertainty in the Bayesian framework. Key issues that arise in uncertainty quantification are discussed for two examples drawn from shock wave physics and modeling of petroleum reservoirs. Solution error models, confidence intervals and Gaussian error statistics based on simulation studies are presented

The interaction of shock waves with fluid interfaces

Enhanced resolution for the computation of the interaction of shock waves with fluid interfaces is achieved through a detailed mathematical analysis of 2-dimensional wave interactions produced during the collision of the waves. This computation is carried to late times, which are characterized by interface instability and chaotic mixing processes. Algorithms for incorporating the wave interaction analysis and the resulting bifurcation of front topology give an important extension of the front tracking method and are presented here. The mathematical analysis shows that the customary theory for oblique 2-dimensional wave interactions is equivalent to a 1-dimensional Riemann problem for steady (supersonic) flow. This analysis, known for polytropic gases, is extended here to a general equation of state. Moreover, the asymptotic limit of a small incident angle is analyzed to obtain a well-conditioned numerical algorithm. This limit is found to define a 1-dimensional unsteady Riemann problem.