Hyunkyung Lim

Nonideal Rayleigh-Taylor mixing

Rayleigh–Taylor mixing is a classical hydrodynamic instability that occurs when a light fluid pushes against a heavy fluid. The two main sources of nonideal behavior in Rayleigh–Taylor (RT) mixing are regularizations (physical and numerical), which produce deviations from a pure Euler equation, scale invariant formulation, and nonideal (i.e., experimental) initial conditions. The Kolmogorov theory of turbulence predicts stirring at all length scales for the Euler fluid equations without regularization. We interpret mathematical theories of existence and nonuniqueness in this context, and we provide numerical evidence for dependence of the RT mixing rate on nonideal regularizations; in other words, indeterminacy when modeled by Euler equations. Operationally, indeterminacy shows up as nonunique solutions for RT mixing, parametrized by Schmidt and Prandtl numbers, in the large Reynolds number (Euler equation) limit. Verification and validation evidence is presented for the large eddy simulation algorithm used here. Mesh convergence depends on breaking the nonuniqueness with explicit use of the laminar Schmidt and Prandtl numbers and their turbulent counterparts, defined in terms of subgrid scale models. The dependence of the mixing rate on the Schmidt and Prandtl numbers and other physical parameters will be illustrated. We demonstrate numerically the influence of initial conditions on the mixing rate. Both the dominant short wavelength initial conditions and long wavelength perturbations are observed to play a role. By examination of two classes of experiments, we observe the absence of a single universal explanation, with long and short wavelength initial conditions, and the various physical and numerical regularizations contributing in different proportions in these two different contexts.

A Conservative Front Tracking Method in N-Dimensions

We propose a fully conservative Front Tracking algorithm for systems of nonlinear conservation laws. The algorithm can be applied uniformly in one, two, three and N dimensions. Implementation details for this algorithm and tests of fully conservative simulations are reported.

Verification and validation of a method for the simulation of turbulent mixing

We present the highlights of and supplementary material related to two recent studies yielding verification and validation of a new approach to the simulation of turbulent mixing. The verification is based on (i) a mesh refinement study of the circular Richtmyer–Meshkov unstable flow, (ii) comparison of the code to a well-documented code and (iii) comparison to a simple analytic model. The validation is based on the simulations agreement with the Rayleigh–Taylor unstable experiments of Smeeton–Youngs and of Mueschke–Andrews. The mesh refinement verification gives convergence for such molecular level variables as the probability density functions for concentration, temperature and a chemical reaction rate. The validation study, beyond obtaining nearly perfect agreement with experiment, explores the various factors in the simulations that lead to the agreement and to differences between the two experiments. The important variables are fluid transport parameters, dimensionless groups (not widely recognized to be significant) to characterize the dominant short-wavelength initial perturbations and experimentally measured (long-wavelength) initial perturbations.

Subgrid models for mass and thermal diffusion in turbulent mixing

We propose a new method for the large eddy simulation (LES) of turbulent mixing flows. The method yields convergent probability distribution functions (PDFs) for temperature and concentration and a chemical reaction rate when applied to reshocked Richtmyer–Meshkov (RM) unstable flows. Because such a mesh convergence is an unusual and perhaps original capability for LES of RM flows, we review previous validation studies of the principal components of the algorithm. The components are (i) a front tracking code, FronTier, to control numerical mass diffusion and (ii) dynamic subgrid scale (SGS) models to compensate for unresolved scales in the LES. We also review the relevant code comparison studies. We compare our results to a simple model based on 1D diffusion, taking place in the geometry defined statistically by the interface (the 50% isoconcentration surface between the two fluids). Several conclusions important to physics could be drawn from our study. We model chemical reactions with no closure approximations beyond those in the LES of the fluid variables itself, and as with dynamic SGS models, these closures contain no adjustable parameters. The chemical reaction rate is specified by the joint PDF for temperature and concentration. We observe a bimodal distribution for the PDF and we observe significant dependence on fluid transport parameters.

Sensitivity of inertial confinement fusion hot spot properties to the deuterium-tritium fuel adiabat

We determine the dependence of key Inertial Confinement Fusion (ICF) hot spot simulation properties on the deuterium-tritium fuel adiabat, here modified by addition of energy to the cold shell. Variation of this parameter reduces the simulation to experiment discrepancy in some, but not all, experimentally inferred quantities. Using simulations with radiation drives tuned to match experimental shots N120321 and N120405 from the National Ignition Campaign (NIC), we carry out sets of simulations with varying amounts of added entropy and examine the sensitivities of important experimental quantities. Neutron yields, burn widths, hot spot densities, and pressures follow a trend approaching their experimentally inferred quantities. Ion temperatures and areal densities are sensitive to the adiabat changes, but do not necessarily converge to their experimental quantities with the added entropy. This suggests that a modification to the simulation adiabat is one of, but not the only explanation of the observed sim...

A Mathematical Theory for LES Convergence

Abstract Practical simulations of turbulent processes are generally cutoff, with a grid resolution that stops within the inertial range, meaning that multiple active regions and length scales occur below the grid level and are not resolved. This is the regime of large eddy simulations (LES), in which the larger but not the smaller of the turbulent length scales are resolved. Solutions of the fluid Navier-Stokes equations, when considered in the inertial regime, are conventionally regarded as solutions of the Euler equations. In other words, the viscous and diffusive transport terms in the Navier-Stokes equations can be neglected in the inertial regime and in LES simulations, while the Euler equation becomes fundamental. For such simulations, significant new solution details emerge as the grid is refined. It follows that conventional notions of grid convergence are at risk of failure, and that a new, and weaker notion of convergence may be appropriate. It is generally understood that the LES or inertial regime is inherently fluctuating and its description must be statistical in nature. Here we develop such a point of view systematically, based on Young measures, which are measures depending on or indexed by space time points. In the Young measure (ξ)x,t, the random variable ξ of the measure is a solution state variable, i.e., a solution dependent variable, representing momentum, density, energy and species concentrations, while the space time coordinates, x, t, serve to index the measure. Theoretical evidence suggests that convergence via Young measures is sufficiently weak to encompass the LES/inertial regime; numerical and theoretical evidence suggests that this notion may be required for passive scalar concentration and thermal degrees of freedom. Our objective in this research is twofold: turbulent simulations without recourse to adjustable parameters (calibration) and extension to more complex physics, without use of additional models or parameters, in both cases with validation through comparison to experimental data.

Turbulent Transport at High Reynolds Numbers in an Inertial Confinement Fusion Context

Abstract : Mix is a critical input to hydro simulations used in modeling chemical or nuclear reaction processes in fluids. It has been identified as a possible cause of performance degradation in inertial confinement fusion (ICF) targets. Mix contributes to numerical solution uncertainty through its dependence on turbulent transport coefficients, themselves uncertain and even controversial quantities. These coefficients are a central object of study in this paper, carried out in an Richtmyer Meshkov unstable circular two-dimensional (2D) geometry suggested by an ICF design. We study a pre-turbulent regime and a fully developed regime. The former, at times between the first shock passage and reshock, is characterized by mixing in the form of interpenetrating but coherent fingers and the latter, at times after reshock, has fully developed turbulent structures. This paper focuses on the scaling of spatial averages of turbulence coefficients under mesh refinement and under variation of molecular viscosity [i.e., Reynolds number (Re)]. We find that the coefficients scale under mesh refinement with a power of spatial grid spacing derived from the Kolmogorov 2/3 law, especially after reshock. We document the dominance of turbulent over molecular transport and convergence of the turbulent transport coefficients in the infinite Re limit. The transport coefficients do not coincide for the pre- and post-reshock flow regimes, with significantly stronger transport coefficients after reshock.

Mathematical, Physical and Numerical Principles Essential for Models of Turbulent Mixing

We propose mathematical, physical and numerical principles which are important for the modeling of turbulent mixing, especially the classical and well studied Rayleigh-Taylor and Richtmyer-Meshkov instabilities which involve acceleration driven mixing of a fluid discontinuity layer, by (respectively) a continuous acceleration or an impulsive delta function force. The fundamental mathematical issue is the nonuniqueness and thus indeterminancy of solutions of the 3D compressible Euler equation. Verification (demonstration that the numerical solution of the equations is mathematically correct) is meaningless for such a model of turbulent mixing. Uniqueness requires physical fluid transport, i.e. the compressible (multifluid) Navier-Stokes equation. The same fundamental issue, formulated in terms of physics, is that the properties of the mixing depend on dimensionless ratios of the transport coefficients, namely the Schmidt number (viscosity/mass diffusion) and Prandtl number (viscosity/thermal conductivity). Validation (meaning that the simulation equations correctly model the problem to be solved) is impossible without specification of fluid transport. It is in effect an effort to validate an answer for 0/0. The fundamental issue, formulated in numerical terms is that the physical transport terms are so small that they cannot be resolved at feasible grid levels. Large eddy simulations (LES) are needed. For these, subgrid scale (SGS) terms must be added to the equations, to correctly reflect the influence of the unresolved transport on the grid scales that are resolved. In the absence of such an approach, numerical artifacts intrude, leading to apparently converged solutions, with answers that depend on the computer code. Plainly, this issue, in its three guises, poses a challenge for verification and validation (V&V), and since V&V is a major scientific enterprise, it is of great importance.

Euler equation existence, non-uniqueness and mesh converged statistics

We review existence and non-uniqueness results for the Euler equation of fluid flow. These results are placed in the context of physical models and their solutions. Non-uniqueness is in direct conflict with the purpose of practical simulations, so that a mitigating strategy, outlined here, is important. We illustrate these issues in an examination of mesh converged turbulent statistics, with comparison to laboratory experiments.

The role of conductivity discontinuities in design of cardiac defibrillation

Fibrillation is an erratic electrical state of the heart, of rapid twitching rather than organized contractions. Ventricular fibrillation is fatal if not treated promptly. The standard treatment, defibrillation, is a strong electrical shock to reinitialize the electrical dynamics and allow a normal heart beat. Both the normal and the fibrillatory electrical dynamics of the heart are organized into moving wave fronts of changing electrical signals, especially in the transmembrane voltage, which is the potential difference between the cardiac cellular interior and the intracellular region of the heart. In a normal heart beat, the wave front motion is from bottom to top and is accompanied by the release of Ca ions to induce contractions and pump the blood. In a fibrillatory state, these wave fronts are organized into rotating scroll waves, with a centerline known as a filament. Treatment requires altering the electrical state of the heart through an externally applied electrical shock, in a manner that precludes the existence of the filaments and scroll waves. Detailed mechanisms for the success of this treatment are partially understood, and involve local shock-induced changes in the transmembrane potential, known as virtual electrode alterations. These transmembrane alterations are located at boundaries of the cardiac tissue, including blood vessels and the heart chamber wall, where discontinuities in electrical conductivity occur. The primary focus of this paper is the defibrillation shock and the subsequent electrical phenomena it induces. Six partially overlapping causal factors for defibrillation success are identified from the literature. We present evidence in favor of five of these and against one of them. A major conclusion is that a dynamically growing wave front starting at the heart surface appears to play a primary role during defibrillation by critically reducing the volume available to sustain the dynamic motion of scroll waves; in contrast, virtual electrodes occurring at the boundaries of small, isolated blood vessels only cause minor effects. As a consequence, we suggest that the size of the heart (specifically, the surface to volume ratio) is an important defibrillation variable.