David J. Horntrop

Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields.

Monte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedures such as the Fourier method and the moving average method; explicit numerical examples are presented which demonstrate the poor convergence of these conventional methods in various regimes with long range velocity correlations. A new method for computing fractal random fields involving wavelets and random plane waves developed recently by two of the authors [J. Comput. Phys. 117, 146 (1995)] is applied to compute pair dispersion over many decades for systematic families of anisotropic fractal velocity fields with the Kolmogorov spectrum. The important associated preconstant for pair dispersion in the Richardson law in these anisotropic settings is compared with the one obtained over many decades recently by two of the authors [Phys. Fluids 8, 1052 (1996)] for an isotropic fractal field with the Kolmogorov spectrum. (c) 1997 American Institute of Physics.

Hierarchical Monte Carlo methods for fractal random fields

Two hierarchical Monte Carlo methods for the generation of self-similar fractal random fields are compared and contrasted. The first technique, successive random addition (SRA), is currently popular in the physics community. Despite the intuitive appeal of SRA, rigorous mathematical reasoning reveals that SRA cannot be consistent with any stationary power-law Gaussian random field for any Hurst exponent; furthermore, there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of Hurst exponentsH, with 0.30<H<0.85. Thus, SRA is inconsistent with a stationary power-law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field. The second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion (MWE) with high-moment cancellation. This method is described briefly, including a demonstration that, unlike SRA, MWE is consistent with a stationary power-law random field over many decades of scaling and has low variance.

Mesoscopic simulation of Ostwald ripening

The self-organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as Ostwald ripening. This phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system. Ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange Ising model. This model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations. The typical cluster size is observed to grow like t1/3 over range of times with faster growth at later times. The results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level.

Temporal Dynamics in Density Relaxation

The density relaxation phenomenon is modeled using both Monte Carlo and discrete element simulations to investigate the effects of regular taps applied to a vessel having a planar floor filled with monodisperse spheres. Results suggest the existence of a critical tap intensity which produces a maximum bulk solids fraction. We find that the mechanism responsible for the relaxation phenomenon is an evolving quasi‐ordered packing structure propagating upwards from the plane floor.

Mesoscopic simulation for self-organization in surface processes

The self-organization of particles in a system through a diffusive mechanism is known as Ostwald ripening. This phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system. The mesoscopic model of this physical situation is a stochastic partial differential equation which can be derived from the appropriate particle system. This model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations. The results included here demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level.

Concentration effects in mesoscopic simulation of coarsening

The self-organization of particles in a two-phase system in the coexistence region through a diffusive mechanism is known as Ostwald ripening. A mesoscopic model derived from the spin exchange Ising model is used here in a computational study of Ostwald ripening. The typical cluster size is observed to grow like Lifshitz-Slyozov growth law over an early range of times with faster growth at later times. The effect of changing the concentration of particles on the morphological evolution of the system is also discussed.

Density Relaxation of Granular Matter through Monte Carlo Simulations

This paper presents Monte Carlo simulations to study the density relaxation process in monodisperse granular materials, in which the effect of discrete taps on the system is modeled by applying normalized vertical lifts of intensity Δ/d to the assembly. Examination of the bulk densities over a wide range of lift intensities suggests the existence of a critical value that optimizes the evolution of the density. Furthermore, the ‘evolution’ of the distribution of particles along the depth reveals an upward progression of organized layers that are induced by the plane floor. Results of ongoing discrete element simulations (presented elsewhere) provide strong support for this finding.

On the calculation of convolutions with Gaussian Kernels

The calculation of convolutions with Gaussian kernels arises in many applications. The accuracy of a family of recently developed numerical integration rules is studied in this paper. In spite of the attractive implementation properties of the method, the poor convergence properties greatly restrict the situations in which the method should be used.

Monte Carlo Simulation of a Random Field Model for Transport

Monte Carlo Simulation is used to study of a random field based model of transport inspired by the problem of the transport of a passive scalar through a porous medium. The Simulation technique employed to generate the random velocity field is the random shearing direction method in conjunction with the randomization method. A comparison of the Simulation results with a theoretical prediction of Koch and Brady (1988) is made. In the case when the constant mean velocity is strong relative to the random part of the velocity field, the Simulation and the theoretical prediction are in good agreement; on the other hand, when the mean velocity is weak relative to the random fluctuations, the Simulation and prediction do not agree.