Stony Brook AMS - Downloadable Preprints, 1995

We introduce the medial axis transform as a tool in the analysis of geometric structure of void space in porous media. The technique is used to study high (5 micron) resolution, three dimensional, synchrotron computed microtomographic data consisting of two drill core samples (Berea sandstone and Danish chalk) and a sample of uniform diameter, packed glass beads.

From analysis of the medial axis, we obtain results for a discrete version of the pore-size distribution, the distribution of disconnected void volumes, and tortuosity distributions between parallel faces.

A hierarchy of enzyme-catalyzed positive feedback loops is examined by mathematical and numerical analysis. Four systems are described, from the simplest, in which an enzyme catalyzes its own information from an inactive precursor, to the most complex, in which two feedback loops act in a cascade analogous to a section of the blood-coagulation system. In the latter we also examine the function of a long-range feedback, in which the final enzyme produced in the second loop activates the initial step in the first loop. When the enzymes generated are subject to inhibition or inactivation, all four systems exhibit threshold properties akin to other excitable systems like neuron firing. For those that are amenable to mathematical analysis, expressions are derived that relate the excitation threshold to the kinetics of enzyme generation and inhibition and the initial conditions. For the most complex system it was expedient to employ numerical simulation to demonstrate threshold behavior, and here long-range feedback was seen to have two distinctive effects. At sufficiently high catalytic rates, this feedback is capable of exciting an otherwise sub-threshold system. At lower catalytic rates, where the long-range feedback does not significantly affect the threshold, it nonetheless has a major effect in potentiating the response above the threshold. In particular oscillatory behavior observed in simulation of sequential feedback loops is abolished when a long-range feedback is present.

The Rayleigh-Taylor instability is a gravity driven instability of a contact surface between fluids of different densities. The growth of this instability is sensitive to numerical or physical mass diffusion. For this reason, high resolution of the contact discontinuity is particularly important. In this paper, we address this problem using a second order TVD finite difference scheme with artificial compression. We describe our numerical simulations of the 3-D Rayleigh-Taylor instability using this scheme. The numerical solutions are compared to (a) the exact 2-D solution in the linear regime and (b) numerical solutions using the TVD scheme and the front tracking method. The computational program is used to study the evolution of a single bubble and 3-D bubble merger, i. e. the nonlinear evolution of a single mode and the process of nonlinear mode-mode interaction.

We develop a model for the dynamics of a fully developed shear band that allows effective computation across several length scales. From a macroscopic point of view, a shear band is a discontinuity in tangential velocity that supports a shear stress. Numerical simulation of the full system of governing equations reveals that the internal structure of the band consists of a quasistatic core surrounded by a thermal layer. We show that the shear band can be modeled as a composite structure whose evolution is governed by an integral equation, coupled to the external flow through jump conditions. We establish the accuracy of the model equations by numerical experiments.

The reduction of porosity is an important requirement in the manufacture of reliable composite components. In the Resin Transfer Molding (RTM) process, porosity results from the formation and growth of gas bubbles during the fill and cure stages of the process. Understanding the dynamics of the formation and motion of these bubbles will allow design of RTM processes that provide optimum quality and performance. In this paper, we demonstrate that an unsaturated flow model for gas bubble migration is in qualitative agreement with experimental results.

The Rayleigh-Taylor instability of an interface separating fluids of distinct density is driven by an acceleration across the interface. Low order statistical moments of fluctuating fluid quantities characterize the hydrodynamics of the mixing zone.

A new model is proposed for the momentum coupling between the two phases. This model is validated against computational data for compressible flows, including flows near the incompressible limit. Our main result is a zero parameter first order closure for ensemble averaged two phase flow equations. We do not, however, fully solve the closure problem, as the equations we derive are missing an (internal) boundary condition along any surface for which either phase goes to zero volume fraction. In this sense, the closure problem is reduced from a volume to a surface condition, rather than being solved completely.

A new understanding of the compressibility dependent loss of universality of the mixing rate is obtained in terms of a one parameter family of solutions of the two phase flow equations. This parameter measures the initial perturbation amplitude in dimensionless units which eliminate this effect in the incompressible limit.

We compare two formulations of the statistical moments, one based on two phase flow and the other on turbulence models. These formulations describe different aspects of the mixing process. For the problem considered, the two phase flow moments appear to be preferable, in that they subsume the turbulence moments but not conversely.

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two states $U_1$ and $U_2$ such that a traveling wave leads from $U_1$ to $U_2$ and another leads from $U_2$ to $U_1$. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.

We study the scale up problem for flow in porous media. The general nature of this problem is outlined, leading to a discussion of assumptions on random fields appropriate for the description of geological heterogeneities.

The main point of this paper is to use direct numerical simulation to evaluate the ensemble averages describing fluid dispersion, for flow in porous media. The relation between ensemble dispersion and single realization dispersion is discussed in the case of linear transport, and the role of plume or channel width is also explored. Finally we consider nonlinear transport, and contrast dispersive to hyperbolic renormalization of the flow equations. For the geological and fluid parameters considered here, the hyperbolic renormalization is trivial, indicating that dispersive renormalization is appropriate in these cases.

Further study of the ideas explored in this paper will be required for a proper understanding of their role in a more complete theory which we hope will follow.

The system of shock wave-induced vapor condensation in fluids of large heat capacity has been investigated theoretically. The wave structure is governed by differences in the time scales associated with various relaxation mechanisms and their coupling to the macroscopic flow. In the low to moderate Mach number range, the viscous and heat-conducting forces are localized within a discontinuous forerunner wave, while the slower nucleation and droplet growth processes are resolved inside a trailing condensation wave. A time-dependent analysis of the fluid motion is required because the phase transition can occur too slowly for the system to reach a steady state on the laboratory time scale. In the high Mach number regime, a steady-state motion is rapidly attained, but the analysis requires a simultaneous treatment of nucleation, viscosity, and heat conduction. It appears that instability mechanisms similar to the ones found in ZND detonation waves are responsible for the shock-front irregularities observed in high Mach-number flows. Based on the rapidity of the supersaturating shock compression and the time delay of nucleation, it is anticipated that metastable states near the vapor spinodal are attainable, especially at high temperatures.

The method of front tracking has been applied to several complex problems in gas dynamics. All interactions between tracked waves should be handled automatically and robustly, and this provides a challenge both analytically and from a programming standpoint. Such issues have formed the basis for the present research, and a description of the relevant algorithms and ideas is included in this thesis.

Three applications are used to produce interesting interactions for study. The first is the bifurcation from regular to Mach reflection of a shock wave at a reflecting surface. The results indicate that, in an idealized limit neglecting many physical effects, front tracking prefers the mechanical equilibrium point as the transition criterion for pseudo-steady flows.

The next application is the Richtmyer-Meshkov instability of an interface separating two gases. The refraction of a shock through the interface causes any perturbations to grow over time, becoming unstable and leading eventually to fingering and mixing of the gases. The focus has been on larger amplitude perturbations, leading to more complex and irregular refraction structures. The simulations show that the overall growth rate of the interface and mixing zone is continuous across the bifurcation to irregular configurations.

The last application is the non-ideal blast phenomenon, which involves the interaction of an expanding blast wave with a layer of warmed air near a reflecting boundary. The thermal layer channels the energy and holds it close to the ground, leading to a cascading effect with many complex wave interactions. Numerous algorithms are discussed for dealing with such phenomena, including the automatic untracking of waves during a simulation. Since it is not possible to foresee every eventuality, the code should be robust enough to do something reasonable when it encounters an unexpected situation. Also needed is the ability to leave sufficiently weak scattered waves untracked during the installation of a complicated structure.

All three applications also involved the implementation of overlapping subdomain boundary conditions, appropriate for periodic, reflecting and parallel processing artificial boundaries. Issues pertinent to the implementation of these algorithms are discussed.

Resin Transfer Molding (RTM), as a method for the manufacture of advanced fiber reinforced composite materials, is attractive because it offers the possibility of lower manufacturing costs and more complex shapes than traditional methods.

A major issue in this new manufacturing process is the elimination of void spaces in the resin fill operation, so that products with high quality are manufactured. Process modeling is particularly useful in understanding, designing, and optimizing the process conditions to achieve this goal.

In this paper we report on our program with the Advanced Technology & Development Center at Northrop Grumman and demonstrate how modeling could improve the manufacturing process and enhance product quality. We review the manufacturing process and related issues and present a manufacturing process model, developed recently by the authors. This model is applied to study the formation and migration of air bubbles in the preform, a major RTM manufacturing problem, and suggests new process strategies to reduce the void content of the finished product.

In this paper we analyze a recently proposed two-phase flow model of the Rayleigh-Taylor mixing zone. We obtain an exact solution of the model equations in the incompressible, large time scaling limit. The scaling limit equations are the renormalization group equations for this problem, and our results show that the large time behavior of the Rayleigh-Taylor mixing layer is determined by an RNG fixed point, for incompressible fluids. Our solution allows independent pressures in each fluid. The pressures do not equilibrate dynamically. Pressure difference boundary conditions are related, on a physical basis, to drag and bouyancy forces at the edge of the two-phase mixing layer.

Richtmyer-Meshkov instability is a fingering instability which occurs at a material interface accelerated by a shock wave. We present an analytic, explicit prediction for the growth rate of the unstable interface. The theoretical prediction agrees, for the first time, with the experimental data on air-$roman SF sub 6$, and is in remarkable agreement with the results of recent full non-linear numerical simulations from early to late times. Previous theoretical predictions of the growth rate for air-SF6 unstable interfaces were about two times larger than the experimental data.

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 improve significantly 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.

Internal boundaries and interfaces are an important part of many fluid and solid modeling problems. Front Tracking contains a general interface framework, closely related to the non-manifold geometry used in CAD solid modeling packages, to support numerical simulation of such fluid problems. It can thus be considered to be a systematic application of the ideas of computational geometry to computational fluid dynamics. It is based on the principle that finite differences behave best when applied to differentiable functions, and that weak derivatives of nondifferentiable functions can be replaced by regularized expressions such as jump conditions. Front Tracking offers superior resolution for fluid problems with important discontinuities and interfaces, and in some cases, it has provided the unique method to obtain correct answers. Here we present Computer Science issues which have contributed to the success of Front Tracking: software design and organization -- modularity, data structures and data hiding.

Non-strictly-hyperbolic and mixed-type conservation laws pose a challenge to computational methods. Specifically, the computation of transitional shock waves, which are dependent on the parabolic regularization of the conservation laws, is sensitive to the numerical viscosity. In this paper, we propose a front tracking algorithm that treats transitional shock waves correctly. The algorithm includes the computation of saddle-to-saddle connections in planar dynamical systems using the multiple shooting method. We compare our results with those of commonly-used shock capturing schemes, including the Lax-Friedrichs, Lax-Wendroff, and higher-order Godunov schemes.