The processes of freezing and melting were present at the beginning of the Earth and continue to affect the natural and industrial worlds. These processes created the Earth's crust and affect the dynamics of magmas and ice floes, which in turn affect the circulation of the oceans and the patterns of climate and weather. A huge majority of commercial solid materials were “born” as liquids and frozen into useful configurations. The systems in which solidification is important range in scale from nanometers to kilometers and couple with a vast spectrum of other physics.

The solidification of a liquid or the melting of a solid involves a complex-interplay of many physical effects. The solid–liquid interface is an active free boundary from which latent heat is liberated during phase transformation. This heat is conducted away from the interface through the solid and liquid, resulting in the presence of thermal boundary layers near the interface. Across the interface, the density changes, say, from ρℓ to ρs. Thus, if ρs > ρℓ, so that the material shrinks upon solidification, a flow is induced toward the interface from “infinity.”

If the liquid is not pure but contains solute, preferential rejection or incorporation of solute occurs at the interface. For example, if a single solute is present and its solubility is smaller in the (crystalline) solid than it is in the liquid, the solute will be rejected at the interface.

Materials Science is an extremely broad field covering metals, semiconductors, ceramics, and polymers, just to mention a few. Its study is dominated by the fabrication of specimens and the characterization of their properties. A relatively small portion of the field is devoted to phase transformation, the dynamic process by which in the present context a liquid is frozen or a solid is melted.

This book is devoted to the study of liquid (melt)-solid transformations of atomically rough materials: metals or semiconductors, including model organics like plastic crystals. The emphasis is on the use of instability behavior as a means of understanding those processes that ultimately determine the micro-structure of a crystalline solid. The fundamental building block of this study is the Mullins–Sekerka instability of a front, which gives conditions for the growth of infinitesimal disturbances of a soild–liquid front. This is generalized in many ways: into the nonlinear regime, including thermodynamic disequilibrium, anisotropic material properties, and effects of convection in the liquid. Cellular, eutectic, and dendritic behaviors are discussed. The emphasis is on dynamic phenomena rather than equilibria. In a sense then, it concerns “physiology” rather than “anatomy.”

The aim of this book is to present in a systematic way the field of continuum solidification theory. This begins with the primitive field equations for diffusion and the derivation of appropriate jump conditions on the interface between the solid and liquid. It then uses such models to explore morphological instabilities in the linearized range and gives physical explanations for the phenomena uncovered.

In all the systems discussed heretofore the solidification front was considered to be a mathematical interface of zero thickness endowed with surface properties deemed appropriate to the physics. In this chapter, another approach is taken. The front is allowed to be diffuse, and the fields of interest, such as T and C, are supposed to have well-defined bulk behaviors away from the interfacial region and rapid, though continuous, variations within it. Minimally, one would wish the model to satisfy the laws of thermodynamics, appropriately extended into the nonequilibrium regions, and regain the interfacial properties and jump conditions appropriate to the thin-interface limit when the interfacial thickness approaches zero.

On the one hand one would anticipate that an infinite number of such models likely exists. On the other hand one would anticipate that the thin-interface limit might be taken a number of ways, each giving distinct properties to the front. Nonetheless, conceptually there are two possible virtues of the diffuse-interface approach. If the models are well chosen on the basis of some underlying framework, then there would be a systematic means of generalizing the models to systems such as rapid solidification. In Chapter 6 high rates of solidification were modeled by appending to the standard model variations k = k(Vn), m = m(Vn) with, for example, the equilibrium Gibbs–Thomson undercooling. A systematic generalization could indicate how the relationships for k and m emerge, and what other alterations to the model should simultaneously be included. Call this “model building.”

This monograph is an attempt to address the theory of turbulence from the points of view of several disciplines. The authors are fully aware of the limited achievements here as compared with the task of understanding turbulence. Even though necessarily limited, the results in this book benefit from many years of work by the authors and from interdisciplinary exchanges among them and between them and others. We believe that it can be a useful guide on the long road toward understanding turbulence.

One of the objectives of this book is to let physicists and engineers know about the existing mathematical tools from which they might benefit. We would also like to help mathematicians learn what physical turbulence is about so that they can focus their research on problems of interest to physics and engineering as well as mathematics. We have tried to make the mathematical part accessible to the physicist and engineer, and the physical part accessible to the mathematician, without sacrificing rigor in either case. Although the rich intuition of physicists and engineers has served well to advance our still incomplete understanding of the mechanics of fluids, the rigorous mathematics introduced herein will serve to surmount the limitations of pure intuition. The work is predicated on the demonstrable fact that some of the abstract entities emerging from functional analysis of the Navier–Stokes equations represent real, physical observables: energy, enstrophy, and their decay with respect to time.

Introduction

As mentioned earlier in this text, we take for granted that the Navier–Stokes equations (NSE), together with the associated boundary and initial conditions, embody all the macroscopic physics of fluid flows. In particular, the evolution of any measured property of a turbulent flow must be relatable to the solutions of those equations. In turbulent flow regimes, the physical properties are universally recognized as randomly varying and characterized by some suitable probability distribution functions. In this and the following chapter, we discuss how those probability distribution functions (also called probability distributions or measures, or Borel measures, in the mathematical terminology; see Appendix A.1) are determined by the underlying Navier–Stokes equations. Although in many cases such distributions may not be known explicitly, their existence and many useful properties may be readily established. For many practical purposes, such partial knowledge may be all that is needed. Thus we note that the issue of an explicit form of the distribution function – in particular, whether this measure is unique or depends on the initial data – is still an incompletely solved mathematical problem. But there are enough firm results available assuring that many of the widely accepted experimental results are meaningful and in consonance with the theory of the Navier–Stokes equations.

For instance, measurements of various aspects of turbulent flows (e.g., the turbulent boundary layer) are actually measurements of time-averaged quantities.

Introduction

In principle, the idea that solutions of the Navier–Stokes equations (NSE) might be adequately represented in a finite-dimensional space arose as a result of the realization that the rapidly varying, high-wavenumber components of the turbulent flow decay so rapidly as to leave the energy-carrying (lower-wavenumber) modes unaffected. With the understanding gained from Kolmogorov's [1941a,b] phenomenological theory (see also Section 3), it appeared that, in 3-dimensional turbulent flows, only wavenumbers up to the cutoff value κd = (∈/ν3)1/4 need be considered. This is the boundary between the inertial range, which is dominated by the inertial term in the equation, and the dissipation range, which is dominated by the viscous term. As explained by Landau and Lifshitz [1971], the question is then reduced to finding the number of resolution elements needed to describe the velocity field in a volume – say, a cube of length ℓ0 on each side. Clearly, if the smallest resolved distance is to be ℓd = 1/κd, then the number of resolution elements is simply (ℓ0/ℓd)3. On adducing some phenomenological and intuitive arguments, it was argued that this ratio is Re9/4, where Re is the Reynolds number. An alternate way to count the number of active modes is as follows: since these modes are those in the inertial range, their frequency κ satisfies κ0 < κ < κd, with κ0 = 1/ℓ0; we conclude that, for κd/κ0 large, that number is of the order of (κd/κ0)3 = (ℓ0/ℓd)3.

Introduction

This long and technical chapter aims at providing some basic connections between the mathematical theory of the Navier–Stokes equations (NSE) and the conventional theory of turbulence. As stated earlier, the conventional theory of turbulence (including the famous Kolmogorov spectrum law) is based principally on physical and scaling arguments, with little reference to the NSE. We believe that it is instructive to connect turbulence more precisely with the Navier–Stokes equations.

It is commonly accepted that turbulent flows are necessarily statistical in nature. Indeed, if a flow is turbulent, then all physical quantities are rapidly varying in space and time and we cannot determine the actual instantaneous values of these quantities. Instead, one usually measures the moments, or some averaged values of physical quantities; that is, only a statistical description of the flow is available. The first task in this chapter is to establish, in a more precise way, the time evolution of the probability distribution functions associated with the fluid flow – that is, the statistical solutions of the Navier–Stokes equations. Although the discussion is relevant to deterministic data (initial values of the velocities and volume forces), we extend our discussion to the case of random data; however, we will not examine the more involved case of very irregular forcing (such as white or colored volume forces), since deterministic or moderately irregular stochastic data suffice, in practice, to generate complex turbulent flows.

Introduction

The purpose of this chapter is to recall some elements of the classical mathematical theory of the Navier–Stokes equations (NSE). We try also to explain the physical background of this theory for the physics-oriented reader.

As they stand, the Navier–Stokes equations are presumed to embody all of the physics inherent in the given incompressible, viscous fluid flow. Unfortunately, this does not automatically guarantee that the solutions to those equations satisfy the given physics. In fact, it is not even guaranteed a priori that a satisfactory solution exists. This chapter addresses the means for specifying function spaces – that is, the ensembles of functions consistent with the physics of the situation (such as incompressibility, boundedness of energy and enstrophy, as well as the prescribed boundary conditions) – that can serve as solutions to the Navier–Stokes equations. An important point is made that the kinematic pressure, p, is determined uniquely by the velocity field up to an additive constant. Hence, one cannot specify independently the initial boundary conditions for the pressure. This observation leads naturally to a representation of the NSE by an abstract differential equation in a corresponding function space for the velocity field.

Two types of boundary conditions are considered: no-slip, which are relevant to flows in domains bounded by solid impermeable walls; and space-periodic boundary conditions, which serve to study some idealized flows (including homogeneous flows) far away from real boundaries.

Introduction

In this chapter we first briefly recall, in Section 1, the derivation of the Navier–Stokes equations (NSE) starting from the basic conservation principles in mechanics: conservation of mass and momentum. Section 2 contains some general remarks on turbulence, and it alludes to some developments not presented in the book. For the benefit of the mathematically oriented reader (and perhaps others), Section 3 provides a fairly detailed account of the Kolmogorov theory of turbulence, which underlies many parts of Chapters III–V. For the physics-oriented reader, Section 4 gives an intuitive introduction to the mathematical perspective and the necessary tools. A more rigorous presentation appears in the first half of Chapter II and thereafter as needed. For each of the aspects that we develop, the present chapter should prove more useful for the nonspecialist than for the specialist.

Viscous Fluids. The Navier–Stokes Equations

Fluids obey the general laws of continuum mechanics: conservation of mass, energy, and linear momentum. They can be written as mathematical equations once a representation for the state of a fluid is chosen. In the context of mathematics, there are two classical representations. One is the so-called Lagrangian representation, where the state of a fluid “particle” at a given time is described with reference to its initial position.

As described in the previous chapter, the term reactive flow applies to a very broad range of physical phenomena. In some cases the equations are not even rigorously known. In this chapter, we first consider the equations of gas-phase reactive flows, which are generally accepted as valid in the continuum regime. This set of time-dependent, coupled, partial differential equations governs the conservation of mass and species density, momentum, and energy. The equations describe the convective motion of the fluid, reactions among the constituent species that may change the molecular composition, and other transport processes such as thermal conduction, molecular diffusion, and radiation transport. Many different situations are described by these equations when they are combined with various initial and boundary conditions. In a later section of this chapter, we discuss interactions among these processes and generalizations of this set of equations to describe multiphase reactive flows.

The material presented in this chapter is somewhat condensed, and is not meant to give an in-depth explanation to those unfamiliar with the individual topics. The purpose is to present the reactive-flow equations, to establish the notation used throughout this book, and then to relate each term in the equations to physical processes important in reactive flows. The chapter can then be used as a reference for the more detailed discussions of numerical methods in subsequent chapters. It would be reasonable to skim this chapter the first time through the book, and then to refer back to it as needed.

This chapter presents and analyzes the properties of the simplest finite-difference methods for simulating four of the main physical processes in reactive flows: chemical reactions, diffusion, convection, and wave motion. The material presented is an overview and short course on solving idealized forms of the equations representing these processes. The discussion highlights the features and weaknesses of these solution methods and brings out numerical difficulties that reappear in solutions of the complete set of reactive-flow conservation equations. Throughout the presentation, we list and describe the major computational and algorithmic trade-offs that arise in simulating each process separately.

The material presented here introduces the more advanced solution techniques described in Chapters 5 through 9. Chapter 11 deals with techniques for solving the coupled set of equations that forms the reactive Navier-Stokes equations discussed in Chapter 2. In particular, Chapter 11 shows how the disparate time and space scales of each type of process can be used to determine a reasonable overall timestep for the computation. The choice of the numerical boundary conditions that are so crucial for correctly defining the physical problem, are discussed in Chapters 5 through 9. Sections 10–1 and 10–2 are devoted to issues of selecting boundary conditions for the reactive-flow equations.

Table 4.1 shows the mathematical representations discussed in this chapter and indicates where the numerical solutions for more complex forms of these equations are discussed elsewhere in this book. There are many references on numerical methods and scientific computation for science and engineering that cover material complementary to this chapter.