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.

Coupled sets of ordinary differential equations (ODEs) are used to describe the evolution of the interactions among chemical species as well as many other local processes. ODEs appear, for example, when spectral and other expansion methods are used to solve timedependent partial differential equations. In these cases, spatial derivatives are converted to algebraic relationships leaving ODEs to be integrated in time. ODEs also describe the motions of projectiles and orbiting bodies, population dynamics, electrical circuits, local temperature equilibration, momentum interchange among phases in multiphase flows, the decomposition of radioactive material, and energy level and species conversion processes in atomic, molecular, and nuclear physics.

Algorithms for integrating ODEs were not originally derived by numerical analysts or applied mathematicians, but by scientists interested in solving specific sets of equations for their particular applications. Bashforth and Adams (1883), for example, developed a method for solving the equations describing capillary action. One of the first algorithms to cope with the difficulties of integrating stiff ODEs was suggested by Curtiss and Hirschfelder (1952) for chemical kinetics studies. Ten years after Curtiss and Hirschfelder identified the stiffness problem in ODEs, Dahlquist (1963) identified numerical instability as the cause of the difficulty and provided basic definitions and concepts that are still helpful in classifying and evaluating algorithms. The importance of the practical applications has spurred active research in developing and testing integration methods for solving coupled ODEs. Continued efforts of applied mathematicians have put the numerical solution of ODEs on a sounder theoretical basis and have provided insights into the constraints imposed by stability, convergence, and accuracy requirements.

Boundary conditions are used to represent an infinitely large region of space using a finite computational domain, to describe boundary layers near walls, and to simulate details of chemical reactions, heat transfer, and other surface effects. Developing the correct boundary conditions to use in a numerical model involves complicated physical and numerical issues that make it relatively easy to make conceptual and programming errors. It is necessary to determine the correct boundary conditions to apply, how they should be implemented numerically, and whether there are inconsistencies between these boundary conditions and the description of the physical system within the computational domain. Although it is not always necessary to understand the solution completely to model the interior of a computational domain, implementing physically reasonable and sufficiently consistent boundary conditions requires a strong understanding of the interior and exterior phenomena and how they interact.

Interfaces are internal boundaries that have structure and can move with the flow. When interfaces are present, they greatly increase the complexity of the simulation. Additional physical processes that are not present within the flow field, such as surface tension, evaporation, condensation, or chemical reactions, can be important at these interfaces. Often the behavior of the interface has to be modeled phenomenologically as part of a much larger overall problem. This occurs when there are orders of magnitude difference in the gradients perpendicular to and parallel to the interface. For example, a shock may have a radius of curvature of centimeters or meters, but it may be only a micron thick. The layer in which ice melts and sublimates is only a fraction of a millimeter thick.

Reactive flows include a broad range of phenomena, such as flames, detonations, chemical lasers, the earth's atmosphere, stars and supernovae, and perhaps even the elementary particle interactions in the very early stages of the universe. There are striking physical differences among these flows, even though the general forms of the underlying equations are all quite similar. Therefore, considerations and procedures for constructing numerical models of these systems are also similar. The obvious and major differences are in the scales of the phenomena, the input data, the mathematical approximations that arise in representing different contributing physical processes, and the strength of the coupling among these processes.

For example, in flames and detonations, there is a close coupling among the chemical reactions, subsequent heat release, and the fluid dynamics, so that all of the processes must be considered simultaneously. In the earth's upper atmosphere, which is a weakly ionized plasma in a background neutral wind, the chemical reactions among ionized gases and the fluid dynamics are weakly coupled. These reactions take place in the background provided by the neutral gas motions. The sun's atmosphere is highly ionized, with reactions among photons, electrons, and ionized and neutral atomic species, all in the presence of strong electromagnetic fields. A Type Ia supernova creates the heavier elements in the periodic table through a series of strongly coupled thermonuclear reactions that occur in nuclear flames and detonations. The types of reactions, the major physical processes, and the degree and type of coupling among the processes vary substantially in these systems. Sometimes reactions are essentially decoupled from the fluid flow, sometimes radiation is important, and sometimes diffusive transport effects are important.

Chapter 8 described numerical algorithms for solving continuity equations and presented straightforward methods to solve sets of continuity equations to simulate fluid systems. This chapter describes CFD methods that seek to improve the solution by incorporating more of the known flow physics. In some cases, additional constraints are added to the solution of the continuity equations. In other cases, the formulation of the problem itself is changed to use variables other than the primary conserved variables ρ, ρv, and E. The result is usually a more complicated algorithm and a less general numerical model. Sometimes, however, it can lead to a more accurate solution method for specific classes of problems.

This chapter first considers methods that exploit approximations based on the the flow speed. As discussed in Chapter 2 (Section 2–2.1, Table 2.2), flow speeds are generally divided into five regimes. In order of increasing Mach number, these are: incompressible, subsonic, transonic, supersonic, and hypersonic flows. The boundaries between these regimes loosely mark the appearance or disappearance of different physical phenomena, and each regime has peculiar features that make certain models and solution algorithms more effective. The material presented below describes methods for solving coupled flow problems in three speed regimes which are composites of those listed in Table 2.2.

• Fast flows (see Section 9–2). In this regime the fluid velocity is at least a significant fraction of the speed of sound or faster. Compressibility effects, such as shocks, must be resolved.

• […]

The previous chapters described techniques for solving the equations used to model different physical terms in the reactive-flow equations using algorithms that seemed most appropriate for each particular type of term. In each case, we identified those methods that would be best to combine and use in a reactive-flow program. This chapter delves into a fundamental issue in numerical simulations of reactive-flows: how to put all of this together in one computer model. How do we couple these separate algorithms in a way that is accurate enough and produces efficient yet flexible programs?

In a reacting flow, the different physical processes occur simultaneously, not separately or sequentially. For example, any temperature increase due to chemical reactions causes a local expansion of the gas at the same time the reactions are occurring, not some finite time later. There are at least two computational problems that result from this. First, the simulations must reproduce the correct physics of the interactions, even if it is not contained in the separate processes treated sequentially. Second, the coupling among parts of the equations representing different physical processes can be mathematically stiff. This is stiffness in the same sense discussed in Chapter 5, where some of the equations representing changes in densities of different reacting species may be mathematically stiff. The problem of coupling different processes becomes very serious if the system is characterized by multiple time and space scales.

The last section in Chapter 4, Section 4–6, gave a brief introduction to the coupling problem and highlighted the two main approaches, global-implicit methods and timestepsplitting methods.

Any attempt to define turbulence in a few words, or even a few lines, would probably invite argument and cause confusion. Turbulence is best described by a few of its characteristics. Turbulent flows are generally high Reynolds-number flows that appear to be irregular or random. Turbulent fluid motions are complex and contain many different time and space scales all coexisting in the same volume of fluid. In the terminology used in Section 11–5.1, turbulence is generally a homogeneous phenomena in the sense that all of the important scales present, microscopic through macroscopic, occupy the same space simultaneously, and it is contiguous in the sense that the relevant spatial and temporal scales are very close or overlapping. Experiments on turbulent flows are not microscopically reproducible from one time to the next.

Perhaps the most important aspect of turbulence for reactive flows is that it provides an efficient way for distinct, initially separate materials to interpenetrate and mix. Turbulence greatly increases the rates of heat, mass, and momentum transfer, as well as interspecies mixing, which is usually a necessary precursor for chemical reactions. This rapid mixing is caused by the spectrum of vortices in the flow, which act to increase the surface area of the interface between different and partially unmixed materials. As the interface surface area increases, proportionately more material diffuses across this interface, so that more molecular-scale mixing occurs. Therefore, a turbulent flame with its convoluted surface area propagates faster than a laminar (nonturbulent) flame because of the resulting faster energy release. On the computational side, the addition of chemical reactions and heat release makes it more expensive to simulate reacting turbulent flows than nonreacting turbulent flows.

This chapter returns to the problems of representing a continuous physical variable by a discrete set of numbers, and then using this representation as a basis for solving the equations of the mathematical model. Here the term computational representation includes:

• the particular discretization (that is, the mesh, grid, or expansion) used to approximate the continuous flow variables,

• the data structures used to present this discretization to the computer, and

• the interpretation procedure used to reconstruct a numerical approximation of the continuous variable from the set of discrete quantities.

Choosing a computational representation is just as important as choosing a mathematical model to describe the system, or choosing the algorithms to implement that model. For example, the choice of either an Eulerian or Lagrangian representation is important because this choice constrains the type of numerical algorithms and gridding methods that can be used.

This chapter describes the basic concepts underpinning different approaches to finding a good computational grid. This topic has received a great deal of attention in the computational fluid dynamics community, and has been the subject of many conferences and reviews. It is extremely important for solving practical problems in realistic geometries or where complex flow patterns develop. Part of the problem is generating the initial grid, another part is modifying it appropriately as the flow or computational domain evolves. Localized improvements in resolution can substantially increase accuracy at relatively little computational cost.

The material covered in this chapter is complicated and covers information from many areas of research in a cursory manner. In fact, each section or even subsection deserves at least an extensive review article.

Reactive flows encompass a very broad range of phenomena, including flames, detonations, chemical lasers, the earth's atmosphere, stars and supernovae, and perhaps even the elementary particle interactions in the very early stages of the universe. Despite the obvious physical differences among these flows, there is a striking similarity in the forms of the descriptive equations. Thus the considerations and procedures for constructing numerical models of these systems are also similar.

There has been an enormous growth in computational capabilities and resources since the first edition of this book appeared in 1987. What were difficult, expensive computations can now be done on desktop computers. Available hardware has improved almost beyond recognition. Supporting software is available for graphics and for handling large amounts of output data. New paradigms, such as parallel and massively parallel computing using distributed or shared memory, have been developed to the point where they are available to most users. Recipes also exist to interconnect desktop computers to build personal parallel computers.

With the explosive growth in available computer technology, there has been concomitant growth in the use of this technology to solve complex reactive-flow problems having numerous physical processes interacting simultaneously on many different time and space scales. The ability to solve these problems is underpinned by significant developments in numerical algorithms for solving the governing equations. With so many practitioners, many new avenues have been explored, and a number have been developed significantly.

The electronic, atomic, and molecular motions associated with internal energy cause materials to emit and absorb electromagnetic radiation continuously. Electromagnetic radiation spans a wide spectrum, ranging from radio waves to cosmic rays, and it is an important energy-transport mechanism. As such, it is an important physical effect and material diagnostic in reactive-flow systems, such as black-hole accretion disks, stellar interiors, large-scale fires, small-scale laboratory flames, rocket propulsion, hypersonic shock layers, and laser-matter interactions. For example, in forest fires or furnaces, radiation can cause ignition at widely separated regions by a phenomenon called flashover. Flashover occurs when the radiation from one combustion region heats a distant surface until it ignites. Radiation can also be important in engine combustion chambers, where temperatures reach two or three thousand degrees Kelvin. Soot particles formed by combustion processes emit and absorb radiation, thereby changing the heat balance and thus the buoyancy of the products.

The energy-exchange mechanisms for conduction and convection differ fundamentally from those of radiation. For example, emitted radiation depends very sensitively on the material temperature and becomes more important as the temperature increases. The net radiant energy transferred generally depends on differences of the absolute temperatures raised to the fourth power, following the Stefan-Boltzmann law. The energy-exchange mechanisms for convection and conduction usually depend linearly on the temperature difference. Another important difference between radiation transport and transport by conduction or convection is that radiant energy, carried by photons, can be transported in a vacuum as well as in a material medium. In convection and conduction, energy is transported by the material medium.

Radiation transport is a major scientific field in its own right.