In this book, it is our intent to equip graduate students in applied mathematics and engineering with a range of classical analytical methods for the solution of partial differential equations. In our research specialties, numerical methods, on the one hand, and perturbation and variational methods, on the other, constitute contemporary tools that are explicitly not covered in this book, since there are significant books that are devoted to those topics specifically.

This book grew not from the authors' desire to write a textbook but rather from many years for each of us in compiling notes to be distributed to our graduate students in courses devoted to the solution of partial differential equations. One of us taught mostly engineers (MRF at The Ohio State University), and the other (IH at Howard University and later at Rensselaer Polytechnic Institute) mostly mathematicians. It surprised us to learn, on becoming re–acquainted at RPI, how similar are our perspectives about this material, and in particular the level of rigor with which it ought to be presented. Further, both of us wanted to create a book that would include many of the techniques that we have learned one way or another but are quite simply not in books.

The topics chosen for the book are those that we have found to be of considerable use in our own research careers. These are topics that are applicable in many areas, such as aeronautics and astronautics; biomechanics; chemical, civil, and mechanical engineering fluid mechanics; and geophysical flows.

INTRODUCTION

The two combustion modes of deflagration (flame) and detonation can be generally distinguished from each other in a number of ways: by their propagation speed, the expansion (deflagration) versus compression (detonation) nature of the wave, subsonic (deflagration) versus supersonic (detonation) speed relative to the mixture ahead of the wave, and the difference in the propagation mechanism. A deflagration wave propagates via the diffusion of heat and mass from the flame zone to effect ignition in the reactants ahead. The propagation speed is governed by heat and mass diffusivity, and the diffusion flux is also dependent on the reaction rate that maintains the steep gradient across the flame. On the other hand, a detonation wave is a supersonic compression shock wave that ignites the mixture by adiabatic heating across the leading shock front. The shock is in turn maintained by the backward expansion of the reacting gases and products relative to the front, thus providing the forward thrust needed to drive the shock. A propagating flame generally has a precursor shock ahead of it, and the flame can therefore propagate at supersonic speeds with respect to a stationary coordinate system. In theory, there is still a pressure drop across the flame itself, but there may still be a net pressure increase in the products across the shock–flame complex with respect to the initial pressure of the reactants.

INTRODUCTION

A solution to the steady conservation equations across a detonation wave alone does not guarantee that a steady CJ wave can be realized experimentally. In the previous chapter, physical arguments (e.g., stability and entropy considerations) were presented in an attempt to rule out certain solutions of the conservation equations. However, solutions to the conservation equations for detonations must also be compatible with the rear boundary conditions in the combustion products, as in the case of deflagrations. For strong detonations where the flow downstream is subsonic, disturbances from the rear can propagate upstream to influence the wave, and thus it is clear that the solution must satisfy the rear boundary condition. Even for a Chapman–Jouguet (CJ) detonation, where the CJ criterion permits the solution of the conservation equations to be determined independently of the downstream flow of the combustion products, the solution for the nonsteady expansion of the products must nevertheless satisfy the sonic condition at the CJ plane. However, this is not always the case. For example, for diverging cylindrical and spherical detonations, the sonic condition behind a CJ detonation results in a singularity, which leads to the question of whether or not diverging CJ detonations can exist. Thus, the existence of a solution to the steady conservation laws also depends on the compatibility of the solution with the dynamics of the combustion products behind the wave.

The Chapman–Jouguet (CJ) theory, formulated over a century ago, provides a simple method for determining the detonation velocity using the conservation equations and the equilibrium thermodynamic properties of the reactants and products. The Zeldovich–von Neumann–Döring (ZND) theory for the detonation structure, developed in the 1940s, permits the variation of the state in the reaction zone to be computed by integrating the chemical-kinetic rate equations simultaneously with the flow equations. The CJ criterion, which chooses the minimum velocity (or tangency) solution on the equilibrium Hugoniot curve, had been shown by von Neumann to be invalid for certain explosives that have a temperature overshoot (or intersecting partially reacted Hugoniot curves). For these explosives, the detonation velocity is higher than the CJ value and corresponds to weak detonation solutions on the equilibrium Hugoniot curve. Experimental evidence from the past 50 years has also confirmed that detonations are intrinsically unstable and have a transient three-dimensional structure. This throws further doubt on the general validity of the steady one-dimensional CJ theory. However, in spite of all the lack of support for the CJ theory, the CJ detonation velocity is found to agree remarkably well with experiments. Even for near-limit mixtures where three-dimensional transient effects are significant, the averaged velocity still generally agrees with the CJ value to within 10%.

It is of importance to first define deflagrations and detonations and give the characteristics that distinguish these two types of combustion waves. Since this book is concerned with a description of the detonation phenomenon, it is of value to first introduce the various topics that are concerned with detonations prior to their detailed description in later chapters. In this manner, a global perspective can be obtained and permit selective reading of the chapters for those who are already familiar with the subject.

In telling a story, it is natural to start from the beginning, and thus the presentation of the various topics follows more or less their historical development. However, no attempt is made here to discuss the extensive early literature. A historical chronology of detonation research covering the period from its first discovery in the late 1800s to the state of knowledge in the mid 1950s has been documented by Manson and co-workers (Bauer et al., 1991; Manson & Dabora, 1993). An extensive bibliography of the early works is given in these two papers for those who want to pursue further the history of detonations. This chapter is in essence a qualitative summary of the material covered in this book.

DEFLAGRATIONS AND DETONATIONS

Upon ignition, a combustion wave propagates away from the ignition source. Combustion waves transform reactants into products, releasing the potential energy stored in the chemical bonds of the reactant molecules, which is then converted into internal (thermal) and kinetic energy of the combustion products.

Explosives are highly energetic substances with fast reaction rates and can be in gaseous, liquid, or solid form. Chemical reactions can propagate through the explosive at high supersonic speeds as a detonation wave: a compression shock with an abrupt increase in the thermodynamic state, initiating chemical reactions that turn the reactants into products. This book is devoted to a description of the detonation phenomenon, explaining the physical and chemical processes responsible for the self-sustained propagation of the detonation wave, the hydrodynamic theory that permits the detonation state to be determined, the influence of boundary conditions on the propagation of the detonation, and how detonations are initiated in the explosive.

The book is concerned only with detonation waves in gaseous explosives, because they aremuch better understood than detonations in condensed phasemedia. There are many similarities between detonations in gaseous and in condensed explosives, in that the detonation pressure of condensed explosives is much higher than the material strength of condensed explosives and the hydrodynamic theory of gaseous detonations is applicable also to condensed phase detonations. However, material properties such as heterogeneity, porosity, and crystalline structure can play important roles in the initiation (and hence sensitivity) of condensed explosives.

It is perhaps impossible to be entirely objective in writing a book, even a scientific one. The choice of the topics, the order of their presentation, and the emphasis placed on each topic, as well as the interpretation of theoretical and experimental observations, are bound to reflect the author's views.

INTRODUCTION

Direct initiation refers to the instantaneous formation of the detonation without going through the predetonation stage of flame acceleration. By “instantaneous,” it is meant that the conditions required for the onset of detonation are generated directly by the ignition source rather than by flame acceleration, as in the transition from deflagration to detonation.

Direct initiation was first used to generate spherical detonations because, in an unconfined geometry, the various flame acceleration mechanisms are ineffective (or absent), and hence the transition from a spherical deflagration to a spherical detonation cannot generally be realized. In 1923, Lafitte (1925) used a powerful igniter consisting of 1 g of mercury fulminate to directly initiate a spherical detonation in a mixture of CS2 + 3O2. He also used a planar detonation emerging from a 7-mm-diameter tube into the center of a spherical flask containing the same mixture of CS2 + 3O2. With this method, Lafitte was not successful in initiating a spherical detonation. Using the same powerful igniter of mercury fulminate, spherical detonations were also initiated in 2H2 + O2 mixtures. From the streak photographs taken by Lafitte, the detonation is observed to form instantaneously by the igniter, without a noticeable predetonation period. The detonation of a condensed explosive charge generates a very strong blast wave that decays rapidly as it expands to form the CJ detonation (hence, this mode of initiation is also referred to as blast initiation).

INTRODUCTION

The detonation velocity from the Chapman–Jouguet (CJ) theory is independent of initial and boundary conditions and depends only on the thermodynamic properties of the explosive mixture. Experimentally, however, it is found that initial and boundary conditions can have strong influences on the propagation of the detonation wave. It is the finite thickness of the reaction zone that renders the detonation vulnerable to boundary effects. Thus, the influence of the boundary can only enter into a theory that considers a finite reaction-zone thickness. In the original paper by Zeldovich (1940) on the detonation structure, he already attempted to include the effect of heat and momentum losses on the propagation of the detonation wave. However, in the one-dimensional model of Zeldovich, the two-dimensional effect of losses at the boundary cannot be treated properly. It was Fay (1959) who later modeled boundary layer effects more accurately as a divergence of the flow in the reaction zone behind the shock front, resulting in a curved detonation front. Thus, heat and momentum losses become similar to curvature effects due to the lateral expansion of the detonation products in a two- or three-dimensional detonation. However, for small curvature this can be modeled within the framework of a quasi- one-dimensional theory.

INTRODUCTION

The heart of the detonation phenomenon is the structure where the detailed ignition and combustion processes take place. The Chapman–Jouguet (CJ) theory does not require any knowledge of the structure, as it is based on steady one-dimensional flow across the transition zone with equilibrium conditions on both sides. The CJ theory also assumes that the flow within the reaction zone itself is one-dimensional and steady. The details of the transition zone are provided by the Zeldovich–von Neumann–Döring (ZND) model for the detonation structure. The ZND model is explicitly based on a steady planar structure and is therefore compatible with the one-dimensional gasdynamic CJ theory. The excellent agreement between experimental values of the detonation velocity and the theoretical predictions from the CJ theory fortuitously confirm the validity of the steady one-dimensional assumption of the CJ theory and also, implicitly, the ZND model for the detonation structure.

Early experimental diagnostics lacked the resolution to disprove the steady one-dimensional ZND structure. The state of detonation theory in the early 1960s was best summarized in a remark by Fay (1962): “… the peculiar disadvantage of detonation research is that it was too successful at too early a date. The quantitative explanation of the velocity of such waves by Chapman and Jouguet has perhaps intimidated further enquiry.” However, it was during the late 1950s and early 1960s when overwhelming experimental evidence was produced to show that the detonation structure is neither steady nor one-dimensional.

INTRODUCTION

In the gasdynamic analysis of detonation waves discussed in Chapter 2, the conservation equations are based on the upstream and downstream equilibrium states only. The detailed transition through the structure of the detonation was not considered. For a given initial upstream state, the locus of possible downstream states is given by the Hugoniot curve. Although the transition from the initial to the final state follows the Rayleigh line, the intermediate states along the Rayleigh line need not be considered if only the relationships between upstream and downstream are desired. It suffices to have the final state correspond to the intersection of the Rayleigh line with the Hugoniot curve.

Because the gasdynamic theory of detonation is concerned with the relationships between the upstream and downstream equilibrium states only, shocks, detonations, and deflagrations can all be analyzed using the conservation equations across the front. The conservation equations do not require the mechanism for the transition across the wave to be specified. However, to describe this transition zone, a model for the structure of the detonation wave must be defined. The model specifies the physical and chemical processes that are responsible for transforming the initial state to the final state. Most of the early pioneers of detonation research proposed the mechanism responsible for the propagation of the detonation implicitly in their discussion of the phenomenon.

INTRODUCTION

For given initial and boundary conditions, the possible combustion waves that can be realized are given by the solutions of the steady one-dimensional conservation equations across the wave. Since the three conservation equations (of mass, momentum, and energy) and the equation of state for the reactants and products constitute only four equations for the five unknown quantities (p1, ρ1, u1, h1, and the wave velocity u0), an extra equation is required to close this set. For non-reacting gases, the solutions to the conservation equations were first investigated by Rankine (1870) and by Hugoniot (1887–1889), who derived the relationship between the upstream and downstream states in terms of a specified wave speed, pressure, or particle velocity downstream of the shock wave. Analysis of the solutions of the conservation equations also provided important information on the stability of shock waves and on the impossibility of rarefaction shocks in most common fluids. For reacting mixtures, similar analyses of the conservation laws were first carried out independently by Chapman (1889), Jouguet (1904), and Crussard (1907), but none of these investigators were aware of similar studies carried out by Mikelson (1890) in Russia. More thorough investigations of the properties of the solutions of these conservation laws were carried out later by Becker (1917, 1922a, 1922b), Zeldovich (1940, 1950), Döring (1943), Kistiakowsky and Wilson (1941), and von Neumann (1942). Their aim was to provide a more rigorous theoretical justification of the Chapman–Jouguet criterion.

INTRODUCTION

Self-propagating one-dimensional ZND detonations are unstable and hence not observed experimentally in general. However, a solution for the laminar structure of a ZND detonation can always be obtained from the steady one-dimensional conservation equations, irrespective of the activation energy, which controls the temperature sensitivity and thus the stability of the detonation. The use of the steady one-dimensional conservation equations excludes any time-dependent multidimensional solution that describes the instability of the detonation wave. The classical method of investigating the stability of a steady solution is to impose small multidimensional perturbations on the solution and see if the amplitude of the perturbations grows. The assumption of small perturbations permits the perturbed equations to be linearized and integrated, and thus the unstable modes to be determined. As with most hydrodynamic stability analyses, the dispersion relation is rather involved and cannot be expressed analytically, which obscures the physical basis of the stability mechanism.

An alternative method is to start with the time-dependent, nonlinear, reactive Euler equations and then integrate them numerically for given initial conditions. Stability is indicated when a steady ZND solution is achieved asymptotically at large times. Unlike linear stability analyses, direct numerical simulations have the advantage that the full nonlinearity of the problem is retained. Furthermore, stability in one, two, and three dimensions can be separately investigated, which facilitates the interpretation of the numerical results. Current numerical techniques and modern computers can readily handle the integration of the multidimensional, time-dependent, reactive Euler equations.