In this chapter, we review selected results from the statistical description of turbulence needed to develop CFD models for turbulent reacting flows. The principal goal is to gain insight into the dominant physical processes that control scalar mixing in turbulent flows. More details on the theory of turbulence and turbulent flows can be found in any of the following texts: Batchelor (1953), Tennekes and Lumley (1972), Hinze (1975), McComb (1990), Lesieur (1997), and Pope (2000). The notation employed in this chapter follows as closely as possible the notation used in Pope (2000). In particular, the random velocity field is denoted by U, while the fluctuating velocity field (i.e., with the mean velocity field subtracted out) is denoted by u. The corresponding sample space variables are denoted by V and v, respectively.

Homogeneous turbulence

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t* the function U(x, t*) varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x*, U(x*, t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of ‘random’ in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier–Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively ‘smooth.’

The material contained in this chapter closely parallels the presentation in Chapter 2. In Section 3.1, we review the phenomenological description of turbulent mixing that is often employed in engineering models to relate the scalar mixing time to the turbulence time scales. In Section 3.2, the statistical description of homogeneous turbulent mixing is developed based on the one-point and two-point probability density function of the scalar field. In Section 3.3, the transport equations for one-point statistics used in engineering models of inhomogeneous scalar mixing are derived and simplified for high-Reynolds-number turbulent flows. Both inert and reacting scalars are considered. Finally, in Section 3.4, we consider the turbulent mixing of two inert scalars with different molecular diffusion coefficients. The latter is often referred to as differential diffusion, and is known to affect pollutant formation in gas-phase turbulent reacting flows (Bilger 1982; Bilger and Dibble 1982; Kerstein et al. 1995; Kronenburg and Bilger 1997; Nilsen and Kosály 1997; Nilsen and Kosály 1998).

Phenomenology of turbulent mixing

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum Eφ(k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998).

In setting out to write this book, my main objective was to provide a reasonably complete introduction to computational models for turbulent reacting flows for students, researchers, and industrial end-users new to the field. The focus of the book is thus on the formulation of models as opposed to the numerical issues arising from their solution. Models for turbulent reacting flows are now widely used in the context of computational fluid dynamics (CFD) for simulating chemical transport processes in many industries. However, although CFD codes for non-reacting flows and for flows where the chemistry is relatively insensitive to the fluid dynamics are now widely available, their extension to reacting flows is less well developed (at least in commercial CFD codes), and certainly less well understood by potential end-users. There is thus a need for an introductory text that covers all of the most widely used reacting flow models, and which attempts to compare their relative advantages and disadvantages for particular applications.

The primary intended audience of this book comprises graduate-level engineering students and CFD practitioners in industry. It is assumed that the reader is familiar with basic concepts from chemical-reaction-engineering (CRE) and transport phenomena. Some previous exposure to theory of turbulent flows would also be very helpful, but is not absolutely required to understand the concepts presented. Nevertheless, readers who are unfamiliar with turbulent flows are encouraged to review Part I of the recent text Turbulent Flows by Pope (2000) before attempting to tackle the material in this book. In order to facilitate this effort, I have used the same notation as Pope (2000) whenever possible.

In this chapter, we present the most widely used methods for closing the chemical source term in the Reynolds-averaged scalar transport equation. Although most of these methods were not originally formulated in terms of the joint composition PDF, we attempt to do so here in order to clarify the relationships between the various methods. A schematic of the closures discussed in this chapter is shown in Fig. 5.1. In general, a closure for the chemical source term must assume a particular form for the joint composition PDF. This can be done either directly (e.g., presumed PDF methods), or indirectly by breaking the joint composition PDF into parts (e.g., by conditioning on the mixture-fraction vector). In any case, the assumed form will be strongly dependent on the functional form of the chemical source term. In Section 5.1, we begin by reviewing the methods needed to render the chemical source term in the simplest possible form. As stated in Chapter 1, the treatment of non-premixed turbulent reacting flows is emphasized in this book. For these flows, it is often possible to define a mixture-fraction vector, and thus the necessary theory is covered in Section 5.3.

Overview of the closure problem

In this section, we first introduce the ‘standard’ form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term.

This chapter is devoted to methods for describing the turbulent transport of passive scalars. The basic transport equations resulting from Reynolds averaging have been derived in earlier chapters and contain unclosed terms that must be modeled. Thus the available models for these terms are the primary focus of this chapter. However, to begin the discussion, we first review transport models based on the direct numerical simulation of the Navier–Stokes equation, and other models that do not require one-point closures. The presentation of turbulent transport models in this chapter is not intended to be comprehensive. Instead, the emphasis is on the differences between particular classes of models, and how they relate to models for turbulent reacting flow. A more detailed discussion of turbulent-flow models can be found in Pope (2000). For practical advice on choosing appropriate models for particular flows, the reader may wish to consult Wilcox (1993).

Direct numerical simulation

Direct numerical simulation (DNS) involves a full numerical solution of the Navier–Stokes equations without closures (Rogallo and Moin 1984; Givi 1989; Moin and Mahesh 1998). A detailed introduction to the numerical methods used for DNS can be found in Ferziger and Perić (2002). The principal advantage of DNS is that it provides extremely detailed information about the flow. For example, the instantaneous pressure at any point in the flow can be extracted from DNS, but is nearly impossible to measure experimentally. Likewise, Lagrangian statistics can be obtained for any flow quantity and used to develop new turbulence models based on Lagrangian PDF methods (Yeung 2002). The application of DNS to inhomogeneous turbulent flows is limited to simple ‘canonical’ flows at relatively modest Reynolds numbers.

For purely temporal disturbances k is real, and if it is given together with the complete set of the flow parameters, then we can solve the complex wave frequency ω from (10.26) as the eigenvalue. The IMSL library routine GVLCG has been used to obtain ω. We characterize the spatial-temporal disturbances of a given wave number kr for a given set of flowparameters with the spatial amplification curves ωr = 0. There are at least two such curves for a given set of flow parameters in the case of convective instability. One corresponds to the sinuous mode and the other to the varicose mode. For each mode, we start with an initial guess of ki for a given kr. Then solve for ωr and ωi using the IMSL routine GVCCG. If ωr = 0 the guess was perfect, if not we find ki by using the Newton integration method with a reduced value of ∣ωr∣. With the new ki and the original kr we update (ωr, ωi) by means of the IMSL routine GVCCG. We repeat this procedure until the IMSL routine gives ωr = 0.

Annular jets are encountered in many industrial processes. Their stability has been studied in the contexts of ink-jet printing (Hertz and Hermanrud, 1983; Sanz and Meseguer, 1985), encapsulation (Lee and Wang, 1989; Kendall, 1986), gas absorption (Baird and Davidson, 1962), and atomization (Crapper, Dombrowski, and Pyott, 1975; Lee and Chen, 1991; Shen and Li, 1996; Villermaux, 1998). Shen and Li analyzed the spatial-temporal instability of an annular liquid jet surrounded by an inviscid gas. Hu and Joseph (1989) investigated the temporal instability of a three-layered liquid core-annular flow. The instability of annular layeres has been used to model the formation of liquid bridges in microairways in lungs (Newhouse and Pozrikidis, 1992). The related problems of liquid bridge instability are reviewed by Alexander (1998). Annular jet instability is also of considerable theoretical interest because it includes many other flow instabilities as special cases (Meyer and Weihs, 1987). Moreover, it serves to establish knowledge of the fluid physics of flows with two distinctive curved fluid-fluid interfaces subjected to different shear forces, capillary forces, and inertial forces under variable gravitational conditions.

An Annular Jet

Consider the flow of a fluid in an annulus enclosing another fluid, which is surrounded by yet another fluid inside a circular pipe of radius Rω as shown in Figure 10.1. The axis of the pipe aligns with the direction of the acceleration due to gravity g. All three fluids are incompressible.

The phenomena of the breakup of liquid sheets and jets are encountered in nature as well as in various industrial applications. A good understanding of these phenomena requires a sound basic scientific knowledge of the dynamics of flows involving interfaces between different fluids. This book is the outcome of the author's inquiry into this fundamental knowledge. My understanding of the subject matter has been consolidated gradually through direct and indirect collaborations with my students and colleagues. The objective and scope of this book in the context of related existing works are explained in Chapter 1. Chapters 2 to 5 are devoted to exposition of the onset of sheet breakup. Chapters 6 to 10 discuss jet breakup. A perspective of the challenging aspects of the subject, including the nonlinear evolution subsequent to the onset of instability and nanojets, is sketched in Chapter 11. Some additional topics related to the breakup of a liquid body into smaller parts are discussed in the epilogue. Readers are expected to have the equivalent of at least an undergraduate background in science or engineering. In the theoretical development I have strived for mathematical rigor, numerical accuracy, and rational approximation. However, mathematics has not been used just for the sake of mathematics. I have depended on comparisons between different theories and experiments to establish physical concepts. Practical applications of the concepts are pointed out in appropriate places. The references relevant to each chapter are listed at the end of the chapter.

Nonuniform liquid sheets are encountered in various industrial applications including radiation cooling in space (Chubb et al., 1994), and paper making (Soderberg and Alfredson, 1998). Many of the works cited below were motivated by applications in surface coating, fuel spray formation, nuclear safety, and other industrial processes. The spatial variation of sheet thicknesses in these applications is necessitated by the conservation of mass flow across the cross section perpendicular to the flow direction. For example, the thickness of a planar sheet of constant width must decrease in the flow direction due to the gravitational acceleration. Consequently the local Weber number, based on the local thickness and velocity, changes spatially. In particular We may be greater than one in part of the sheet and smaller than one in the rest. If one locally applies the concept of absolute and convective instability in a uniform sheet, then part of the sheet may experience convective instability, while the remaining part may experience absolute instability. Depending on the relative location of the regions of We > 1 and We < 1, one would expect different physical consequences to the entire flow. The objective of this chapter is to elucidate the effect of the spatial variation of We on the dynamics of sheet breakup by properly applying the concept developed in the previous chapter for a sheet of uniform thickness.

A liquid jet emanating from a nozzle or orifice exhibits richly varied phenomena that depend on the orifice geometry, the inlet condition before the jet is emanated, and the environmental situation into which the jet is issued. A liquid jet cannot escape the ultimate fate of breakup because of hydro-dynamic instability. The breakup possesses two major regimes: large drop formation and fine spray formation. These two regimes are controlled by distinctively different physical forces, and between them there exist intermediate regimes. All the regimes arise from a subtle dynamic response of the jet to the disturbances.

Geometry of Liquid Jets

Citing the experiment of Bidone, Rayleigh (1945, p. 355) stated, “Thus in the case of an elliptical aperture, with major axis horizontal, the sections of the jet taken at increasing distances gradually lose their ellipticity until at a certain distance the section is circular. Further out the section again assumes ellipticity, but now with major axis vertical.” This statement is illustrated in Figure 6.1, which was taken from Taylor (1960), who also carried out the experiment. The phenomenon was understood as the vibration of a jet enclosed in an envelope of constant tension about its equilibrium configuration with a circular cross section. However, Taylor (1960) demonstrated that the phenomenon can still be predicted without the surface tension in the absence of gravity. With gravity, if the jet is issued vertically downward, it will accelerate.

Overview

When a dense fluid is ejected into a less dense fluid from a narrow slit whose thickness is much smaller than its width, a sheet of fluid can form. When the fluid is ejected not from a slit but from a hole, a jet forms. The linear scale of a sheet or jet can range from light years in astrophysical phenomena (Hughes, 1991) to nanometers in biological applications (Benita, 1996). The fluids involved range from a complex charged plasma under strong electromagnetic and gravitational forces to a small group of simple molecules moving freely with little external force. The fluid sheet and jet are inherently unstable and breakup easily. The dynamics of liquid sheets was first investigated systematically by Savart (1833). Platou (1873) sought the nature of surface tension through his inquiry of jet instability. Rayleigh (1879) illuminated his jet stability analysis results with acoustic excitation of the jet. In some modern applications of the instability of sheets and jets, it is advantageous to hasten the breakup, but in other applications suppression of the breakup is essential. Hence knowledge of the physical mechanism of breakup, aside from its intrinsic scientific value, is very useful when one needs to exploit the phenomenon to the fullest extent. Recent applications include film coating, nuclear safety curtain formation, spray combustion, agricultural sprays, ink jet printing, fiber and sheet drawing, powdered milk processing, powder metallurgy, toxic material removal, and encapsulation of biomedical materials.