In Chapter 6 we reviewed the theory underlying transported PDF methods. In order to apply this theory to practical flow problems, numerical algorithms are required to ‘solve’ the PDF transport equation. In general, solving the PDF transport equation using standard finite-difference (FD) or finite-volume (FV) methods is computationally intractable for a number of reasons. For example, the velocity, composition PDF transport equation ((6.19), p. 248) has three space variables (x), three velocity variables (V), Ns composition variables (ψ), and time (t). Even for a statistically two-dimensional, steady-state flow with only one scalar, a finite-difference grid in at least five dimensions would be required! Add to this the problem of developing numerical techniques that ensure fU, φ remains non-negative and normalized to unity at every space/time point (x, t), and the technical difficulties quickly become insurmountable.

A tractable alternative to ‘solving’ the PDF transport equation is to use statistical or Monte-Carlo (MC) simulations. Unlike FV methods, MC simulations can handle a large number of independent variables, and always ensure that the resulting estimate of fu, φ is well behaved. As noted in Section 6.8, MC simulations employ representative samples or so-called ‘notional’ particles. The principal challenge in constructing an MC algorithm is thus to define appropriate rules for the rates of change of the notional-particle variables so that they have statistical properties identical to fU,φ(V, ψ; x, t). The reader should, however, keep in mind that the necessarily finite ensemble of notional particles provides only a (poor) estimate of fu, φ. When developing MC algorithms, it will thus be important to consider the magnitude of the estimation errors and to develop ways to control them.

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.

General properties

From the class of flows that are termed geophysical, there are three that are distinct and these more than illustrate the salient properties that such flows possess when viewed from the basis of perturbations. First, there is stratified flow. In this case there is a mean density variation and it plays a dominant role in the physics because there is a body force due to gravity. At the same time, the fluid velocity, to a large degree of approximation, remains solenoidal and therefore the motion is incompressible. The net result leads to the production of anisotropic waves, known as internal gravity waves, and such motions exist in both the atmosphere and the ocean.

Second, because of the spatial scales involved, motion at many locations of the earth, such as the northern or southern latitudes, are present in an environment where the effects of the earth's rotation cannot be taken as constant. On the contrary, rotation plays a dominant role. Again, this combination of circumstances leads to the generation of waves.

Viscous effects can be neglected in the analysis for both the stratified flow and the problem with rotation but the presence of a mean shear in either flow – as we have already seen so often – does lead to important consequences for the dynamics when determining the stability of the system.

Third, there is the modeled geophysical boundary layer where the rotation is present but taken as constant and the surface is flat.

This text has covered some historical and more advanced theoretical and computational techniques to predict the onset of transitional flows with linear methods, the amplification and interaction of these linear modes in the nonlinear regime, and the matching of these predictions with empirical models. Furthermore, some methods of control have been developed and discussed in the chapter on flow control. Here, we address issues associated with investigating hydrodynamic instabilities using experimental techniques. These issues include the experimental facility, model configuration, and instrumentation, all of which impact the understanding of hydrodynamic instabilities.

Because the authors have primary expertise in theory and computation, we readily acknowledge the topics in this chapter are based on literature from leading scientists and engineers in the field of transitional flows. This chapter serves as an introduction to the experimental process. The content of this chapter is primarily based on the review by Saric (1994b) and a text by Smol'yakov & Tkachenko (1983).

Experimental Facility

Because the theoretical and computational modeling of a hydrodynamic instability process is the goal, two key aspects of the flow must be carefully documented in the experiment before studying the instabilities. First, the physical properties of the flow environment must be understood within the experimental facility. The makeup of the facility dictates the background (or freestream) disturbances and the spatial-temporal characteristics of the flow environment.

The incoming freestream environment should be understood and characterized before commencing with a discussion of the use of artificial disturbances, that are typically the manner hydrodynamic instabilities are investigated.

Polar coordinates

As has been mentioned earlier, there are many flows that require the formulation of the stability problem to be cast in coordinate systems other than Cartesian. For example, pipe flow is perhaps the most notable example. Then, there is the case that is now referred to as stability of Couette flow and has been examined both theoretically and experimentally by Taylor (1921, 1923). In this case, concentric cylinders rotate relative to each other to produce the flow. Free flows, such as the jet and wake, can be thought of as round rather than plane. If the boundary layer occurs on a curved wall, then Görtler vortices result. All of these examples can be described in terms of polar coordinates. And, at the outset, it should be recognized that, not only will the governing mathematics be different from that that has been used up to this point but the resulting physics may have novel characteristics as well.

The prevailing basis for the flows that have been examined heretofore has been that the flows are parallel or almost parallel. Then, the solutions for the disturbances were all of the form of plane waves that propagate in the direction of the mean flow or, more generally, obliquely to the mean flow. And, as we have learned, if solid boundaries are present in the flow, viscosity is a cause for instability. Now there are flows that will have curved streamlines and this leads to the possibility of a centrifugal force and its influence must be considered.

History, background, and rationale

In examining the dynamics of any physical system the concept of stability becomes relevant only after first establishing the possibility of equilibrium. Once this step has been taken, the concept becomes ubiquitous, regardless of the actual system being probed. As expressed by Betchov & Criminale (1967), stability can be defined as the ability of a dynamical system to be immune to small disturbances. It is clear that the disturbances need not necessarily be small in magnitude but the fact that the disturbances become amplified as a result and then there is a departure from any state of equilibrium the system had is implicit. Should no equilibrium be possible, then it can already be concluded that that particular system in question is statically unstable and the dynamics is a moot point.

Such tests for stability can be and are made in any field, such as mechanics, astronomy, electronics and biology, for example. In each case from this list, there is a common thread in that only a finite number of discrete degrees of freedom are required to describe the motion and there is only one independent variable. Like tests can be made for problems in continuous media but the number of degrees of freedom becomes infinite and the governing equations are now partial differential equations instead of the ordinary variety. Thus, conclusions are harder to obtain in any general manner but it is not impossible.

Introduction

The consideration of flows when the fluid is compressible presents a great many difficulties. The basic mathematics requires far more detail in order to make a rational investigation. The number of dependent variables is increased. And, regardless of the specific mean flow that is under scrutiny, the boundary conditions can be quite involved. Such observations will become more than obvious as the bases for examining the stability of such flows are established.

By their nature, the physics of compressible flows implies that there are now fluctuations in the density as well as the velocity and pressure. And, the density can be altered by pressure forces and the temperature. As a result, the laws of thermodynamics must be considered along with the equations for the conservation of mass and momentum. Consequently, a new set of governing equations must be derived. Moreover, this set of equations must be valid for flows than range from slightly supersonic to those that are hypersonic; i.e., M, the Mach number defined for the flow is of order one or larger. Flows that are characterized by a Mach number that is small compared to 1 and simply have a mean density that is inhomogeneous are those flows that satisfy what is known as the Boussinesq approximation and will be examined in Chapter 7. Suffice it for now to say that it is the force of gravity that plays a key role in such cases.