Turbulence inhomogeneity

The purpose of this paper is to review the use of laser induced fluorescence spectroscopy (LIFS) for studying combustion processes. The study of such processes imposes severe constraints on diagnostic instrumentation. High velocities and temperatures are common, as well as turbulent inhomogeneities, and there is a need to make space and time resolved species concentration and temperature measurements. The development of LIFS has reached the point where it is capable of making significant contributions to experimental combustion studies. 

The generating random process we used is based on a rather subtle mathematical technique that we cannot describe here. Basically, we start from a symmetric, positive definite, correlation matrix A from which we deduce an accessory matrix B using the Cholesky method. The required vector U whose the components are the correlated velocity fluctuations is then equal to the matrix B multiplied by a vector whose components are uncorrelated, centered, normal variables of variances unity. The procedure first designed for an lD formulation has been extended to 2D-problems. Mean turbulence inhomogeneities can be accounted for in the process. Details can be found in Desjonqu res, 1987, Berlemont, 1987, Gouesbet et al, 1987, Berlemont et al, 1987, Desjonqu res et al, 1987. 

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

The form of (1.15) is identical to the balance equation that is used in finite-volume CFD codes for passive scalar mixing.17 The principal difference between a zone model and a finite-volume CFD model is that in a zone model the grid can be chosen to optimize the capture of inhomogeneities in the scalar fields independent of the mean velocity and turbulence fields.18 Theoretically, this fact could be exploited to reduce the number of zones to the minimum required to resolve spatial gradients in the scalar fields, thereby greatly reducing the computational requirements. 

At high Reynolds number, this would also apply to inhomogeneous turbulent flows. 

The first two terms on the right-hand side of this expression are the spatial transport terms. For homogeneous turbulence, these terms will be exactly zero. For inhomogeneous turbulence, the molecular transport term vV2e will be negligible (order Re,1). Spatial transport will thus be due to the unclosed velocity fluctuation term (u, e), and the unclosed... 

Table 2.4. The turbulence statistics and unclosed quantities appearing in the transport equations for high-Reynolds-number inhomogeneous turbulent flows. 

For convenience, the turbulence statistics used in engineering calculations of inhomogeneous, high-Reynolds-number turbulent flows are summarized in Table 2.4 along with the unclosed terms that appear in their transport equations. Models for the unclosed terms are discussed in Chapter 4. 

Obviously, a successful model for turbulent mixing must, at a minimum, be able to account for the time dependence of inhomogeneous turbulence.2 However, the situation... 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

In general, liquid-phase reactions (Sc > 1) and fast chemistry are beyond the range of DNS. The treatment of inhomogeneous flows (e.g., a chemical reactor) adds further restrictions. Thus, although DNS is a valuable tool for studying fundamentals,4 it is not a useful tool for chemical-reactor modeling. Nonetheless, much can be learned about scalar transport in turbulent flows from DNS. For example, valuable information about the effect of molecular diffusion on the joint scalar PDF can be easily extracted from a DNS simulation and used to validate the micromixing closures needed in other scalar transport models. 

Figure 4.5. Sketch of how LEM can be applied to an inhomogeneous flow. At fixed time intervals, sub-domains from neighboring grid cells are exchanged to mimic advection and turbulent diffusion.

One common difficulty when applying the E-model is the need to know the turbulent dissipation rate e for the flow. Moreover, because e will have an inhomogeneous distribution in most chemical reactors, the problem of finding e a priori is non-trivial. In most... 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Despite the ability of the GLM to reproduce any realizable Reynolds-stress model, Pope (2002b) has shown that it is not consistent with DNS data for homogeneous turbulent shear flow. In order to overcome this problem, and to incorporate the Reynolds-number effects observed in DNS, a stochastic model for the acceleration can be formulated (Pope 2002a Pope 2003). However, it remains to be seen how well such models will perform for more complex inhomogeneous flows. In particular, further research is needed to determine the functional forms of the coefficient matrices in both homogeneous and inhomogeneous turbulent flows. 

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