Inhomogeneous turbulence

Owing to the complexity of multi-point descriptions, almost all CFD models for complex turbulent flows are based on one-point turbulence statistics. As shown in Section 2.1, one-point turbulence statistics are found by integrating over the velocity sample space, e.g., 

Likewise, important joint velocity, composition statistics can be computed from the one-point joint velocity, scalar PDF. For example, the scalar flux is defined by 

Note that, unlike derivations that rely on time averages,15 for which 

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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. 

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. 

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. 

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. 

Erratum Application of the velocity-dissipation probability density function model to inhomogeneous turbulent flows [Phys. Fluids A 3, 1947 (1991)]. Physics of Fluids A Fluid Dynamics 4, 1088. 

Van Slooten, P. R. and S. B. Pope (1997). PDF modeling of inhomogeneous turbulence with exact representation of rapid distortions. Physics of Fluids 9, 1085-1105. 

Equilibrium Fint = PeXt at every point always in equilibrium Inhomogeneities turbulence sticky piston no well-defined Pint not in equilibrium at every point... 

Hunt, J.C.R., Eames, I., and Westerweel, J. (2005) Mechanics of inhomogeneous turbulence and interfacial layers, Journal of Fluid Mechanics, in press. 

Lumley, J.L. (1967) The structure of inhomogeneous turbulent flows, in A.M. Yaglom and V.I. Tatarsky (eds.), Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, 166-178. 

Ermak, D.L. and Nasstrom. J.S.. 2000. A Lagrangian stochastic diffusion method for inhomogeneous turbulence. Atmos. Environ., 34, pp. 1059-1068. 

The present LES concept has in most cases been used as a research tool to study isotropic and homogeneous turbulence within the more theoretical fields of science. Note that the residual-stress models for homogeneous turbulence are not adequate describing industrial non-isotropic inhomogeneous turbulent flows ([137] [121], chap 13). 

Reeks, M. W. 1983 The transport of discrete particles in inhomogeneous turbulence. Journal of Aerosol Science 14 (6), 729-739. 

Reeks, M. W. 2005a On model equations for particle dispersion in inhomogeneous turbulence. International Journal of Multiphase Flow 31, 93-114. 

Zaichik, L. I. Aupchenkov, V. M. 1998 Kinetic equation for the probability density function of velocity and temperature of particles in an inhomogeneous turbulent flow analysis of flow in a shear layer. High Temperature 36 (4), 572-582. 

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