Homogenization turbulence

In homogeneous turbulence, turbulence properties are independent of spatial position. The kinetic energy of turbulence k is given by... 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

Shy, S.S., Jang, R.H., and Tang, C.Y., Simulation of turbulent burning velocities using aqueous autocatalytic reactions in a near-homogeneous turbulence. Combust. Flame, 105, 54, 1996. 

In order to implement the PDF equations into a LES context, a filtered version of the PDF equation is required, usually denoted as filtered density function (FDF). Although the LES filtering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damkohler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re = 4,000. 

Van Vliet, E., Derksen, J. J., and Van den Akker, H. E. A., Modelling of Parallel Competitive Reactions in Isotropic Homogeneous Turbulence Using a Filtered Density Function Approach for Large Eddy Simulations . Proc. PVP01 3rd Int. Symp. on Comput. Techn. for Fluid/Thermal/Chemical Systems with Industrial Appl., Atlanta, GE, USA (2001). 

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t the function U(x, D varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x lJ(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. ... 

Much of the theoretical work in turbulent flows has been concentrated on the description of statistically homogeneous turbulence. In a statistically homogeneous turbulent flow, measurable statistical quantities such as the mean velocity2 or the turbulent kinetic energy are the same at every point in the flow. Among other things, this implies that the turbulence... 

More precisely, the spatial gradient of the mean velocity is independent of position in a homogeneous turbulent shear flow. 

Unlike simple random variables that have no space or time dependence, the statistics of the random velocity field in homogeneous turbulence can be described at many different levels of complexity. For example, a probabilistic theory could be formulated in terms of the set of functions U(x, t) (x, t) e R3 x R However, from a CFD modeling perspective, such a theory would be of little practical use. Thus, we will consider only one-point and two-point formulations that describe a homogeneous turbulent flow by the velocity statistics at one or two fixed points in space and/or time. 

In homogeneous turbulence, the one-point joint velocity PDF can be written as /u(V t), and can be readily measured using hot-wire anemometry or laser Doppler velocimetry (LDV). 

In fully developed homogeneous turbulence,7 the one-point joint velocity PDF is nearly Gaussian (Pope 2000). A Gaussian joint PDF is uniquely defined by a vector of expected values (j, and a covariance matrix C ... 

The reader familiar with turbulence modeling will recognize the covariance matrix as the Reynolds stresses. Thus, for fully developed homogeneous turbulence, knowledge of the mean velocity and the Reynolds stresses completely determines the one-point joint velocity PDF. 

The two-point description of homogeneous turbulence begins with the two-point joint velocity PDF /u,u (V, V x, x. t) defined by... 

In simulations of homogeneous turbulence, R,. is often used to characterize the magnitude of the turbulence. 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

Pope (2000) developed the following model turbulent energy spectrum to describe fully developed homogeneous turbulence 12... 

A homogeneous turbulent shear flow will have an additional unclosed production term on the right-hand side (Pope 2000). 

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... 

For high-Reynolds-number homogeneous turbulent flows,22 the right-hand side of the dissipation-rate transport equation thus reduces to the difference between two large terms 23... 

Combining (3.122) and (3.127), the scalar dissipation rate for homogeneous turbulent mixing can be expressed as31... 

For a fully developed scalar spectrum in stationary, homogeneous turbulence, = ej,. 

Using (3.130), the transport equation for the scalar dissipation rate in high-Reynolds-number homogeneous turbulence becomes... 

In homogeneous turbulence, spectral transport can be quantified by the scalar cospectral energy transfer rate Tap(ic, t). We can also define the wavenumber that separates the viscous-convective and the viscous-diffusive sub-ranges nf by introducing the arithmetic-mean molecular diffusivity Tap defined by... 

For inert-scalar mixing in homogeneous turbulence, ( a4> p)v can be found from37... 

Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. For the simplest case of two inert scalars, this implies F / and y 2 > 1 (see (3.140)). In homogeneous turbulence, one effect of differential diffusion is to de-correlate the scalars. This occurs first at the diffusive scales, and then backscatters to larger scales until the energy-containing scales de-correlate. Thus, one of the principal difficulties of modeling differential diffusion is the need to account for this length-scale dependence. 

In homogeneous turbulence, the governing equations for the scalar covariance, (3.137), and the joint scalar dissipation rate, (3.166), reduce, respectively, to... 

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