Spatial transport

This transport equation cannot be solved directly because it involves several unclosed terms. The SGS flux ucj> represents the spatial transport of by the unresolved velocity fluctuations. Models for this term can generally be written in the form of a generalized transport equation ... 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

This term represents the spatial transport of e by turbulent velocity fluctuations. 

The pressure term is thus responsible for spatial transport due to the fluctuating pressure field. 

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... 

The firsttwo terms on the right-hand side of this expression are responsible for spatial transport of scalar dissipation. In high-Reynolds-number turbulent flows, the scalar-dissipation flux (iijC ) is the dominant term. The other terms on the right-hand side are similar to the corresponding terms in the dissipation transport equation ((2.125), p. 52), and are defined as follows. 

Spatial transport of scalar covariance is described by the triple-correlation term u,(p a(p p), and the molecular-transport term defined by... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Inter-cell spatial transport by the mean velocity and turbulent diffusivity change the memberships of the sets 0 /, but not the particle composition vectors 4> n). 

Keeping only the accumulation and spatial-transport terms, the FV code solves a discretized form of... 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

The spatial-transport algorithm is relatively easy to implement on orthogonal grids using information provided by the FV code. 

Spatial transport is limited to first-order, up-wind schemes, and is thus strongly affected by numerical diffusion. 

Relative to Eulerian PDF codes, the spatial-transport algorithm has much higher accuracy. The number of grid cells required for equivalent accuracy is thus considerably smaller. 

The total computational cost is proportional to the number of notional particles (A p), and the spatial-transport algorithm is trivial to parallelize. 

Contrary to the experimental techniques discussed above, spatial transport is important in flames. However, the laminar flame presents fewer difficulties than most other spatially varying combustion problems, because the relevant transport parameters are fairly well defined [427], Heat transport takes place primarily by thermal conduction, while transport of chemical species is dominated by molecular diffusion. 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

The power spectrum of the local magnetic field fluctuations shows different features at different Larmor frequency and temperature. The interpretation of the results is that the dominant source of these fluctuations is spatial transport of the electrons along the polymer chains and the frequency and temperature dependences reflect the details of their motions. 

This transient-state equation connects the one-dimensional flow of soil-water with the temporal and spatial transports of solutes as influenced by sorption and degradation (Scott 2000). 

To conclude the book, spatially ID numerical examples with different types of GPBE are solved using QBMM combined with KBFVM for spatial transport (Vikas et al, 2011a) in Chapter 8. We provide the exact formulas used in the numerical implementation, since it is the authors hope that the reader will attempt to reproduce some of these examples and, thereby, gain valuable experience in simulating polydisperse multiphase systems that can be applied to the reader s own applications. 

Note that the sign of the source term will depend on whether particles are created or destroyed in the system. Note also that the spatial transport term in Eq. (4.46) will generally not be closed unless, for example, all particles have identical velocities. The transport equation in Eq. (4.46) is mainly used for systems with particle aggregation and breakage (i.e. when N(t, x) is not constant). In such cases, it will typically be coupled to a system of moment-transport equations involving higher-order moments. 

When using moment methods for inhomogeneous systems, the moment set is transported in physical space due to advection, diffusion, and free transport. Since the moment-transport equations are derived from a transport equation for the NDE, the problem of moment transport is closely related to the problem of transporting the NDF. Denoting the NDE by n(t, X, ), the process of spatial transport involves changes in n(t, x, ) for fixed values... 

In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

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