Higher-order moment transport

The moment-transport equations that we have derived up to this point are of first order in the velocity variables. In order to describe fluctuations about the first-order moments, it is necessary to derive transport equations for second- and sometimes higher-order moments. Just as before, this is accomplished by starting from Eq. (4.39) with a particular choice for g. In order to illustrate how this is done, we will consider the function g = fp2V which results in the particle-mass-average moment 

Second-order moment of disperse-phase velocity 

Starting from Eq. (4.39), the moment-transport equation corresponding to Eq. (4.107) is (no summation over i is implied) 

The other terms involving -]p are defined similarly. Note that, as is usually the case in moment methods, the convection term is unclosed and involves even higher-order moments. The remaining terms on the right-hand side of Eq. (4.109) are defined in a manner very similar to their counterparts in Eq. (4.85). Thus we will not discuss them in detail except to say that they will usually not appear in closed form. Only in the special case discussed in Section 4.4 where the velocity NDF is nearly Gaussian can we provide an accurate closure for Eq. (4.111) in terms of the velocity moments of up to second order. 

The second-order moment of the fluid-phase velocity 

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As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

In Chapter 5, we will review models referred to as moment methods, which attempt to close the chemical source term by expressing the unclosed higher-order moments in terms of lower-order moments. However, in general, such models are of limited applicability. On the other hand, transported PDF methods (discussed in Chapter 6) treat the chemical source term exactly. 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

This implies that higher order moments are introduced, thus the system of PDEs cannot be closed analytically. It is possible to show that similar effects will occur for the other source terms as well. This problem limits the application of the exact method of moments to the particular case where we have constant kernels only. In other cases one has to introduce approximate closures in order to eliminate the higher order moments ensuring that the transport equations for the moments of the particle size distribution can be expressed in terms of the lower order moments only (i.e., a modeling process very similar to turbulence modeling). 

This example illustrates the classical problem faced when working with macroscale models (Struchtrup, 2005). No matter how the transport equations for the moments are derived, they will always contain unclosed terms that depend on higher-order moments (e.g. Up depends on 0p, etc.). In comparison, the solution to the kinetic equation for the NDF contains information about all possible moments. In other words, if we could compute n t, x, v) directly, it would not be necessary to work with the macroscale model equations. The obvious question then arises Why don t we simply solve the kinetic equation for the mesoscale model instead of working with the macroscale model ... 

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. 

Obviously, conditional moments of higher order could also be modeled. However, as with moment closures, the unclosed terms in the higher-order transport equation are more and more difficult to close. 

The process of finding moment-transport equations starting from the PBE can be continued to arbitrary order. We should note that in most applications the resulting moment-transport equations will not be closed. In other words, the moment-transport equation of order k will involve moments of order higher than k. We will discuss moment-closure methods in Chapters 7 and 8 in the context of quadrature-based moment methods. 

In general, simulations carried out starting from realizable moment sets should result in realizable moment sets. Nevertheless, the moment-transport equations are integrated numerically with some finite discretization errors. As Wright (2007) clearly reports, most of the problems are caused by the approximation of the convective term, in particular with higher-order discretization schemes, which can turn a realizable set of moments into... 

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