Phase-space integration moments

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

In Section 4.3, example macroscale transport equations are derived for selected moments of the NDF. Having introduced the precise forms of the mesoscale advection models in Eq. (5.2), it is of interest to derive explicitly some example moment source terms resulting from these models. In order to do so, we will use the rules presented in Section 4.3.1 for phase-space integration. For simplicity, we consider only the advection term involving (Afp)i and assume that the only phase-space variables of interest are v and Vf, and that the model in Eq. (5.2) reduces to... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

In summary, computing the moment-transport equations starting from Eq. (4.39) involves integration over phase space using the mles described above for particular choices of g. In the following, we will assume that the flux term at the boundary of phase space can be neglected. However, the reader should keep in mind that this assumption must be verified for particular cases. 

The extension of PTC to higher spatial dimensions is straightforward and many examples can be found in the literature (see, for example, Desjardins et al. (2008)). In Figure 8.1 we show a 2D example with two jets crossing. An important question that arises with PTC is whether or not the moment-transport equations can predict it. To answer this question, we return to Eq. (8.3) and integrate over phase space to find the moment-transport equation ... 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

These methods are appealing since the fundamental equation of motion is for the phase-space distribution itself rather than for individual trajectories. The structure of the Fokker-Planck equation in effect carries out a number of averages that must otherwise be performed by generating suitable trajectory ensembles. A preliminary application of the Fokker-Planck method to gas-surface scattering has been made [3.37]. In this application it was assumed that the full phase-space distribution was Gaussian in character with time-dependent first and second moments. Consequently the Fokker-Planck equation produced a set of first-order differential equations for these moments [3.48]. Integration of these equations was essentially... 

By performing a derivate moment expansion of the rate constants appearing in the phase space master equation, one can convert this integral equation to an equivalent differential equationcalledthe generahzed Fokker-Planck equation ... 

Eq.(2.1) represents a differential first-order equation in which unique unknown magnitude is speed of a corpuscle v, and argument—a time t. Speed of substance of a master phase in all space points is necessary known. In the capacity of initial data, except the size and properties of a corpuscle, its mle during the initial moment of a time is set. It is underlined also that should occur at collision of a corpuscle with a wall or with other corpuscle. For performance of calculation the members containing are transferred to the left-hand side of Eq. (2.1). Speed and a corpuscle position during each subsequent moment of a time is defined by a numerical integration on atime with some step At with all other members of Eq.(2.1). 

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