Probability density function transported

Lou H, Miller RS. On the scalar probability density function transport equation for binary mixing in isotropic turbulence at supercritical pressure. Phys Fluids 2001 13(l) 3386-3399. 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

Janicka, J., W. Kolbe, and W. Kollmann (1979). Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics 4, 47-66. 

To account for the effect of a sufficiently broad, statistical distribution of heterogeneities on the overall transport, we can consider a probabilistic approach that will generate a probability density function in space (5) and time (t), /(i, t), describing key features of the transport. The effects of multiscale heterogeneities on contaminant transport patterns are significant, and consideration only of the mean transport behavior, such as the spatial moments of the concentration distribution, is not sufficient. The continuous time random walk (CTRW) approach is a physically based method that has been advanced recently as an effective means to quantify contaminant transport. The interested reader is referred to a detailed review of this approach (Berkowitz et al. 2006). 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

Local voidages for FCC catalyst at various radial positions were measured with an optical fiber probe in a Type A apparatus, from which radial volidage profiles and their probability density functions were computed by Li et al. (1980b), as shown in Figs 20 and 21. When gas velocity is less than the incipient fast fluidization velocity of 1.25 m/s, the radial voidage profile is relatively flat when gas velocity increases further, this profile becomes steeper high in the center. As flow is transformed into pneumatic transport, the... 

The limitations associated with (7) are essentially a consequence of the stochastic nature of atmospheric transport and diffusion. Because the wind velocities are random functions of space and time, the airborne pollutant concentrations are random variables in space and time. Thus, the determination of the Cj, in the sense of being a specified quantity at any time, is not possible, but we can at best derive the probability density functions satisfied by the c. The complete specification of the probability density function for a stochastic process as complex as atmospheric diffusion is almost never possible. Instead, we must adopt a less desirable but more feasible approach, the determination of certain statisical moments of Ci, notably its mean, . (The mean concentration can be... 

Zaichik, L., Oesterle, B. Aupchenkov, V. 2004 On the probability density function model for the transport of particles in anisotropic turbulent flow. Physics of Eluids 16 (6), 1956-1964. 

The residence time is the time spent in a reservoir by an individual atom or molecule. It is also the age of a molecule when it leaves the reservoir. If the pathway of the atom from the source to the sink is characterized by a physical transport, the term transit time can be used as an alternative. Even for the same element (or compound), different atoms (or molecules) will have different residence times in a given reservoir. The probability density function of residence times is denoted by 0(r), where (r)dr describes the fraction of the atoms (molecules) having a residence time in the interval to r to r + dr. The probability density function may have very... 

In the absence of sources, pure advection conserves the concentrations within fluid elements and only rearranges their distribution in space leaving the probability density function of the concentrations unchanged. Therefore to capture the gradual homogenization of an initially non-uniform concentration field under mixing a second transport process - diffusion - also needs to be included in the description. 

As shown by Eq. (4), the rate of reactions involving electrons depends on the EVDF, /(r, V, f). Determination of the distribution function is one of the central problems in understanding plasma chemistry. The EVDF is defined in the phase-space element dydr such that /(r, v, f) dy dx is the number of electrons dn at time t located between r and r + dr which have velocities between v and v -I- d. When normalized by the total number of electrons n, it is a probability density function. The EVDF is obtained by solving the Boltzmann transport equation [42, 43, 48, 49]... 

To compute tracer discharge J, we recognize that y is a conditional solution of transport since rand j are random variables. Let f(r,P) denote the joint probability density function for r and p. Then the Laplace solution for 7 at a discharge surface is... 

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333], The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes standard diffusion, tfansport with inertia, and anomalous diffusion. 

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