One-point probability density function

For a fixed point in space x and a given instant t, the random velocity field Ui(x, t) can be characterized by a one-point probability density function (PDF) fufiVp x, t) defined by4 

In words, the right-hand side is the probability that the random variable U (x,t) falls between the sample space values 1 and V + dlj for different realizations of the turbulent flow.5 In a homogeneous flow, this probability is independent of x, and thus we can write the one-point PDF as only a function of the sample space variable and time f(V t). 

The above definition can be extended to include an arbitrary number of random variables. For example, the one-point joint velocity PDF fv( x, t) describes all three velocity 

4 A semi-colon is used in the argument list to remind us that V is an independent (sample space) variable, while x and t are fixed parameters. Some authors refer to (V x, t) as the one-point, one-time velocity PDF. Here we use point to refer to a space-time point in the four-dimensional space (x, t). 

5 Because x and t are fixed, experimentally this definition implies that U (x, t) is the first velocity component measured at a fixed location in the flow at a fixed time instant from the start of the experiment. 

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In considering issues related to combustion, malodor, or toxicity or chemical reactions, one generally requires (at least) the one-point probability density function. For example, the probability of ignition (PI) of, say, a methane gas cloud is given by the probability that, at a position located by vector x at time t, concentrations between the lower 0 and upper 0 flammability limits are encountered. That is. 

Since one is only rarely interested in the density at a precise point on the z-axis, the cumulative probability (cumulative frequency) tables are more important in effect, the integral from -oo to +z over the probability density function for various z > 0 is tabulated again a few entries are given in Fig. 1.13. 

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. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

After P(Z) or P(Z) is obtained at each point in the flow by the methods that have now been described, it is usually possible to calculate any desired averages involving temperature and species concentrations at each point. As an illustration, let us assume that equations (3-82), (3-83) and (3-84) hold true. These relationships are shown schematically in Figure 10.3. A convenient approach is to use equation (9) to generate probability-density functions for Yq, and T from P(Z). If the shape of P(Z) is the second one shown for the jet in Figure 10.1, then the probability-density functions obtained are those illustrated in Figure 10.4. For the fuel, the contribution to P Z) for 0 < Z < Z collapses into a delta function at 7 = 0, as is readily... 

One of the simplest models of queuing is the following one. Let customers arrive at a service point in a Poisson process [see 2.2-3] of rate X [customers arriving per unit time]. Suppose that customers can be served only one at a time and that customers arriving to find the server busy queue up in the order of arrival until their turn for service comes. Such a queuing policy is called First In First Out (FIFO). Further, suppose that the length of time taken to serve a customer is a random variable with the exponential p.d.f. = (probability density function) given by... 

A set of observed data points is assumed to be available as samples from an unknown probability density function. Density estimation is the construction of an estimate of the density function from the observed data. In parametric approaches, one assumes that the data belong to one of a known family of distributions and the required function parameters are estimated. This approach becomes inadequate when one wants to approximate a multi-model function, or for cases where the process variables exhibit nonlinear correlations [127]. Moreover, for most processes, the underlying distribution of the data is not known and most likely does not follow a particular class of density function. Therefore, one has to estimate the density function using a nonparametric (unstructured) approach. 

The quantity/ (x) dr is, by definition, the probability of occurrence of the random variable within the interval of width dr around point x. In practical terms, this means that if we randomly extract an x value, the likelihood that it falls within the infinitesimal interval from x to x+dr is f(x)dx. To obtain probabilities corresponding to finite intervals — the only ones that have physical meaning — we must integrate the probability density function between the appropriate limits. The integral is the area below the f(x) curve between these limits, which implies that Fig. 2.3 is also a histogram. Since the random variable is now continuous, the... 

Probability density functions are spherically symmetric for the s electrons. They vanish at one point forn = 1, at two points at n = 2, at three points at n = 3 and so on. 

With the cost/benefit model fully defined, one next must forecast levels of attributes or, in other words, benefits and costs. Thus, for each alternative investment, one must forecast the stream of benefits and costs that will result if this investment is made. Quite often, these forecasts involve probability density functions rather than point forecasts. Utility theory models can easily incorporate the impact of such uncertainties on stakeholders risk aversions. On the other hand, information on probability density functions may not be available, or may be prohibitively expensive. In these situations, beliefs of stakeholders and subject matter experts can be employed, perhaps coupled with sensitivity analysis (see Step 7) to determine where additional data collection may be warranted. 

We have already indicated that the square of the electromagnetic wave is interpreted as the probability density function for finding photons at various places in space. We now attribute an analogous meaning to for matter waves. Thus, in a one-dimensional problem (for example, a particle constrained to move on a line), the probability that the particle will be found in the interval dx around the point xi is taken to be ir x ) dx. If i/ is a complex fimction, then the absolute square, is used instead of... 

Perhaps the most detained lagrandian model is the one of Durbin and coworkers [31], first devised in order to simulate the turbulent diffusion only,. and later extended to take into account a two species bimolecular reaction [32]. It is based on a stochastic simulation of the random walk of fluid particles, but is able also to provide the probability density function of the position (and then of the composition), within the entrance section of the reactor, of two fluid particles which would be at the same later time at a given point within the reactor. Owen to this new... 

In this section, we discuss how one, guided by the principles of nonequilibrium thermodynamics, can use the Monte Carlo technique to drive an ensemble of system configurations to sample statistically appropriate steady-state nonequilibrium phase-space points corresponding to an imposed external field [161,164,193-195]. For simplicity, we limit our discussion to the case of an unentangled polymer melt. The starting point is the probability density function of the generalized canonical... 

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