Probability-density functionals

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)] 3t (x, t)  

An avenue that has received exploration is the development of equations for evolution of probability-density functions. If, for example, attention is restricted entirely to particular, fixed values of x and t, then the variable whose value may be represented by v becomes a random variable instead of a random function, and its statistics are described by a probability-density function. The probability-density function for v may be denoted by P(v where P(v) dv is the probability that the random variable lies in the range dv about the value v. By definition P(v) 0, and P(v) dv = 1, One approach to obtaining an equation of evolution for P(v) is to introduce the ensemble average of a fine-grained density, as described by O Brien in [27], for example another is formally to perform suitable integrations in 

Because of these difficulties with moment methods for reacting flows, we shall not present them here. A number of reviews are available [22], [25], [27], [32]. There are classes of turbulent combustion problems for which moment methods are reasonably well justified [40]. Since the computational difficulties in use of moment methods tend to be less severe than those for many other techniques (for example, techniques involving evolution equations for probability-density functions), they currently are being applied to turbulent combustion in relatively complex geometrical configurations [22], [31], [32]. Many of the aspects of moment methods play important roles in other approaches, notably in those for turbulent diffusion flames (Section 10.2). We shall develop those aspects later, as they are needed. 

Nevertheless, definitions of averages per se require further discussion here because they arise in one way or another in all approaches, and there are a number of different ways to define averages. 

Favre defines the mass-weighted average of a variable vasv = (pv)/p 

Favre defines the mass-weighted average of a variable vasv = (pv)/p and denotes the departure of v from v s v = v — v. Then the average v is not zero, but pv = 0 by definition. In this notation, equation (1) becomes 

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When providing input for the STOMP calculation a range of values of porosity (and all of the other input parameters) should be provided, based on the measured data and estimates of how the parameters may vary away from the control points. The uncertainty associated with each parameter may be expressed in terms of a probability density function, and these may be combined to create a probability density function for STOMP. 

It is common practice within oil companies to use expectation curves to express ranges of uncertainty. The relationship between probability density functions and expectation curves is a simple one. 

Figure 6.6 The probability density function and the expectation curve...

Bruce A D 1981 Probability density functions for collective coordinates in Ising-like systems J. Phys. O Soiid State Phys. 14 3667-88... 

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

Figure 6.6 Wave functions (dashed lines) and probability density functions (solid...

Figure 4 Sample spatial restraint m Modeller. A restraint on a given C -C , distance, d, is expressed as a conditional probability density function that depends on two other equivalent distances (d = 17.0 and d" = 23.5) p(dld, d"). The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from many triple alignments of protein structures. In practice, more complicated restraints are used that depend on additional information such as similarity between the proteins, solvent accessibility, and distance from a gap m the alignment.

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

The shape of the Normal distribution is shown in Figure 3 for an arbitrary mean, /i= 150 and varying standard deviation, ct. Notice it is symmetrical about the mean and that the area under each curve is equal representing a probability of one. The equation which describes the shape of a Normal distribution is called the Probability Density Function (PDF) and is usually represented by the term f x), or the function of A , where A is the variable of interest or variate. 

Nagode, M. and Fajdiga, M. 1998 A General Multi-Modal Probability Density Function Suitable for the Rainflow Ranges of Stationary Random Processes. Int. Journal of Fatigue, 20(3), 211-223. 

Distributions also are called probability density functions (pdf). 

Confidence limits are partial integrations over a probability density function. There are two special cases failure with time and failure with demand. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

Human actions can initiate accident sequences or cause failures, or conversely rectify or mitigate an accident sequence once initiated. The current methodology lacks nuclear-plant-based data, an experience base for human factors probability density functions, and a knowledge of how this distribution changes under stress. 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

Introducing a concept of gradient diffusion for particles and employing a mixture fraction for the non-reacting fluid originating upstream, / = c Vc O) and a probability density function for the statistics of the fluid elements, /(/), equation (2.100) becomes... 

MSMPR Mixed Suspension, Mixed Product Removal PDF Probability Density Function... 

The function px derived from Fx is called a first order probability density function. The reason for this terminology can be appreciated by noting the form that Eqs. (3-10), (3-12), and (3-13) assume when they are written in terms of px instead of Fx... 

We conclude this section with examples of some particularly important probability density functions that will be used in later applications. In each of these examples, the reader should verify that the function px is a probability density function by showing that it is non-negative and has unit area. All of the integrals and sums involved are elementary except perhaps in the case of the gaussian distribution, for which the reader is referred to Cramer.7... 

Fig. 3-3. Some Important Probability Density Functions and Their Corresponding Distribution Functions. Arrows are used to indicate Dirac delta functions with the height of the arrow indicating the area under the delta function.

The thermally produced noise voltage X(t) appearing across the terminals of a hot resistor is often modeled by assuming that the probability density function for X(t) is gaussian,... 

The probability density function pY can differentiation with the result... 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

Here we have a case where all, not only most, of the area under the probability density function is located within V 2 standard deviations of the mean, but where this fact alone gives a very misleading picture of the arcsine distribution, whose area is mainly concentrated at the edges of the distribution. Quantitatively, this is borne out by the easily verified fact that one half of the area is located outside of the interval [—0.9,0.9]. 

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