Velocity probability density function

It was further noticed that the value of the coefficient was sensitive to the density ratio between the continuous and dispersed phases, [p /p ). Moreover, in a recent study Brenn et al [9] investigated unsteady bubbly flow with very low void fractions and concluded that the velocity probability density functions of bubbles in liquid are better described using two superimposed Gaussian functions. 

In three dimensions, the velocity distribution function is proportional to the velocity probability density function ... 

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. 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

Since the histogram gives a probability density function of the particle position, the correlation in the velocities Vy and V2j in the j-direction causes the change in the shape of the histogram plotted against Vy and V2j, due to the different coefficient y — Pj in... 

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) fufiVi x, t) defined by4... 

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. 

Anand, M. S., A. T. Hsu, and S. B. Pope (1997). Calculations of swirl combustors using joint velocity-scalar probability density function methods.. 47.4.4 Journal 35, 1143-1150. 

Erratum Application of the velocity-dissipation probability density function model to inhomogeneous turbulent flows [Phys. Fluids A 3, 1947 (1991)]. Physics of Fluids A Fluid Dynamics 4, 1088. 

The present study is to elaborate on the computational approaches to explore flame stabilization techniques in subsonic ramjets, and to control combustion both passively and actively. The primary focus is on statistical models of turbulent combustion, in particular, the Presumed Probability Density Function (PPDF) method and the Pressure-Coupled Joint Velocity-Scalar Probability Density Function (PC JVS PDF) method [23, 24]. 

The pavement modelling allows to introduce into the model the temporal evolution of the size distribution of materials at the bed surface. By a progressive decrease of the probability density function of the lift force, this model successfully predicts the temporal decrease in mass flux that occurs with the presence of coarse particles at the surface. The rate of this decrease depends on the flow velocity and the characteristics of the particles. In order to improve the accuracy of the estimation of fugitive particle emissions with a wide size distribution, it is necessary to take into account this temporal decrease. 

The expression shows that the rate is determined by the component of the system point velocity that is perpendicular to the chosen surface S, an intuitively reasonable result. The velocity is multiplied by a probability density function p and a geometric factor V5 / 35/3gi, and the integrand is integrated over all momenta and coordinates q2, qsn—S where qi is chosen such that dS/dqi = 0. 

Figure 5. Probability density function (pdf or histogram) for temperature X velocity for turbulent diffusion flame. These data correspond to a test zone along the axis, 50 fuel-tip diameters downstream from the fuel line tip.

Probability density functions, or histograms, of the product of instantaneous temperature x velocity were obtained through use of this combustion probe system for a variety of downsteam and radial flame test positions. A typical histogram is shown in Fig. 3, while Fig. 4 displays the same data (as well as data for a test position further downsteam) in a "scattergram" format i.e., in a plot of velocity vs. temperature. Here, each datum corresponds to a specific shot, while the histogram bins correspond to integrated results from numbers of shots. 

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... 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

The averaging operation for the liquid droplet velocity described in the previous section introduces a particle velocity deviation from the mean (or correlated) velocity, noted as m" = Up — ui, and named the random uncorrelated velocity [280]. By definition, the statistical average (based on the particle probability density function) of this uncorrelated velocity is zero < u" >= 0. A conservation equation can be written for the associated kinetic energy 59i =< Up pip > /2 ... 

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) ... 

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... 

In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in configuration angular-velocity space for linear molecules in the same form as for fixed-axis rotators—that is, the form of the FKKE suggested by Barkai and Silbey [30] for one-dimensional translational Brownian motion. For rotators in space, the FKKE becomes... 

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