Turbulence, deterministic

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

The Warner function has all the desired asymptotical characteristics, i.e. a linear dependence of f(r) on r at small deformation and a finite length Nlp in the limit of infinite force (Fig. 3). In a non-deterministic flow such as a turbulent flow, it was found useful to model f(r) with an anharmonic oscillator law which permits us to account for the deviation of f(r) from linearity in the intermediate range of chain deformation [34] ... 

In the preceding categories of flow, the velocity field is deterministic since it can be calculated (at least in principle) from the constitutive equation of the fluid and the experimental boundary conditions. Turbulent flow, on the other hand, is distinctively unpredictable, as was pointed out a century ago by Osborne Reynolds. 

The models of Chapter 9 contain at least one empirical parameter. This parameter is used to account for complex flow fields that are not deterministic, time-invariant, and calculable. We are specifically concerned with packed-bed reactors, turbulent-flow reactors, and static mixers (also known as motionless mixers). We begin with packed-bed reactors because they are ubiquitous within the petrochemical industry and because their mathematical treatment closely parallels that of the laminar flow reactors in Chapter 8. 

In a single realization of a turbulent how, hie corresponding fluid-particle fluxes (A(X+, t) and (X+, )) are deterministic. However, they will be different for separate realizations. The noise components thus represent the variability between different realizations in a large ensemble of turbulent hows. 

Specifying 5 as a deterministic function implies that the inevitable randomness in S is negligible when compared to the mean emission rate and to the effect of the randomness of the turbulent field on c.)... 

Improvements in deterministic (photochemical/diffusion) methods are based largely on accounting for more physicochemical effects in the structure of the model. Specific research subjects for improved models include photochemical aerosol formation and the effects of turbulence on chemical reaction rates. The challenge to the researcher is to incorporate the study of these subjects without needlessly complicating already complex models. How accurate a mathematical simulation is required What, roughly, will be the effect of omitting some particular chemical or physical component What is the sensitivity of model outputs to inaccuracies in the inputs ... 

There are many ways of categorizing air quality models. One differentiation is between statistical and deterministic models. The structure of statistical models is based on the patterns that appear in the extensive measured data. The structure of deterministic models is based on mechanistic principles wherever possible. Most deterministic models contain some degree of empiricism. For example, few models, if any, use turbulent-diffiision formulations that are based on first principles, but rather use measured values of dispersion. The same is true in regard... 

The present investigation applies deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase. It also applies stochastic methods to describe the evolution of polydispersed particles and fluctuations of parameters [4]. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations obeys the deterministic laws. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

In this section we emphasize a parallelism between the Danckwerts statistical model and a deterministic representation of the turbulent mass transfer [116], The starting point was the observation that an almost... 

Random motion is ubiquitous. At the molecular level, the thermal motions of atoms and molecules are random. Further, motions in macroscopic systems are often described by random processes. For example, the motion of stirred coffee is a turbulent flow that can be characterized by random velocity components. Randomness means that the movement of an individual portion of the medium (i.e., a molecule, a water parcel, etc.) cannot be described deterministically. However, if we analyze the average effect of many individual random motions, we often end up with a simple macroscopic law that depicts the mean motion of the random system (see Box 18.1). 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

Two basic trajectory models, i.e., the deterministic trajectory model [Crowe etal., 1977] and the stochastic trajectory model [Crowe, 1991], are introduced in this section. The deterministic approach, which neglects the turbulent fluctuation of particles, specifically, the turbulent diffusion of the mass, momentum, and energy of particles, is considered the most... 

The description of small scale turbulent fields in confined spaces by fundamental approaches, based on statistical methods or on the concept of deterministic chaos, is a very promising and interesting research task nevertheless, at the authors knowledge, no fundamental approach is at the moment available for the modeling of large-scale confined systems, so that it is necessary to introduce semi-empirical models to express the tensor of turbulent stresses as a function of measurable quantities, such as geometry and velocity. Therefore, even in this case, a few parameters must be adjusted on the basis of independent measures of the fluid dynamic behavior. In any case, it must be underlined that these models are very complex and, therefore, well suited for simulation of complex systems but neither for identification of chemical parameters nor for online control and diagnosis [5, 6],... 

We have now to illustrate the dynamic model used to derive v /(t). It rests on the physics of intermittent processes, namely, on a model adopted to account for turbulence [39]. This is a popular prototype of deterministic approach to turbulence given by the Manneville map [40], This map reads... 

P. Berge. Y. Pomeau, and C. Vidal, Order within Chaos Towards a Deterministic Approach to Turbulence, John Wiley Sons, New York, 1986. 

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

Since atmospheric flows are turbulent, we may represent the wind velocities Uj as the sum of a deterministic and a stochastic component, Uj - - u j. The deterministic component includes that portion of the velocity which is known either from measurement or computation while the stochastic component represents everything not included in the deterministic component. Replacing Uj by Uj - - u j in (1) gives... 

---