Turbulence Eulerian approach

The Eulerian approach to turbulent diffusion was shown to lead to the atmospheric diffusion equation (2.19) ... 

The mathematical models used to infer rates of water motion from the conservative properties and biogeochemical rates from nonconservative ones were flrst developed in the 1960s. Although they require acceptance of several assumptions, these models represent an elegant approach to obtaining rate information from easily measured constituents in seawater, such as salinity and the concentrations of the nonconservative chemical of interest. These models use an Eulerian approach. That is, they look at how a conservative property, such as the concentration of a conservative solute C, varies over time in an infinitesimally small volume of the ocean. Since C is conservative, its concentrations can only be altered by water transport, either via advection and/or turbulent mixing. Both processes can move water through any or all of the three dimensions... 

It must be noted here that most industrial fluidized bed reactors operate in a turbulent flow regime. Trajectory simulations of individual particles in a turbulent field may become quite complicated and time consuming. Details of models used to account for the influence of turbulence on particle trajectories are discussed in Chapter 4. These complications and constraints on available computational resources may restrict the number of particles considered in DPM simulations. Eulerian-Eulerian approaches based on the kinetic theory of granular flows may be more suitable to model such cases. Application of this approach to simulations of fluidized beds is discussed below. 

Simonin O, Elour I (1992) An Eulerian Approach for Turbulent Reactive Two-Phase Plows Loaded with Discrete Particles. In Sommerfeld M Sixth Workshop on Two-phase Plow Predictions. Erlangen, pp 61-62... 

In the Lagrangian approach, the elemental control volume is considered to be moving with the fluid as a whole. In the Eulerian approach, in contrast, the control volume is assumed fixed in the space, the fluid is assumed to flow through and pass the control volume. The particle-phase equations are formulated in Lagrangian form, and the coupling between the two phases is introduced through particle sources in the Eulerian gas-phase equations. The standard k-e turbulence model, finite rate chemistry, and DTRM (discrete transfer radiation model) radiation model are used. 

The techniques for describing the statistical properties of the concentrations of marked particles, such as trace gases, in a turbulent fluid can be divided into two categories Eulerian and Lagrangian. The Eulerian methods attempt to formulate the concentration statistics in terms of the statistical properties of the Eulerian fluid velocities, that is, the velocities measured at fixed points in the fluid. A formulation of this type is very useful not only because the Eulerian statistics are readily measurable (as determined from continuous-time recordings of the wind velocities by a fixed network of instruments) but also because the mathematical expressions are directly applicable to situations in which chemical reactions are taking place. Unfortunately, the Eulerian approaches lead to a serious mathematical obstacle known as the closure problem, for which no generally valid solution has yet been found. 

By extrapolation of the computed autocorrelation coefficient to zero time the square of the llagrangian fluctuating velocity can be obtained. The square root of this quantity can then be compared to the Eulerian rms velocity for the same column compartment. The two velocities should be equal in case of homogeneous turbulence. Devanathan (1991) shows that this is not the case in bubble columns but that homogeneous turbulence is approached at the highest gas velocities in the largest diameter column. 

The Eulerian approach relies on solving a transport equation for the particle concentration or number-density. The particles are not considered individually but as a continuous field akin to heat or chemical species, but furthermore characterized by inertia phenomena. The transport equation contains a convection term in which the mean velocities are known from the turbulence model, and also a dispersion term which is at the core of the problem. This dispersion term involves a dispersion tensor or, in a ID-formulation, a dispersion coefficient that we have to determine. 

The main advantage of the Eulerian approach is that it is not time-consuming in terms of CPU and consequently not costly. As a matter of fact, the code DISCO-2, including turbulence predictions, can run on a personal computer in a reasonable time. This efficiency is certainly the consequence of a previous formal effort which permits to carry out statistical averages on trajectories, in a implicit way, through pure physics and mathematics. However, this formal effort has been successful only by stating restrictive... 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

For simulating computationally the spatial and temporal evolution of both physical and chemical processes in mixing devices operated in a turbulent singlephase mode, two essentially different approaches are available the Lagrangian approach and the Eulerian technique. These will be explained briefly. 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

Aerosol production and transport over the oceans are of interest in studies concerning cloud physics, air pollution, atmospheric optics, and air-sea interactions. However, the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere is not well known. In an effort to investigate these phenomena, Edson et al.[12l used an interactive Eulerian-Lagrangian approach to simulate the generation, turbulent transport and evaporation of droplets. The k-e turbulence closure model was incorporated in the Eulerian-Lagrangian model to accurately simulate... 

The Eulerian continuum approach is basically an extension of the mathematical formulation of the fluid dynamics for a single phase to a multiphase. However, since neither the fluid phase nor the particle phase is actually continuous throughout the system at any moment, ways to construct a continuum of each phase have to be established. The transport properties of each pseudocontinuous phase, or the turbulence models of each phase in the case of turbulent gas-solid flows, need to be determined. In addition, the phase interactions must be expressed in continuous forms. 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

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-... 

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