Eulerian approaches

The problem of determining the concentration distribution resulting from a continuous source of strength q at the origin in an infinite isotropic fluid with a velocity u in the x direction can be formulated by the Eulerian approach as 

If we invoke the slender plume approximation, we are interested only in the solution close to the plume centerline. Thus, as in (17.63), (17.71) can be approximated by 

An Alternate Derivation of (17.73) Equation (17.73) is based on the slender plume approximation as expressed by (17.72). We will now show that the slender plume approximation is equivalent to neglecting diffusion in the direction of the mean flow in the atmospheric diffusion equation. Thus (c) is governed by 

Before we proceed to solve (17.74) a few comments about the boundary conditions are useful. When the x diffusion term is dropped in the atmospheric diffusion equation, the equation becomes first order in x, and the natural point for the single boundary condition on jr is at X = 0. Since the source is also at jc = 0 we have an option of whether to place the source on the R.H.S. of the equation, as in (17.74), or in the. v = 0 boundary condition. If we follow the latter course, then the jc = 0 boundary condition is obtained by equating material fluxes across the plane at jc = 0. The result is 

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The geometrical flexibility of the VOF scheme can be significantly improved if in its formulation, instead of using a fixed framework, a combination of a Lagrangian-Eulerian approach is adopted. The most common approach to develop such a combined framework is the application of the Arbitrary... 

Equation (12.43) is called an Eulerian approach because the behavior of the species is described relative to a fixed coordinate system. The equation can also be considered to be a transport equation for particles when they are... 

The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

Boemer, A., Qi, H., Hannes, J., and Renz, U. Modelling of solids circulation in a fluidised bed with Eulerian approach. 29th IEA-FBC Meeting in Paris, France, Nov. 24-26, 1994 (1994). 

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

Either Eq. (5.7) or (5.8) can be used in Eq. (5.6) to give the complete distribution in the cases of totally reflecting and totally absorbing surfaces, respectively. The case of a partially absorbing surface cannot be treated by the same image source approach since some material particles are reflected and some are absorbed. The Eulerian approach offers a convenient way to determine the form of the vertical distribution over the range of situations. 

Now we turn to the Eulerian approach to obtain the same results as we have just obtained by the Lagrangian approach. 

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... 

Using an Eulerian approach, the description of fluid motion requires the determination of the thermodynamic state, in terms of sensible fluid properties, pressure, P, density, p, and temperature, T. and of the velocity field u(x, t) [25-29],... 

Since a multiphase flow usually takes place in a confined volume, the desire to have a mathematical description based on a fixed domain renders the Eulerian method an ideal one to describe the flow field. The Eulerian approach requires that the transport quantities of all phases be continuous throughout the computational domain. As mentioned before, in reality, each phase is time-dependent and may be discretely distributed. Hence, averaging theorems need to be applied to construct a continuum for each phase so that the existing Eulerian description of a single-phase flow may be extended to a multiphase flow. 

Equation (8.22) constitutes the means whereby the configuration-specific kinematic viscosity of the suspension may be computed from the prescribed spatially periodic, microscale, kinematic viscosity data v(r) by first solving an appropriate microscale unit-cell problem. Its Lagrangian derivation differs significantly from volume-average Eulerian approaches (Zuzovsky et al, 1983 Nunan and Keller, 1984) usually employed in deriving such suspension-scale properties. 

Eulerian approach can be reduced significantly (Sokolichin et al, 1997). Especially for dispersed flows with a high volume fraction of the dispersed phase, the increased computational requirements of mixed Eulerian-Lagrangian approaches should be mentioned as a disadvantage. 

Kumar et al. (1995) used the CFDLIB code developed at Los Alamos Scientific Laboratory to simulate the gas-liquid flow in bubble columns. Their model, which is based on the Eulerian approach, could successfully predict the experimentally observed von Karman vortices (Chen et al., 1989) in a 2D bubble colunm with large aspect ratio (i.e., ratio of colunm height and colunm diameter). 

In the recent years different numerical models for the conversion of wood in a packed bed have been presented, e. g, [3-6], Existing models mostly describe the as a porous media by an Eulerian approach, with the cons vation equations for the solid and the gas phase solved with the same mesh. This approach implies that heat and mass transfer can only be taken into account according to the dimensions of the bed but not within the particles itself. Temperature and species distributions are assumed to be homogenous over the fuel particles. Thus, the influence of the particle dimensions on the conversion process can only be captured by simplified assumptions or macrokinetic data. 

FIGURE 4.1 Modeling approaches for multiphase flows, (a) Volume of fluid approach, (b) Eulerian-Lagrangian approach, (c) Eulerian-Eulerian approach. 

With this approach, even the dispersed phase is treated as a continuum. All phases share the domain and may interpenetrate as they move within it. This approach is more suitable for modeling dispersed multiphase systems with a significant volume fraction of dispersed phase (> 10%). Such situations may occur in many types of reactor, for example, in fluidized bed reactors, bubble column reactors and multiphase stirred reactors. It is possible to represent coupling between different phases by developing suitable interphase transport models. It is, however, difficult to handle complex phenomena at particle level (such as change in size due to reactions/evaporation etc.) with the Eulerian-Eulerian approach. 

With a Eulerian-Lagrangian approach, processes occurring at the particle surface can be modeled when simulating particle trajectories (for example, the process of dissolution or evaporation can be simulated). However, as the volume fraction of dispersed phase increases, the Eulerian-Lagrangian approach becomes increasingly computation intensive. A Eulerian-Eulerian approach more efficiently simulate such dispersed multiphase flows. 

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