Eulerian method

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. 

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). 

Eulerian method The method investigates the features of fluid (e.g., velocity, pressure, density) at an arbitrary time (t) and point (.x, y, z). x, y, z, and t are independent variables and velocity, pressure, density, and so on are dependent variables. 

Attempts to extend RANS formulation to LES of two-phase combustion may be found in [318 354 317 255 292]. They are all based on a Euler-Lagrange (EL) description of the dispersed phase in which the flow is solved using an Eulerian method and the particles are tracked with a Lagrangian approach. An alternative is the Euler-Euler (EE) description, also called two-fluid approach, in which both the gas and the dispersed phases are... 

It is possible to convert one method of description to the other using the Reynolds transport theorem attributed to Osborne Reynolds. In other words, the theorem converts the Eulerian method of description of fluid flow to the Lagrangian method of description of fluid flow and vice versa. 

A Eulerian-Eulerian method is applied that uses locally averaged variables for the particle and fluid phases. The optimum scale of averaging was obtained by running the same simulation with successively better grid and time step resolutions until the numerical solution became sufficiently insensitive to further increases in the number of grid points and time steps. 

Donea J, Huerta A, Ponthot JP, Rodrfguez-Ferran A (2004) Arbitrary Lagrangian-Eulerian Methods. Encyclopedia of Computational Mechanics. 

Donea J, Huerta A, Ponthot J-P, Rodrfguez-Ferran A (2004) Arbitrary Lagrangian-Eulerian Methods. In Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of Computational Mechanics, Volume 1 Fundamentals, Chapter 14, pp 1-25, John Wiley Sons Ltd ISBN 0-470-84699-2... 

Ferry, J. Balachandar, S. 2001 A fast Eulerian method for two-phase flow. International Journal of Multiphase Flow 27, 1199-1226. 

To discuss effects of transported chemicals at a fixed location, methods of thinking similar to the Lagrangian and Eulerian methods should be followed. In other words they must cover ... 

Commonalities and differences between these methods are highlighted by grouping the methods not by scale, but according to the term(s) in Eq. (1) used to infer (/ . Table 1 shows such a grouping, in which four classes of Eulerian method are identified ... 

Eulerian methods based on spatial gradients Gradient, aerodynamic, Bowen ratio in the direction of diffusion Surface flux Steady, horizontally uniform X = 1 km, T = 30 min... 

Eulerian methods based on spatial gradients in the direction of advection ... 

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. 

The single-fluid approach has been undertaken utilizing Eulerian methods such as VOF, LS, and the mixed Eulerian-Lagrangian methods [11, 12], For most interface capturing schemes, which use single-fluid formulation, an additional equation is solved to obtain the interface evolution and topology. This equation governs the advection of a variable that can be attributed to the interface. The equation of interface motion is... 

The Eulerian-Eulerian method is more popular because memory storage requirements and computer power demand depend on the number of computational cells considered instead of the number of particles. Hence, the Eulerian-Eulerian approach can be applied to cases for low and high superficial gas velocities. The disadvantage of using the Eulerian-Eulerian method is that the bubble-bubble and bubble-liquid interactions cannot be considered as straightforward as the Eulerian-Lagrangian method, and models for these interactions are typically applied. 

The DNS methods are the most detailed they typically advance the gas-liquid interface through the flow field in an Eulerian mesh and do not require empirical constimtive equations. However, the DNS methods are limited to a very small number of particles/bubbles (e.g., typically 10s of bubbles in the flow field) due to computational limitations. Most industrial applications require high superficial gas velocities, and therefore the Eulerian-Eulerian method is preferred (Dudukovic, 2002 Pan et al., 2000) and will be summarized later. 

A1 Quddus N, Moussa WA, Bhattacharjee S (2008) Motion of a spherical particle in a cylindrical channel using arbitrary Lagrangian-Eulerian method. J Colloid Interface Sci 17 20-630... 

The single-fluid approach has been undertaken utilizing Eulerian methods such as the VOF, level set, and mixed Eulerian-Lagrangian methods [10, 11]. 

Eulerian methods perform weU for a variety of moving boundary problems. However, in these problems, particularly when surface forces are to be included in the flow calculations, the interface is diffused and occupies a few grid cells in practical calculations. This is undesirable in many problems both from an accuracy and physical realizability/modeling standpoint. 

By means of the Eulerian method it is possible to simulate many applications of ga -solid systems. However, further development is required to model the complete fluidized bed combustion process. This includes faster and more... 

The Lagrangian method is limited by the mesh distortion and the need for a time-consuming remeshing, and hence the Eulerian method is favored for our purpose. We will describe below three different approaches based on the Eulerian method. 

The finite element procedure begins with the division of the computational domain Q into a set of non-overlapping elements, for example, a collection of tetrahedral elements Q,- Since the Eulerian method is used, the elements are fixed in space. Methods such as the VOF or the level set method is used to determine the moving flow front. 

Two methods are commonly used to study a fluid, the Lagrangian and the Eu-lerian picture. The Lagrangian description considers the trajectories v t) of fluid points as a function of time, which would be observed if very small iron filings particles were present in the fluid. Although intuitive, this description leads generally to a rather cumbersome analysis and is not the more appropriate one. In lieu of this approach the Eulerian method is preferred, investigating the motion of the fluid at fixed points of the space as a function of time. In this theory the velocity field v(r, /.) is the basic studied quantity. 

The method adopted for the determination of the cyclic variation of film-thickness between cam and follower has been well established (e.g. 1). An initial value of the nominal central film thickness (h ) at some instant during the cycle was estimated, and the normal velocity (dh /dt) determined such that applied load balanced°the sum of the asperity and hydrodynamic components. A time step could then be marched out according to the Eulerian method such that,... 

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