Eulerian properties

Eames, I., Hunt, J.C.R., and Belcher, S.E. (2003) Lagrangian and Eulerian properties of steady flows through groups of obstacles, Journal of Fluid Mechanics, [Summary version in ERCOFTAC Bulletin No. 56.]... 

The relative shock velocity t/ = (7 — Uj is the Eulerian shock velocity often used because it is a material property and is independent of the motion of the... 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

In summary, we have commented briefly on the microscopic applications of NMR velocity imaging in complex polymer flows in complex geometries, where these applications have been termed Rheo-NMR [23]. As some of these complex geometries can be easily established in small scales, NMR velocimetry and visc-ometry at microscopic resolution can provide an effective means to image the entire Eulerian velocity field experimentally and to measure extensional properties in elastic liquids non-invasively. 

There appears to be some confusion on this point in hie literature. In an Eulerian PDF code, the notional particles do not represent fluid particles, rather they are a discrete representation of the composition PDF (e.g., a histogram). Thus, the number of notional particles needed in a grid cell is solely determined by the statistical properties of the PDF. For example, if the PDF is a delta function, then only one particle is required to represent it. Note, however, that the problem of determining the number of particles needed in each grid cell for a particular flow is non-trivial (Pfilzner et al. 1999). 

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... 

This equation provides the relationship between the rate of change of an extensive property N for a system (a specific, but possibly flowing, mass) and the substantial derivative of the associated intensive variable r) in an Eulerian control volume 8V that is fixed in space. 

In the Eulerian view, a fluid is characterized by fields of intensive variables or properties r). For example, the internal energy (or temperature, for a constant specific heat) is assumed to be a continuous function of time and space, r)(t,x). Because rj is a continuous differentiable function, the following expansion is generally valid ... 

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],... 

In the present study, two-dimensional Two-Fluid Eulerian model was used to describe the steady state, dilute phase flow of a wet dispersed phase (wet solid particles) in a continuous gas phase through a pneumatic dryer. The predictions of the numerical solutions were compared successfully with the results of other one-dimensional numerical solutions and experimental data of Baeyens et al. [5] and Rocha [13], Axial and the radial distributions of the characteristic properties were examined. 

Evaluating the performance of a gas-solid transport system usually requires a means of macroscopic field description of the distribution of basic flow properties such as pressure, mass fluxes, concentrations, velocities, and temperatures of phases in the system. To conduct such an evaluation, the Eulerian continuum or multifluid approach is usually the best choice among the available approaches. 

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. 

Usually, the intrinsic average reflects the real physical property or quantity such as density and velocity, while the phase average gives a pseudoproperty or quantity based on the selection of control volume. Phase averages are used to construct the continuum of each phase to which Eulerian description can be applied. 

The calculation of all physical properties and of the reaction temperature has to be related to an accurate equation of state in respect to the exact calculations of the reaction equilibrium, so that a connection to the Eulerian system will eventually be made, at least by the interdependence of average density, residence time and the volume of the reactor. 

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 derivative— The rate of change of a property when the system is open plus the rate of change of the property as the masses cross the boundary of the system. 

Discussions in Chapter 2 may be referred to for explanations of the various symbols. It is straightforward to apply such conservation equations to single-phase flows. In the case of multiphase flows also, in principle, it is possible to use these equations with appropriate boundary conditions at the interface between different phases. In such cases, however, density, viscosity and all the other relevant properties will have to change abruptly at the location of the interface. These methods, which describe and track the time-dependent behavior of the interface itself, are called front tracking methods. Numerical solution of such a set of equations is extremely difficult and enormously computation intensive. The main difficulty arises from the interaction between the moving interface and the Eulerian grid employed to solve the flow field (more discussion about numerical solutions is given in Chapters 6 and 7). 

According to Taylor [159] [160] [161], the properties of the Lagrangian scales are similar to the Eulerian correlation scales. Although the statistical turbulence theory is derived in terms of the Eulerian correlation functions, accurate measurements of the Lagrangian scales and correlations are easier and direct. In contrast, the measurements of Eulerian correlations requires two probes simultaneously working at two different locations. 

An important aspect of Eulerian reactor models is the truncation errors caused by the numerical approximation of the convection/advection terms [82], Very different numerical properties are built into the various schemes proposed for solving these operators. The numerical schemes chosen for a particular problem must be consistent with and reflect the actual physics represented by the model equations. 

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